+
+ | \(x\) |
+ \(P(X=x)\) |
+ \(xP(X=x)\) |
+ \(x^2P(X=x)\) |
+
+
+
+ | -1 |
+ 0.2 |
+ \{ NAMED_ANS_RULE("xp1",12) \} |
+ \{ NAMED_ANS_RULE("x2p1",12) \} |
+
+
+
+ | 0 |
+ 0.4 |
+ \{ NAMED_ANS_RULE("xp2",12) \} |
+ \{ NAMED_ANS_RULE("x2p2",12) \} |
+
+
+
+ | 1 |
+ 0.3 |
+ \{ NAMED_ANS_RULE("xp3",12) \} |
+ \{ NAMED_ANS_RULE("x2p3",12) \} |
+
+
+
+ | 2 |
+ 0.1 |
+ \{ NAMED_ANS_RULE("xp4",12) \} |
+ \{ NAMED_ANS_RULE("x2p4",12) \} |
+
+
+
+ | Totals |
+
+ \{ NAMED_ANS_RULE("EX",12) \}
+ (this total is \(E(X)\))
+ |
+
+ \{ NAMED_ANS_RULE("EX2",12) \}
+ (this total is \(E(X^2)\))
+ |
+
+
+$ECENTER
+END_TEXT
+
+Section::End();
+
+# Register evaluators immediately after the table (top-to-bottom, left-to-right)
+NAMED_ANS("xp1", $cmp_xp1);
+NAMED_ANS("x2p1", $cmp_x2p1);
+
+NAMED_ANS("xp2", $cmp_xp2);
+NAMED_ANS("x2p2", $cmp_x2p2);
+
+NAMED_ANS("xp3", $cmp_xp3);
+NAMED_ANS("x2p3", $cmp_x2p3);
+
+NAMED_ANS("xp4", $cmp_xp4);
+NAMED_ANS("x2p4", $cmp_x2p4);
+
+NAMED_ANS("EX", $cmp_EX);
+NAMED_ANS("EX2", $cmp_EX2);
+
+Section::Begin("Compute Var(X) and sigma_X");
+BEGIN_PGML
+Now use your two totals from the table:
+
+1) Compute the variance:
+[`` \mathrm{Var}(X)=E(X^2)-[E(X)]^2 = ``] [____]{$cmp_var}
+
+2) Compute the standard deviation:
+[`` \sigma_X=\sqrt{\mathrm{Var}(X)} = ``] [____]{$cmp_sig}
+END_PGML
+Section::End();
+
+Section::Begin("Final Answer");
+BEGIN_PGML
+Enter your final results.
+END_PGML
+
+BEGIN_TEXT
+$PAR
+$BR \(E(X)=\) \{ NAMED_ANS_RULE("finalEX",12) \}
+$BR \(\mathrm{Var}(X)=\) \{ NAMED_ANS_RULE("finalVar",12) \}
+$BR \(\sigma_X=\) \{ NAMED_ANS_RULE("finalSig",12) \}
+END_TEXT
+
+Section::End();
+
+NAMED_ANS("finalEX", $cmp_final_EX);
+NAMED_ANS("finalVar", $cmp_final_var);
+NAMED_ANS("finalSig", $cmp_final_sig);
+
+# -------------------------------------------------------
+# Feedback (rating 1–5) (hidden in LB)
+# -------------------------------------------------------
+if ($ENABLE_GP_RATING && !$inLibraryBrowser) {
+ Section::Begin("Feedback");
+ BEGIN_PGML
+How useful was this guided problem for learning expected value and variance?
+
+(1 = not useful, 2 = slightly useful, 3 = useful, 4 = very useful, 5 = extremely useful)
+Note: Consider using the Email Instructor button below to suggest improvements.
+
+Rating (enter 1-5): [___]{$cmp_rating}
+END_PGML
+ Section::End();
+}
+
+Scaffold::End();
+
+ENDDOCUMENT();
diff --git a/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/DiscreteRandomVariables/DiscreteRandomVariables_ExVar_GuidedProblem2.pg b/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/DiscreteRandomVariables/DiscreteRandomVariables_ExVar_GuidedProblem2.pg
new file mode 100644
index 0000000000..6fc4af8fac
--- /dev/null
+++ b/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/DiscreteRandomVariables/DiscreteRandomVariables_ExVar_GuidedProblem2.pg
@@ -0,0 +1,429 @@
+## DESCRIPTION
+## WeBWorK Guided Problems Project — Positive Difference of Two Dice: Distribution, Mean, Std Dev, and P(X<=2) (Guided / Scaffolded)
+## ENDDESCRIPTION
+##
+## Institution(Dawson College)
+## Author(Mehdi Moodi)
+## DBsubject(Statistics)
+## DBchapter(Probability)
+## DBsection(Random Variables)
+## Level(2)
+## KEYWORDS('probability distribution','discrete random variable','two dice','positive difference','expected value','variance','standard deviation')
+## Static(1)
+
+DOCUMENT();
+
+loadMacros(
+ "PGstandard.pl",
+ "PGanswermacros.pl",
+ "MathObjects.pl",
+ "PGML.pl",
+ "scaffold.pl",
+ "answerHints.pl",
+);
+
+TEXT(beginproblem());
+
+# ----------------------------------------------------------------
+# PILOT: usefulness rating toggle (ACTIVE)
+# ----------------------------------------------------------------
+$ENABLE_GP_RATING = 1;
+
+# ----------------------------------------------------------------
+# Library Browser flag (hide rating in LB)
+# ----------------------------------------------------------------
+$inLibraryBrowser = $main::inLibraryBrowser || 0;
+
+# =======================================================
+# Context
+# =======================================================
+Context("Numeric");
+
+# =======================================================
+# True distribution for X = |D1 - D2|
+# Possible values: 0,1,2,3,4,5
+# Counts out of 36: 6,10,8,6,4,2
+# =======================================================
+$P0 = Compute("6/36");
+$P1 = Compute("10/36");
+$P2 = Compute("8/36");
+$P3 = Compute("6/36");
+$P4 = Compute("4/36");
+$P5 = Compute("2/36");
+
+$sumP = Compute("1");
+
+# Work-table columns (exact)
+$xp0 = Compute("0*(6/36)");
+$xp1 = Compute("1*(10/36)");
+$xp2 = Compute("2*(8/36)");
+$xp3 = Compute("3*(6/36)");
+$xp4 = Compute("4*(4/36)");
+$xp5 = Compute("5*(2/36)");
+
+$x2p0 = Compute("(0^2)*(6/36)");
+$x2p1 = Compute("(1^2)*(10/36)");
+$x2p2 = Compute("(2^2)*(8/36)");
+$x2p3 = Compute("(3^2)*(6/36)");
+$x2p4 = Compute("(4^2)*(4/36)");
+$x2p5 = Compute("(5^2)*(2/36)");
+
+# Totals
+$EX = Compute("35/18"); # sum xP
+$EX2 = Compute("35/6"); # sum x^2P
+$Var = Compute("665/324"); # EX2 - (EX)^2
+$Sig = Compute("sqrt(665)/18");
+
+$P_le2 = Compute("2/3");
+
+# =======================================================
+# Evaluators (stable; hints are guidance only)
+# =======================================================
+
+# Outcomes / values
+$cmp_n_outcomes = Real(36)->cmp;
+$cmp_minX = Real(0)->cmp;
+$cmp_maxX = Real(5)->cmp;
+$cmp_kvals = Real(6)->cmp;
+
+# Distribution entries (Part a)
+$cmp_p0 = $P0->cmp->withPostFilter(AnswerHints(
+ sub { abs($_[0] - (1/6)) < 1e-6 } => "Difference 0 happens when the dice match (a double). Count those outcomes, then divide by 36.",
+));
+$cmp_p1 = $P1->cmp->withPostFilter(AnswerHints(
+ sub { abs($_[0] - (1/3)) < 1e-6 } => "For X=1, think: the dice differ by 1 (like 2 and 3). Count ordered pairs with |i-j|=1, then divide by 36.",
+));
+$cmp_p2 = $P2->cmp->withPostFilter(AnswerHints(
+ sub { abs($_[0] - (2/9)) < 1e-6 } => "For X=2, count ordered pairs (i,j) with |i-j|=2, then divide by 36.",
+));
+$cmp_p3 = $P3->cmp->withPostFilter(AnswerHints(
+ sub { abs($_[0] - (1/6)) < 1e-6 } => "For X=3, count ordered pairs with |i-j|=3, then divide by 36.",
+));
+$cmp_p4 = $P4->cmp->withPostFilter(AnswerHints(
+ sub { abs($_[0] - (1/9)) < 1e-6 } => "For X=4, the dice must be far apart (like 1 and 5). Count ordered pairs with |i-j|=4, then divide by 36.",
+));
+$cmp_p5 = $P5->cmp->withPostFilter(AnswerHints(
+ sub { abs($_[0] - (1/18)) < 1e-6 } => "For X=5, only the most extreme faces work (1 and 6 in either order). Divide the count by 36.",
+));
+
+# Check sum to 1
+$cmp_sumP = $sumP->cmp->withPostFilter(AnswerHints(
+ sub { abs($_[0] - (2/3)) < 1e-6 } => "2/3 is P(X<=2). Here you are checking that ALL probabilities add to 1.",
+));
+
+# Work-table: copied probabilities (to avoid AnSwEr warnings, these are NAMED and graded)
+$cmp_pw0 = $P0->cmp;
+$cmp_pw1 = $P1->cmp;
+$cmp_pw2 = $P2->cmp;
+$cmp_pw3 = $P3->cmp;
+$cmp_pw4 = $P4->cmp;
+$cmp_pw5 = $P5->cmp;
+
+# Work-table computed columns
+$cmp_xp0 = $xp0->cmp; $cmp_x2p0 = $x2p0->cmp;
+$cmp_xp1 = $xp1->cmp; $cmp_x2p1 = $x2p1->cmp;
+$cmp_xp2 = $xp2->cmp; $cmp_x2p2 = $x2p2->cmp;
+$cmp_xp3 = $xp3->cmp; $cmp_x2p3 = $x2p3->cmp;
+$cmp_xp4 = $xp4->cmp; $cmp_x2p4 = $x2p4->cmp;
+$cmp_xp5 = $xp5->cmp; $cmp_x2p5 = $x2p5->cmp;
+
+# Totals in work table
+$cmp_EX = $EX->cmp;
+$cmp_EX2 = $EX2->cmp;
+
+# Var and sigma
+$cmp_var = $Var->cmp->withPostFilter(AnswerHints(
+ sub { abs($_[0] - $_[1]) < 1e-6 } =>
+ "Use Var(X) = E(X^2) - [E(X)]^2, using the two totals from your work table.",
+));
+
+$cmp_sig = $Sig->cmp->withPostFilter(AnswerHints(
+ sub { abs($_[0] - $_[1]) < 1e-6 } =>
+ "Standard deviation is sigma_X = sqrt(Var(X)).",
+));
+
+# P(X<=2)
+$cmp_Ple2 = $P_le2->cmp->withPostFilter(AnswerHints(
+ sub { abs($_[0] - (24/36)) < 1e-6 } =>
+ "Good setup: P(X<=2)=P(0)+P(1)+P(2). You can simplify 24/36.",
+));
+
+# Final answers
+$cmp_final_EX = $EX->cmp;
+$cmp_final_sig = $Sig->cmp;
+$cmp_final_Ple2 = $P_le2->cmp;
+
+# =======================================================
+# Rating (1–5)
+# =======================================================
+$cmp_rating = Real(3)->cmp(
+ checker => sub {
+ my ($correct, $student, $ansHash) = @_;
+ my $v = eval { $student->value };
+ if (defined($v) && $v == int($v) && $v >= 1 && $v <= 5) {
+ $ansHash->{score} = 1;
+ $ansHash->{ans_message} = "Thanks!";
+ } else {
+ $ansHash->{score} = 0;
+ $ansHash->{ans_message} = "Please enter an integer from 1 to 5.";
+ }
+ return $ansHash->{score};
+ }
+);
+
+# =======================================================
+# Scaffold
+# =======================================================
+Scaffold::Begin();
+
+Section::Begin("Problem Statement");
+BEGIN_PGML
+A pair of fair dice is rolled. Let [`` X ``] denote the **positive difference** of the number of dots on the top faces.
+(For example, if the dice show 2 and 5, then [`` X=3 ``].)
+
+a) Construct the probability distribution of [`` X ``].
+b) Find [`` E(X) ``].
+c) Find [`` \sigma_X ``].
+d) Find [`` P(X \le 2) ``].
+END_PGML
+Section::End();
+
+Section::Begin("Outcomes and possible values of X");
+BEGIN_PGML
+When two dice are rolled, outcomes can be represented as ordered pairs [`` (i,j) ``] where
+[`` i ``] is the first die and [`` j ``] is the second die.
+
+1) How many equally likely outcomes are there?
+
+[____]{$cmp_n_outcomes}
+
+The random variable is:
+[`` X = |i-j| ``]
+
+2) What is the smallest possible value of [`` X ``]?
+
+[____]{$cmp_minX}
+
+3) What is the largest possible value of [`` X ``]?
+
+[____]{$cmp_maxX}
+
+4) How many possible values can [`` X ``] take?
+
+[____]{$cmp_kvals}
+END_PGML
+Section::End();
+
+Section::Begin("Construct the probability distribution (Part a)");
+BEGIN_PGML
+Fill in the probability distribution table for [`` X=|i-j| ``].
+
+(You can enter probabilities as fractions, like 10/36.)
+
+After you fill it in, you will check that the probabilities add to 1.
+END_PGML
+
+BEGIN_TEXT
+$BCENTER
+
+
+ | \(x\) |
+ \(P(X=x)\) |
+ \(xP(X=x)\) |
+ \(x^2P(X=x)\) |
+
+
+
+ | 0 |
+ \{ NAMED_ANS_RULE("pw0",10) \} |
+ \{ NAMED_ANS_RULE("xp0",12) \} |
+ \{ NAMED_ANS_RULE("x2p0",12) \} |
+
+
+ | 1 |
+ \{ NAMED_ANS_RULE("pw1",10) \} |
+ \{ NAMED_ANS_RULE("xp1",12) \} |
+ \{ NAMED_ANS_RULE("x2p1",12) \} |
+
+
+ | 2 |
+ \{ NAMED_ANS_RULE("pw2",10) \} |
+ \{ NAMED_ANS_RULE("xp2",12) \} |
+ \{ NAMED_ANS_RULE("x2p2",12) \} |
+
+
+ | 3 |
+ \{ NAMED_ANS_RULE("pw3",10) \} |
+ \{ NAMED_ANS_RULE("xp3",12) \} |
+ \{ NAMED_ANS_RULE("x2p3",12) \} |
+
+
+ | 4 |
+ \{ NAMED_ANS_RULE("pw4",10) \} |
+ \{ NAMED_ANS_RULE("xp4",12) \} |
+ \{ NAMED_ANS_RULE("x2p4",12) \} |
+
+
+ | 5 |
+ \{ NAMED_ANS_RULE("pw5",10) \} |
+ \{ NAMED_ANS_RULE("xp5",12) \} |
+ \{ NAMED_ANS_RULE("x2p5",12) \} |
+
+
+
+ | Totals |
+
+ \{ NAMED_ANS_RULE("EX",12) \}
+ (this total is \(E(X)\))
+ |
+
+ \{ NAMED_ANS_RULE("EX2",12) \}
+ (this total is \(E(X^2)\))
+ |
+
+
+$ECENTER
+END_TEXT
+
+Section::End();
+
+# Grade the copied probabilities (prevents AnSwEr00xx warnings)
+NAMED_ANS("pw0", $cmp_pw0);
+NAMED_ANS("pw1", $cmp_pw1);
+NAMED_ANS("pw2", $cmp_pw2);
+NAMED_ANS("pw3", $cmp_pw3);
+NAMED_ANS("pw4", $cmp_pw4);
+NAMED_ANS("pw5", $cmp_pw5);
+
+# Grade computed columns + totals
+NAMED_ANS("xp0", $cmp_xp0); NAMED_ANS("x2p0", $cmp_x2p0);
+NAMED_ANS("xp1", $cmp_xp1); NAMED_ANS("x2p1", $cmp_x2p1);
+NAMED_ANS("xp2", $cmp_xp2); NAMED_ANS("x2p2", $cmp_x2p2);
+NAMED_ANS("xp3", $cmp_xp3); NAMED_ANS("x2p3", $cmp_x2p3);
+NAMED_ANS("xp4", $cmp_xp4); NAMED_ANS("x2p4", $cmp_x2p4);
+NAMED_ANS("xp5", $cmp_xp5); NAMED_ANS("x2p5", $cmp_x2p5);
+
+NAMED_ANS("EX", $cmp_EX);
+NAMED_ANS("EX2", $cmp_EX2);
+
+Section::Begin("Compute Var(X) and sigma_X");
+BEGIN_PGML
+Now compute:
+
+1) Variance:
+[`` \mathrm{Var}(X)=E(X^2)-[E(X)]^2 = ``] [____]{$cmp_var}
+
+2) Standard deviation:
+[`` \sigma_X=\sqrt{\mathrm{Var}(X)} = ``] [____]{$cmp_sig}
+END_PGML
+Section::End();
+
+Section::Begin("Compute P(X <= 2) (Part d)");
+BEGIN_PGML
+Use your probability distribution:
+
+[`` P(X \le 2)=P(X=0)+P(X=1)+P(X=2)= ``] [____]{$cmp_Ple2}
+END_PGML
+Section::End();
+
+Section::Begin("Final Answer");
+BEGIN_PGML
+Enter your final results.
+END_PGML
+
+BEGIN_TEXT
+$PAR
+$BR \(E(X)=\) \{ NAMED_ANS_RULE("finalEX",12) \}
+$BR \(\sigma_X=\) \{ NAMED_ANS_RULE("finalSig",12) \}
+$BR \(P(X\le 2)=\) \{ NAMED_ANS_RULE("finalP",12) \}
+END_TEXT
+
+Section::End();
+
+NAMED_ANS("finalEX", $cmp_final_EX);
+NAMED_ANS("finalSig", $cmp_final_sig);
+NAMED_ANS("finalP", $cmp_final_Ple2);
+
+# -------------------------------------------------------
+# Feedback (rating 1–5) (hidden in LB)
+# -------------------------------------------------------
+if ($ENABLE_GP_RATING && !$inLibraryBrowser) {
+ Section::Begin("Feedback");
+ BEGIN_PGML
+How useful was this guided problem for learning probability distributions and expected value?
+
+(1 = not useful, 2 = slightly useful, 3 = useful, 4 = very useful, 5 = extremely useful)
+Note: Consider using the Email Instructor button below to suggest improvements.
+
+Rating (enter 1-5): [___]{$cmp_rating}
+END_PGML
+ Section::End();
+}
+
+Scaffold::End();
+
+ENDDOCUMENT();
diff --git a/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/DiscreteRandomVariables/DiscreteRandomVariables_ExVar_GuidedProblem3.pg b/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/DiscreteRandomVariables/DiscreteRandomVariables_ExVar_GuidedProblem3.pg
new file mode 100644
index 0000000000..bc8eca7cde
--- /dev/null
+++ b/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/DiscreteRandomVariables/DiscreteRandomVariables_ExVar_GuidedProblem3.pg
@@ -0,0 +1,387 @@
+## DESCRIPTION
+## WeBWorK Guided Problems Project — Number of Even Results on Two 4-Sided Dice: Distribution, Mean, Std Dev, and P(X<=1) (Guided / Scaffolded)
+## ENDDESCRIPTION
+##
+## Institution(Dawson College)
+## Author(Mehdi Moodi)
+## DBsubject(Statistics)
+## DBchapter(Probability)
+## DBsection(Random Variables)
+## Level(2)
+## KEYWORDS('probability distribution','discrete random variable','four-sided dice','expected value','variance','standard deviation','even outcomes')
+## Static(1)
+
+DOCUMENT();
+
+loadMacros(
+ "PGstandard.pl",
+ "PGanswermacros.pl",
+ "MathObjects.pl",
+ "PGML.pl",
+ "scaffold.pl",
+ "answerHints.pl",
+);
+
+TEXT(beginproblem());
+
+# ----------------------------------------------------------------
+# PILOT: usefulness rating toggle (ACTIVE)
+# ----------------------------------------------------------------
+$ENABLE_GP_RATING = 1;
+
+# ----------------------------------------------------------------
+# Library Browser flag (hide rating in LB)
+# ----------------------------------------------------------------
+$inLibraryBrowser = $main::inLibraryBrowser || 0;
+
+# =======================================================
+# Context
+# =======================================================
+Context("Numeric");
+
+# =======================================================
+# True distribution for X = number of even results (2 or 4) on two fair 4-sided dice
+# Each die: P(even)=2/4=1/2, P(odd)=1/2
+# X in {0,1,2} with probabilities 1/4, 1/2, 1/4
+# =======================================================
+$P0 = Compute("1/4");
+$P1 = Compute("1/2");
+$P2 = Compute("1/4");
+
+$sumP = Compute("1");
+
+# Work-table columns (exact)
+$xp0 = Compute("0*(1/4)");
+$xp1 = Compute("1*(1/2)");
+$xp2 = Compute("2*(1/4)");
+
+$x2p0 = Compute("(0^2)*(1/4)");
+$x2p1 = Compute("(1^2)*(1/2)");
+$x2p2 = Compute("(2^2)*(1/4)");
+
+# Totals
+$EX = Compute("1"); # 0*1/4 + 1*1/2 + 2*1/4
+$EX2 = Compute("3/2"); # 0 + 1/2 + 1
+$Var = Compute("1/2"); # EX2 - EX^2
+$Sig = Compute("sqrt(2)/2");
+
+$P_le1 = Compute("3/4"); # P(0)+P(1)
+
+# =======================================================
+# Evaluators
+# =======================================================
+
+# Outcomes / values
+$cmp_n_outcomes = Real(16)->cmp->withPostFilter(AnswerHints(
+ sub { abs($_[0] - 4) < 1e-6 } =>
+ "One 4-sided die has 4 outcomes. With two dice, think in ordered pairs (first die, second die).",
+));
+$cmp_minX = Real(0)->cmp;
+$cmp_maxX = Real(2)->cmp;
+$cmp_kvals = Real(3)->cmp->withPostFilter(AnswerHints(
+ sub { abs($_[0] - 2) < 1e-6 } =>
+ "X can be 0, 1, or 2. That's 3 possible values.",
+));
+
+# Distribution entries (Part a) — conceptual hints (no numeric giveaways)
+$cmp_p0 = $P0->cmp->withPostFilter(AnswerHints(
+ sub { abs($_[0] - (1/2)) < 1e-6 } =>
+ "1/2 is the probability a single die is even (or odd). Here X=0 means BOTH dice are odd.",
+));
+$cmp_p1 = $P1->cmp->withPostFilter(AnswerHints(
+ sub { abs($_[0] - (1/4)) < 1e-6 } =>
+ "1/4 would be the probability of both dice odd OR both dice even. X=1 means one even and one odd.",
+));
+$cmp_p2 = $P2->cmp->withPostFilter(AnswerHints(
+ sub { abs($_[0] - (1/2)) < 1e-6 } =>
+ "1/2 is not P(X=2). X=2 means BOTH dice are even (2 or 4).",
+));
+
+# Check sum to 1
+$cmp_sumP = $sumP->cmp->withPostFilter(AnswerHints(
+ sub { abs($_[0] - (3/4)) < 1e-6 } =>
+ "3/4 is P(X<=1). Here you are checking that ALL probabilities add to 1.",
+));
+
+# Work-table: copied probabilities (graded so no AnSwEr warnings)
+$cmp_pw0 = $P0->cmp;
+$cmp_pw1 = $P1->cmp;
+$cmp_pw2 = $P2->cmp;
+
+# Work-table computed columns
+$cmp_xp0 = $xp0->cmp; $cmp_x2p0 = $x2p0->cmp;
+$cmp_xp1 = $xp1->cmp; $cmp_x2p1 = $x2p1->cmp;
+$cmp_xp2 = $xp2->cmp; $cmp_x2p2 = $x2p2->cmp;
+
+# Totals in work table
+$cmp_EX = $EX->cmp;
+$cmp_EX2 = $EX2->cmp;
+
+# Var and sigma
+$cmp_var = $Var->cmp->withPostFilter(AnswerHints(
+ sub { abs($_[0] - $_[1]) < 1e-6 } =>
+ "Use Var(X) = E(X^2) - [E(X)]^2, using the two totals from your work table.",
+));
+
+$cmp_sig = $Sig->cmp->withPostFilter(AnswerHints(
+ sub { abs($_[0] - $_[1]) < 1e-6 } =>
+ "Standard deviation is sigma_X = sqrt(Var(X)).",
+));
+
+# P(X<=1)
+$cmp_Ple1 = $P_le1->cmp->withPostFilter(AnswerHints(
+ sub { abs($_[0] - (1/4)) < 1e-6 } =>
+ "1/4 is P(X=0) (both odd) or P(X=2) (both even). For X<=1 you need P(0)+P(1).",
+));
+
+# Final answers
+$cmp_final_EX = $EX->cmp;
+$cmp_final_sig = $Sig->cmp;
+$cmp_final_Ple1 = $P_le1->cmp;
+
+# =======================================================
+# Rating (1–5)
+# =======================================================
+$cmp_rating = Real(3)->cmp(
+ checker => sub {
+ my ($correct, $student, $ansHash) = @_;
+ my $v = eval { $student->value };
+ if (defined($v) && $v == int($v) && $v >= 1 && $v <= 5) {
+ $ansHash->{score} = 1;
+ $ansHash->{ans_message} = "Thanks!";
+ } else {
+ $ansHash->{score} = 0;
+ $ansHash->{ans_message} = "Please enter an integer from 1 to 5.";
+ }
+ return $ansHash->{score};
+ }
+);
+
+# =======================================================
+# Scaffold
+# =======================================================
+Scaffold::Begin();
+
+Section::Begin("Problem Statement");
+BEGIN_PGML
+Two fair 4-sided dice are rolled. (Each die has faces [`` 1,2,3,4 ``].)
+
+Let [`` X ``] be the number of dice that show an even number.
+(Here “even” means [`` 2 ``] or [`` 4 ``].)
+
+a) Construct the probability distribution of [`` X ``].
+b) Find [`` E(X) ``].
+c) Find [`` \sigma_X ``].
+d) Find [`` P(X \le 1) ``].
+END_PGML
+Section::End();
+
+Section::Begin("Outcomes and possible values of X");
+BEGIN_PGML
+When two 4-sided dice are rolled, outcomes can be represented as ordered pairs [`` (i,j) ``] where
+[`` i ``] is the first die and [`` j ``] is the second die.
+
+1) How many equally likely outcomes are there?
+
+[____]{$cmp_n_outcomes}
+
+The random variable counts how many of the two dice are even:
+[`` X = \text{(number of even results among the two dice)} ``]
+
+2) What is the smallest possible value of [`` X ``]?
+
+[____]{$cmp_minX}
+
+3) What is the largest possible value of [`` X ``]?
+
+[____]{$cmp_maxX}
+
+4) How many possible values can [`` X ``] take?
+
+[____]{$cmp_kvals}
+END_PGML
+Section::End();
+
+Section::Begin("Construct the probability distribution (Part a)");
+BEGIN_PGML
+Fill in the probability distribution table for [`` X ``].
+
+(You can enter probabilities as fractions, like 1/4.)
+
+After you fill it in, you will check that the probabilities add to 1.
+END_PGML
+
+BEGIN_TEXT
+$BCENTER
+
+
+ | \(x\) |
+ \(P(X=x)\) |
+ \(xP(X=x)\) |
+ \(x^2P(X=x)\) |
+
+
+
+ | 0 |
+ \{ NAMED_ANS_RULE("pw0",10) \} |
+ \{ NAMED_ANS_RULE("xp0",12) \} |
+ \{ NAMED_ANS_RULE("x2p0",12) \} |
+
+
+ | 1 |
+ \{ NAMED_ANS_RULE("pw1",10) \} |
+ \{ NAMED_ANS_RULE("xp1",12) \} |
+ \{ NAMED_ANS_RULE("x2p1",12) \} |
+
+
+ | 2 |
+ \{ NAMED_ANS_RULE("pw2",10) \} |
+ \{ NAMED_ANS_RULE("xp2",12) \} |
+ \{ NAMED_ANS_RULE("x2p2",12) \} |
+
+
+
+ | Totals |
+
+ \{ NAMED_ANS_RULE("EX",12) \}
+ (this total is \(E(X)\))
+ |
+
+ \{ NAMED_ANS_RULE("EX2",12) \}
+ (this total is \(E(X^2)\))
+ |
+
+
+$ECENTER
+END_TEXT
+
+Section::End();
+
+# Grade copied probabilities (prevents AnSwEr warnings)
+NAMED_ANS("pw0", $cmp_pw0);
+NAMED_ANS("pw1", $cmp_pw1);
+NAMED_ANS("pw2", $cmp_pw2);
+
+# Grade computed columns + totals
+NAMED_ANS("xp0", $cmp_xp0); NAMED_ANS("x2p0", $cmp_x2p0);
+NAMED_ANS("xp1", $cmp_xp1); NAMED_ANS("x2p1", $cmp_x2p1);
+NAMED_ANS("xp2", $cmp_xp2); NAMED_ANS("x2p2", $cmp_x2p2);
+
+NAMED_ANS("EX", $cmp_EX);
+NAMED_ANS("EX2", $cmp_EX2);
+
+Section::Begin("Compute Var(X) and sigma_X");
+BEGIN_PGML
+Now compute:
+
+1) Variance:
+[`` \mathrm{Var}(X)=E(X^2)-[E(X)]^2 = ``] [____]{$cmp_var}
+
+2) Standard deviation:
+[`` \sigma_X=\sqrt{\mathrm{Var}(X)} = ``] [____]{$cmp_sig}
+END_PGML
+Section::End();
+
+Section::Begin("Compute P(X <= 1) (Part d)");
+BEGIN_PGML
+Use your probability distribution:
+
+[`` P(X \le 1)=P(X=0)+P(X=1)= ``] [____]{$cmp_Ple1}
+END_PGML
+Section::End();
+
+Section::Begin("Final Answer");
+BEGIN_PGML
+Enter your final results.
+END_PGML
+
+BEGIN_TEXT
+$PAR
+$BR \(E(X)=\) \{ NAMED_ANS_RULE("finalEX",12) \}
+$BR \(\sigma_X=\) \{ NAMED_ANS_RULE("finalSig",12) \}
+$BR \(P(X\le 1)=\) \{ NAMED_ANS_RULE("finalP",12) \}
+END_TEXT
+
+Section::End();
+
+NAMED_ANS("finalEX", $cmp_final_EX);
+NAMED_ANS("finalSig", $cmp_final_sig);
+NAMED_ANS("finalP", $cmp_final_Ple1);
+
+# -------------------------------------------------------
+# Feedback (rating 1–5) (hidden in LB)
+# -------------------------------------------------------
+if ($ENABLE_GP_RATING && !$inLibraryBrowser) {
+ Section::Begin("Feedback");
+ BEGIN_PGML
+How useful was this guided problem for learning probability distributions and expected value?
+
+(1 = not useful, 2 = slightly useful, 3 = useful, 4 = very useful, 5 = extremely useful)
+Note: Consider using the Email Instructor button below to suggest improvements.
+
+Rating (enter 1-5): [___]{$cmp_rating}
+END_PGML
+ Section::End();
+}
+
+Scaffold::End();
+
+ENDDOCUMENT();
diff --git a/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/DiscreteRandomVariables/DiscreteRandomVariables_ExVar_GuidedProblem4.pg b/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/DiscreteRandomVariables/DiscreteRandomVariables_ExVar_GuidedProblem4.pg
new file mode 100644
index 0000000000..2f00e21955
--- /dev/null
+++ b/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/DiscreteRandomVariables/DiscreteRandomVariables_ExVar_GuidedProblem4.pg
@@ -0,0 +1,378 @@
+## DESCRIPTION
+## WeBWorK Guided Problems Project — Raffle: Probability Distribution, P(win), and Expected Value (Guided / Scaffolded)
+## ENDDESCRIPTION
+##
+## Institution(Dawson College)
+## Author(Mehdi Moodi)
+## DBsubject(Statistics)
+## DBchapter(Probability)
+## DBsection(Random Variables)
+## Level(2)
+## KEYWORDS('probability distribution','expected value','net gain','raffle','discrete random variable')
+## Static(1)
+
+DOCUMENT();
+
+loadMacros(
+ "PGstandard.pl",
+ "PGanswermacros.pl",
+ "MathObjects.pl",
+ "PGML.pl",
+ "scaffold.pl",
+ "answerHints.pl",
+);
+
+TEXT(beginproblem());
+
+# ----------------------------------------------------------------
+# PILOT: usefulness rating toggle (ACTIVE)
+# ----------------------------------------------------------------
+$ENABLE_GP_RATING = 1;
+
+# ----------------------------------------------------------------
+# Library Browser flag (hide rating in LB)
+# ----------------------------------------------------------------
+$inLibraryBrowser = $main::inLibraryBrowser || 0;
+
+# =======================================================
+# Context & Data
+# =======================================================
+Context("Numeric");
+
+# Ticket + prizes
+$N = Real(1000);
+
+# Net gains (prize - cost)
+$x1 = Compute("300-1"); # 299
+$x2 = Compute("200-1"); # 199
+$x3 = Compute("100-1"); # 99
+$x0 = Compute("0-1"); # -1 (no prize)
+
+# Probabilities
+$p1 = Compute("1/1000");
+$p2 = Compute("1/1000");
+$p3 = Compute("1/1000");
+$p0 = Compute("997/1000");
+
+$Pwin = Compute("3/1000");
+$sumP = Compute("1");
+
+# Work-table xP values and expected value
+$xp1 = Compute("(300-1)*(1/1000)");
+$xp2 = Compute("(200-1)*(1/1000)");
+$xp3 = Compute("(100-1)*(1/1000)");
+$xp0 = Compute("(0-1)*(997/1000)");
+
+$EX = Compute("-2/5"); # -0.4
+$Loss = Compute("2/5"); # expected loss per ticket (positive amount)
+
+# =======================================================
+# Evaluators
+# =======================================================
+
+# Counts
+$cmp_N = $N->cmp->withPostFilter(AnswerHints(
+ sub { abs($_[0] - 1000) < 1e-6 } =>
+ "There are 1000 tickets, and each ticket has an equal chance of being the one you bought.",
+));
+
+$cmp_winners = Real(3)->cmp->withPostFilter(AnswerHints(
+ sub { abs($_[0] - 1) < 1e-6 } =>
+ "There are three prizes, so there are three winning tickets (one for each prize).",
+));
+
+# Net gains
+$cmp_x1 = $x1->cmp->withPostFilter(AnswerHints(
+ sub { abs($_[0] - 300) < 1e-6 } =>
+ "Net gain means prize minus cost. You must subtract the $1 ticket cost.",
+));
+$cmp_x2 = $x2->cmp->withPostFilter(AnswerHints(
+ sub { abs($_[0] - 200) < 1e-6 } =>
+ "Net gain means prize minus cost. Subtract 1 from the prize.",
+));
+$cmp_x3 = $x3->cmp->withPostFilter(AnswerHints(
+ sub { abs($_[0] - 100) < 1e-6 } =>
+ "Net gain means prize minus cost. Subtract 1 from the prize.",
+));
+$cmp_x0 = $x0->cmp->withPostFilter(AnswerHints(
+ sub { abs($_[0] - 0) < 1e-6 } =>
+ "If you win nothing, your net gain is not 0. You still paid $1 for the ticket.",
+));
+
+# Probabilities for distribution table
+$cmp_p1 = $p1->cmp->withPostFilter(AnswerHints(
+ sub { abs($_[0] - (3/1000)) < 1e-6 } =>
+ "3/1000 is the probability of winning any prize. Here you need the probability of FIRST prize only.",
+));
+$cmp_p2 = $p2->cmp;
+$cmp_p3 = $p3->cmp;
+$cmp_p0 = $p0->cmp->withPostFilter(AnswerHints(
+ sub { abs($_[0] - (997/1000)) < 1e-6 } =>
+ "There are 3 winning tickets out of 1000. The remaining tickets are not winners.",
+ sub { abs($_[0] - (1/1000)) < 1e-6 } =>
+ "1/1000 is the probability of a specific prize. For 'no prize' you need the complement.",
+));
+
+# Check sum to 1
+$cmp_sumP = $sumP->cmp->withPostFilter(AnswerHints(
+ sub { abs($_[0] - (3/1000)) < 1e-6 } =>
+ "3/1000 is P(winning any money). Here you are checking that ALL probabilities add to 1.",
+));
+
+# Part (b)
+$cmp_Pwin = $Pwin->cmp->withPostFilter(AnswerHints(
+ sub { abs($_[0] - (997/1000)) < 1e-6 } =>
+ "997/1000 is the probability of winning nothing (no prize). Here you want the probability of winning any prize.",
+));
+
+# Work-table copied probabilities (to avoid AnSwEr warnings)
+$cmp_pw1 = $p1->cmp;
+$cmp_pw2 = $p2->cmp;
+$cmp_pw3 = $p3->cmp;
+$cmp_pw0 = $p0->cmp;
+
+# Work-table xP entries
+$cmp_xp1 = $xp1->cmp;
+$cmp_xp2 = $xp2->cmp;
+$cmp_xp3 = $xp3->cmp;
+$cmp_xp0 = $xp0->cmp;
+
+# Expected value + interpretation numeric
+$cmp_EX = $EX->cmp->withPostFilter(AnswerHints(
+ sub { abs($_[0] - (597/1000)) < 1e-6 } =>
+ "That adds the three winning net gains but forgets the big negative contribution from the 'no prize' outcome.",
+));
+$cmp_Loss = $Loss->cmp->withPostFilter(AnswerHints(
+ sub { abs($_[0] - $_[0]) < 0 } =>
+ "If E(X) is negative, the expected loss per ticket is -E(X).",
+));
+
+# Final answers (expression entries)
+$cmp_final_EX = $EX->cmp;
+$cmp_final_Pwin = $Pwin->cmp;
+
+# =======================================================
+# Rating (1–5)
+# =======================================================
+$cmp_rating = Real(3)->cmp(
+ checker => sub {
+ my ($correct, $student, $ansHash) = @_;
+ my $v = eval { $student->value };
+ if (defined($v) && $v == int($v) && $v >= 1 && $v <= 5) {
+ $ansHash->{score} = 1;
+ $ansHash->{ans_message} = "Thanks!";
+ } else {
+ $ansHash->{score} = 0;
+ $ansHash->{ans_message} = "Please enter an integer from 1 to 5.";
+ }
+ return $ansHash->{score};
+ }
+);
+
+# =======================================================
+# Scaffold
+# =======================================================
+Scaffold::Begin();
+
+Section::Begin("Problem Statement");
+BEGIN_PGML
+A service organization in a large town organizes a raffle each month.
+
+One thousand raffle tickets are sold for \$1 each. Each has an equal chance of winning.
+First prize is \$300, second prize is \$200, and third prize is \$100.
+
+Let [`` X ``] denote the **net gain** from the purchase of one ticket.
+
+a) Construct the probability distribution of [`` X ``].
+b) Find the probability of winning any money in the purchase of one ticket.
+c) Find the expected value of [`` X ``], and interpret its meaning.
+END_PGML
+Section::End();
+
+Section::Begin("Outcomes and possible values of X");
+BEGIN_PGML
+Think of the outcome as “which ticket number wins.”
+
+1) How many equally likely outcomes are there?
+
+[____]{$cmp_N}
+
+2) How many winning tickets are there in total?
+
+[____]{$cmp_winners}
+
+Now compute the possible values of [`` X ``] (net gain = prize minus the \$1 cost):
+
+First prize net gain: [`` 300-1 = ``] [____]{$cmp_x1}
+Second prize net gain: [`` 200-1 = ``] [____]{$cmp_x2}
+Third prize net gain: [`` 100-1 = ``] [____]{$cmp_x3}
+No prize net gain: [`` 0-1 = ``] [____]{$cmp_x0}
+END_PGML
+Section::End();
+
+Section::Begin("Construct the probability distribution (Part a)");
+BEGIN_PGML
+Each ticket has probability [`` 1/1000 ``] of being the one you bought.
+
+Fill in the probability distribution table for [`` X ``].
+END_PGML
+
+BEGIN_TEXT
+$BCENTER
+
+
+ | \(x\) |
+ \(P(X=x)\) |
+ \(xP(X=x)\) |
+
+
+
+ | \(299\) |
+ \{ NAMED_ANS_RULE("pw1",10) \} |
+ \{ NAMED_ANS_RULE("xp1",12) \} |
+
+
+
+ | \(199\) |
+ \{ NAMED_ANS_RULE("pw2",10) \} |
+ \{ NAMED_ANS_RULE("xp2",12) \} |
+
+
+
+ | \(99\) |
+ \{ NAMED_ANS_RULE("pw3",10) \} |
+ \{ NAMED_ANS_RULE("xp3",12) \} |
+
+
+
+ | \(-1\) |
+ \{ NAMED_ANS_RULE("pw0",10) \} |
+ \{ NAMED_ANS_RULE("xp0",12) \} |
+
+
+
+ | Total |
+
+ \{ NAMED_ANS_RULE("EX",12) \}
+ (this total is \(E(X)\))
+ |
+
+
+$ECENTER
+END_TEXT
+
+Section::End();
+
+# Grade copied probabilities (prevents AnSwEr warnings)
+NAMED_ANS("pw1", $cmp_pw1);
+NAMED_ANS("pw2", $cmp_pw2);
+NAMED_ANS("pw3", $cmp_pw3);
+NAMED_ANS("pw0", $cmp_pw0);
+
+# Grade xP column + total
+NAMED_ANS("xp1", $cmp_xp1);
+NAMED_ANS("xp2", $cmp_xp2);
+NAMED_ANS("xp3", $cmp_xp3);
+NAMED_ANS("xp0", $cmp_xp0);
+NAMED_ANS("EX", $cmp_EX);
+
+Section::Begin("Interpret the expected value (Part c)");
+BEGIN_PGML
+If your expected value [`` E(X) ``] is negative, that means the raffle is unfavorable to the buyer.
+
+Compute the **expected loss per ticket** (a positive number):
+[`` \text{expected loss} = -E(X)= ``] [____]{$cmp_Loss}
+END_PGML
+Section::End();
+
+Section::Begin("Final Answer");
+BEGIN_PGML
+Enter the final results.
+END_PGML
+
+BEGIN_TEXT
+$PAR
+$BR \(P(\text{win any money})=\) \{ NAMED_ANS_RULE("finalPwin",12) \}
+$BR \(E(X)=\) \{ NAMED_ANS_RULE("finalEX",12) \}
+END_TEXT
+
+Section::End();
+
+NAMED_ANS("finalPwin", $cmp_final_Pwin);
+NAMED_ANS("finalEX", $cmp_final_EX);
+
+# -------------------------------------------------------
+# Feedback (rating 1–5) (hidden in LB)
+# -------------------------------------------------------
+if ($ENABLE_GP_RATING && !$inLibraryBrowser) {
+ Section::Begin("Feedback");
+ BEGIN_PGML
+How useful was this guided problem for learning expected value?
+
+(1 = not useful, 2 = slightly useful, 3 = useful, 4 = very useful, 5 = extremely useful)
+Note: Consider using the Email Instructor button below to suggest improvements.
+
+Rating (enter 1-5): [___]{$cmp_rating}
+END_PGML
+ Section::End();
+}
+
+Scaffold::End();
+
+ENDDOCUMENT();
diff --git a/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/DiscreteRandomVariables/DiscreteRandomVariables_ExVar_GuidedProblem5.pg b/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/DiscreteRandomVariables/DiscreteRandomVariables_ExVar_GuidedProblem5.pg
new file mode 100644
index 0000000000..6589104eac
--- /dev/null
+++ b/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/DiscreteRandomVariables/DiscreteRandomVariables_ExVar_GuidedProblem5.pg
@@ -0,0 +1,319 @@
+## DESCRIPTION
+## WeBWorK Guided Problems Project — Life Insurance: Probability Distribution and Expected Value (Guided / Scaffolded)
+## ENDDESCRIPTION
+##
+## Institution(Dawson College)
+## Author(Mehdi Moodi)
+## DBsubject(Statistics)
+## DBchapter(Probability)
+## DBsection(Random Variables)
+## Level(2)
+## KEYWORDS('expected value','probability distribution','net gain','life insurance','discrete random variable')
+## Static(1)
+
+DOCUMENT();
+
+loadMacros(
+ "PGstandard.pl",
+ "PGanswermacros.pl",
+ "MathObjects.pl",
+ "PGML.pl",
+ "scaffold.pl",
+ "answerHints.pl",
+);
+
+TEXT(beginproblem());
+
+# ----------------------------------------------------------------
+# PILOT: usefulness rating toggle (ACTIVE)
+# ----------------------------------------------------------------
+$ENABLE_GP_RATING = 1;
+
+# ----------------------------------------------------------------
+# Library Browser flag (hide rating in LB)
+# ----------------------------------------------------------------
+$inLibraryBrowser = $main::inLibraryBrowser || 0;
+
+# =======================================================
+# Context & Data
+# =======================================================
+Context("Numeric");
+
+# Given probabilities
+$p_survive = Compute("9997/10000"); # 0.9997
+$p_die = Compute("3/10000"); # 0.0003
+$sumP = Compute("1");
+
+# Net gain to the company:
+# - If survive: company keeps premium
+# - If die: company pays 200000 but still collected premium
+$x_survive = Compute("195"); # +195
+$x_die = Compute("195-200000"); # -199805
+
+# Expected value
+$xp_survive = Compute("195*(9997/10000)");
+$xp_die = Compute("(195-200000)*(3/10000)");
+$EX = Compute("135");
+
+# Interpretation helper (positive amount; profit if EX>0, loss if EX<0)
+$absEX = Compute("135");
+
+# =======================================================
+# Evaluators
+# =======================================================
+
+$cmp_p_die = $p_die->cmp->withPostFilter(AnswerHints(
+ sub { abs($_[0] - 0.9997) < 1e-6 } =>
+ "0.9997 is the probability of surviving. The probability of dying is the complement.",
+ sub { abs($_[0] - 0.0003) < 1e-6 } =>
+ "Good. P(die) = 1 - P(survive).",
+));
+
+$cmp_x_survive = $x_survive->cmp->withPostFilter(AnswerHints(
+ sub { abs($_[0] - 0) < 1e-6 } =>
+ "If the person survives, the company still collected the premium. Net gain is not 0.",
+ sub { abs($_[0] - 195) < 1e-6 } =>
+ "Good. If no claim is paid, the company keeps the premium.",
+));
+
+$cmp_x_die = $x_die->cmp->withPostFilter(AnswerHints(
+ sub { abs($_[0] - (-200000)) < 1e-6 } =>
+ "You included the payout but forgot the company also collected the premium.",
+ sub { abs($_[0] - (-199805)) < 1e-6 } =>
+ "Good. Net gain = premium - payout.",
+));
+
+# Distribution-table probabilities
+$cmp_ps = $p_survive->cmp;
+$cmp_pd = $p_die->cmp;
+
+# Check sum
+$cmp_sumP = $sumP->cmp->withPostFilter(AnswerHints(
+ sub { abs($_[0] - 0.9997) < 1e-6 } =>
+ "That is only P(survive). Here you are checking that BOTH probabilities add to 1.",
+));
+
+# Work-table copied probabilities (avoid AnSwEr warnings)
+$cmp_pws = $p_survive->cmp;
+$cmp_pwd = $p_die->cmp;
+
+# Work-table xP
+$cmp_xps = $xp_survive->cmp;
+$cmp_xpd = $xp_die->cmp;
+
+# Expected value
+$cmp_EX = $EX->cmp->withPostFilter(AnswerHints(
+ sub { abs($_[0] - 195) < 1e-6 } =>
+ "That would be the gain if the person ALWAYS survived. You must weight by probabilities.",
+ sub { abs($_[0] - (-199805)) < 1e-6 } =>
+ "That would be the gain if the person ALWAYS died. You must weight by probabilities.",
+));
+
+# Interpretation numeric (positive amount)
+$cmp_absEX = $absEX->cmp->withPostFilter(AnswerHints(
+ sub { abs($_[0] - 135) < 1e-6 } =>
+ "Good. If E(X) is positive, this is an expected profit. If E(X) were negative, this would be the expected loss = -E(X).",
+));
+
+# Final answers
+$cmp_final_EX = $EX->cmp;
+
+# =======================================================
+# Rating (1–5)
+# =======================================================
+$cmp_rating = Real(3)->cmp(
+ checker => sub {
+ my ($correct, $student, $ansHash) = @_;
+ my $v = eval { $student->value };
+ if (defined($v) && $v == int($v) && $v >= 1 && $v <= 5) {
+ $ansHash->{score} = 1;
+ $ansHash->{ans_message} = "Thanks!";
+ } else {
+ $ansHash->{score} = 0;
+ $ansHash->{ans_message} = "Please enter an integer from 1 to 5.";
+ }
+ return $ansHash->{score};
+ }
+);
+
+# =======================================================
+# Scaffold
+# =======================================================
+Scaffold::Begin();
+
+Section::Begin("Problem Statement");
+BEGIN_PGML
+A life insurance company will sell a \$200,000 one-year term life insurance policy to an individual in a particular risk group for a premium of \$195.
+
+A person in this risk group has a 99.97% chance of surviving one year.
+
+Let [`` X ``] be the **net gain to the company** from a single policy.
+
+a) Construct the probability distribution of [`` X ``].
+b) Find [`` E(X) ``].
+c) Interpret the meaning of [`` E(X) ``] for the company.
+END_PGML
+Section::End();
+
+Section::Begin("Outcomes and net gain values");
+BEGIN_PGML
+There are two outcomes for the year:
+
+- The person **survives** (no payout)
+- The person **dies** (the company pays \$200,000)
+
+You are given:
+[`` P(\text{survive})=0.9997 ``]
+
+1) Compute:
+[`` P(\text{die})=1-P(\text{survive})= ``] [____]{$cmp_p_die}
+
+Now compute the possible values of [`` X ``] (net gain to the company):
+
+2) If the person survives, the company keeps the premium:
+[`` X=\text{premium}= ``] [____]{$cmp_x_survive}
+
+3) If the person dies, the company pays \$200,000 but still collected the premium:
+[`` X=\text{premium}-200000= ``] [____]{$cmp_x_die}
+END_PGML
+Section::End();
+
+Section::Begin("Probability distribution of X (Part a)");
+BEGIN_PGML
+Fill in the probability distribution table for [`` X ``].
+(You may enter probabilities as decimals or fractions.)
+END_PGML
+
+BEGIN_TEXT
+$BCENTER
+
+
+ | \(x\) |
+ \(P(X=x)\) |
+ \(xP(X=x)\) |
+ \(x^2P(X=x)\) |
+
+
+
+ | 0 |
+ \{ NAMED_ANS_RULE("pw0",10) \} |
+ \{ NAMED_ANS_RULE("xp0",12) \} |
+ \{ NAMED_ANS_RULE("x2p0",12) \} |
+
+
+ | 1 |
+ \{ NAMED_ANS_RULE("pw1",10) \} |
+ \{ NAMED_ANS_RULE("xp1",12) \} |
+ \{ NAMED_ANS_RULE("x2p1",12) \} |
+
+
+ | 2 |
+ \{ NAMED_ANS_RULE("pw2",10) \} |
+ \{ NAMED_ANS_RULE("xp2",12) \} |
+ \{ NAMED_ANS_RULE("x2p2",12) \} |
+
+
+ | 3 |
+ \{ NAMED_ANS_RULE("pw3",10) \} |
+ \{ NAMED_ANS_RULE("xp3",12) \} |
+ \{ NAMED_ANS_RULE("x2p3",12) \} |
+
+
+ | 4 |
+ \{ NAMED_ANS_RULE("pw4",10) \} |
+ \{ NAMED_ANS_RULE("xp4",12) \} |
+ \{ NAMED_ANS_RULE("x2p4",12) \} |
+
+
+
+ | Totals |
+
+ \{ NAMED_ANS_RULE("EX",12) \}
+ (this total is \(E(X)\))
+ |
+
+ \{ NAMED_ANS_RULE("EX2",12) \}
+ (this total is \(E(X^2)\))
+ |
+
+
+$ECENTER
+END_TEXT
+
+Section::End();
+
+# Grade copied probabilities (prevents AnSwEr warnings)
+NAMED_ANS("pw0", $cmp_pw0);
+NAMED_ANS("pw1", $cmp_pw1);
+NAMED_ANS("pw2", $cmp_pw2);
+NAMED_ANS("pw3", $cmp_pw3);
+NAMED_ANS("pw4", $cmp_pw4);
+
+# Grade computed columns + totals
+NAMED_ANS("xp0", $cmp_xp0); NAMED_ANS("x2p0", $cmp_x2p0);
+NAMED_ANS("xp1", $cmp_xp1); NAMED_ANS("x2p1", $cmp_x2p1);
+NAMED_ANS("xp2", $cmp_xp2); NAMED_ANS("x2p2", $cmp_x2p2);
+NAMED_ANS("xp3", $cmp_xp3); NAMED_ANS("x2p3", $cmp_x2p3);
+NAMED_ANS("xp4", $cmp_xp4); NAMED_ANS("x2p4", $cmp_x2p4);
+
+NAMED_ANS("EX", $cmp_EX);
+NAMED_ANS("EX2", $cmp_EX2);
+
+Section::Begin("Compute Var(X) and sigma_X");
+BEGIN_PGML
+Now compute:
+
+1) Variance:
+[`` \mathrm{Var}(X)=E(X^2)-[E(X)]^2 = ``] [____]{$cmp_var}
+
+2) Standard deviation:
+[`` \sigma_X=\sqrt{\mathrm{Var}(X)} = ``] [____]{$cmp_sig}
+END_PGML
+Section::End();
+
+Section::Begin("Compute P(X >= 2) (Part d)");
+BEGIN_PGML
+Use your probability distribution:
+
+[`` P(X \ge 2)=P(X=2)+P(X=3)+P(X=4)= ``] [____]{$cmp_Pge2}
+END_PGML
+Section::End();
+
+Section::Begin("Final Answer");
+BEGIN_PGML
+Enter your final results.
+END_PGML
+
+BEGIN_TEXT
+$PAR
+$BR \(E(X)=\) \{ NAMED_ANS_RULE("finalEX",12) \}
+$BR \(\sigma_X=\) \{ NAMED_ANS_RULE("finalSig",12) \}
+$BR \(P(X\ge 2)=\) \{ NAMED_ANS_RULE("finalP",12) \}
+END_TEXT
+
+Section::End();
+
+NAMED_ANS("finalEX", $cmp_final_EX);
+NAMED_ANS("finalSig", $cmp_final_sig);
+NAMED_ANS("finalP", $cmp_final_Pge2);
+
+# -------------------------------------------------------
+# Feedback (rating 1–5) (hidden in LB)
+# -------------------------------------------------------
+if ($ENABLE_GP_RATING && !$inLibraryBrowser) {
+ Section::Begin("Feedback");
+ BEGIN_PGML
+How useful was this guided problem for learning probability distributions and expected value?
+
+(1 = not useful, 2 = slightly useful, 3 = useful, 4 = very useful, 5 = extremely useful)
+Note: Consider using the Email Instructor button below to suggest improvements.
+
+Rating (enter 1-5): [___]{$cmp_rating}
+END_PGML
+ Section::End();
+}
+
+Scaffold::End();
+
+ENDDOCUMENT();
diff --git a/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/DiscreteRandomVariables/DiscreteRandomVariables_PD_GuidedProblem1.pg b/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/DiscreteRandomVariables/DiscreteRandomVariables_PD_GuidedProblem1.pg
new file mode 100644
index 0000000000..ba36db46de
--- /dev/null
+++ b/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/DiscreteRandomVariables/DiscreteRandomVariables_PD_GuidedProblem1.pg
@@ -0,0 +1,372 @@
+## DESCRIPTION
+## WeBWorK Guided Problems Project — Probability Distribution of a Discrete Random Variable (Guided / Scaffolded)
+## ENDDESCRIPTION
+##
+## Institution(Dawson College)
+## Author(Mehdi Moodi)
+## DBsubject(Statistics)
+## DBchapter(Probability)
+## DBsection(Random Variables)
+## Level(2)
+## KEYWORDS('probability distribution','discrete random variable','coin toss','probability table','at least one')
+## Static(1)
+
+DOCUMENT();
+
+loadMacros(
+ "PGstandard.pl",
+ "PGanswermacros.pl",
+ "MathObjects.pl",
+ "PGML.pl",
+ "scaffold.pl",
+ "answerHints.pl",
+ "parserRadioButtons.pl",
+);
+
+TEXT(beginproblem());
+
+# ----------------------------------------------------------------
+# PILOT: usefulness rating toggle (ACTIVE)
+# ----------------------------------------------------------------
+$ENABLE_GP_RATING = 1;
+
+# =======================================================
+# Context & Data
+# =======================================================
+Context("Numeric");
+Context()->flags->set(implicitMultiplication => 1);
+
+$k_vals = Compute("3"); # X can be 0,1,2 (3 values)
+$P0 = Compute("1/4");
+$P1 = Compute("1/2");
+$P2 = Compute("1/4");
+$sumP = Compute("1");
+$Patleast = Compute("3/4");
+
+# =======================================================
+# Evaluators
+# =======================================================
+$cmp_k = $k_vals->cmp->withPostFilter(AnswerHints(
+ sub { abs($_[0] - 2) < 1e-6 } =>
+ "If X is the number of heads in 2 tosses, it can be 0, 1, or 2 (that is 3 values).",
+));
+
+$cmp_P0 = $P0->cmp->withPostFilter(AnswerHints(
+ sub { abs($_[0] - (1/2)) < 1e-6 } =>
+ "X=0 means no heads. That happens only for TT (one outcome out of 4).",
+));
+
+$cmp_P1 = $P1->cmp->withPostFilter(AnswerHints(
+ sub { abs($_[0] - (1/4)) < 1e-6 } =>
+ "X=1 happens in HT and TH (two outcomes out of 4).",
+));
+
+$cmp_P2 = $P2->cmp->withPostFilter(AnswerHints(
+ sub { abs($_[0] - (1/2)) < 1e-6 } =>
+ "X=2 means two heads. That happens only for HH (one outcome out of 4).",
+));
+
+$cmp_sumP = $sumP->cmp->withPostFilter(AnswerHints(
+ sub { abs($_[0] - (3/4)) < 1e-6 } =>
+ "3/4 is P(at least one head). Here you are checking the sum P(X=0)+P(X=1)+P(X=2).",
+));
+
+$cmp_Patleast = $Patleast->cmp->withPostFilter(AnswerHints(
+ sub { abs($_[0] - (1/4)) < 1e-6 } =>
+ "1/4 is P(X=0) (no heads). 'At least one head' is the complement of that.",
+ sub { abs($_[0] - (1/2)) < 1e-6 } =>
+ "1/2 is P(X=1). 'At least one head' means X=1 or X=2.",
+));
+
+# =======================================================
+# RadioButtons (fixed-looking random order, correct not always A)
+# =======================================================
+
+# Definition MC (correct is NOT A)
+$rb_def = RadioButtons(
+ [
+ "Only the possible values of X (without probabilities).",
+ "A list of each possible value of X together with the probability that X takes that value.",
+ "A list of outcomes like HH, HT, TH, TT.",
+ "Only the most likely value of X.",
+ "A formula that always looks the same for every experiment.",
+ ],
+ 1, # correct is option index 1
+ labels => "ABC",
+ displayLabels => 0,
+ checked => 0, # prevents empty submission (avoids parserRadioButtons warning)
+);
+
+# Section 3 MCs: Is this a probability distribution table?
+$rb_pd1 = RadioButtons(
+ [
+ "Yes",
+ "No, because the probabilities do not add to 1",
+ "No, because one probability is not between 0 and 1",
+ "No, because the table does not list probabilities for all possible values of X",
+ ],
+ 0, # Yes
+ labels => "ABC",
+ displayLabels => 0,
+ checked => 0,
+);
+
+$rb_pd2 = RadioButtons(
+ [
+ "Yes",
+ "No, because one probability is not between 0 and 1",
+ "No, because the table does not list probabilities for all possible values of X",
+ "No, because the probabilities do not add to 1",
+ ],
+ 3, # sum not 1
+ labels => "ABC",
+ displayLabels => 0,
+ checked => 0,
+);
+
+$rb_pd3 = RadioButtons(
+ [
+ "Yes",
+ "No, because one probability is not between 0 and 1",
+ "No, because the probabilities do not add to 1",
+ "No, because the table does not list probabilities for all possible values of X",
+ ],
+ 1, # prob outside [0,1]
+ labels => "ABC",
+ displayLabels => 0,
+ checked => 0,
+);
+
+$rb_pd4 = RadioButtons(
+ [
+ "Yes",
+ "No, because one probability is not between 0 and 1",
+ "No, because the probabilities do not add to 1",
+ "No, because the table does not list probabilities for all possible values of X",
+ ],
+ 3, # missing value(s) of X
+ labels => "ABC",
+ displayLabels => 0,
+ checked => 0,
+);
+
+# =======================================================
+# Rating (1–5)
+# =======================================================
+$cmp_rating = Real(3)->cmp(
+ checker => sub {
+ my ($correct, $student, $ansHash) = @_;
+ my $v = eval { $student->value };
+
+ if (defined($v) && $v == int($v) && $v >= 1 && $v <= 5) {
+ $ansHash->{score} = 1;
+ $ansHash->{ans_message} = "Thanks!";
+ } else {
+ $ansHash->{score} = 0;
+ $ansHash->{ans_message} = "Please enter an integer from 1 to 5.";
+ }
+ return $ansHash->{score};
+ }
+);
+
+# =======================================================
+# Scaffold
+# =======================================================
+Scaffold::Begin();
+
+Section::Begin("Problem Statement");
+BEGIN_PGML
+A fair coin is tossed twice. Let [`` X ``] be the number of heads that are observed.
+
+a. Construct the probability distribution of [`` X ``].
+b. Find the probability that at least one head is observed.
+END_PGML
+Section::End();
+
+Section::Begin("Definition");
+BEGIN_PGML
+The probability distribution of a discrete random variable [`` X ``] is a list of each possible value of [`` X ``] together with the probability that [`` X ``] takes that value in one trial of the experiment.
+
+**Checkpoint:** Which statement best matches the definition of a probability distribution?
+
+[@ $rb_def->buttons() @]*
+END_PGML
+ANS($rb_def->cmp);
+Section::End();
+
+Section::Begin("Two properties");
+BEGIN_PGML
+Any probability distribution for a discrete random variable must satisfy:
+
+1. Each probability [`` P(x) ``] must be between 0 and 1: [`` 0 \le P(x) \le 1 ``]
+2. The sum of all the probabilities is 1: [`` \sum P(x) = 1 ``]
+
+Now decide whether each table is a valid probability distribution table.
+
+**Table 1**
+
+[`` \begin{array}{c|ccc}
+x & 0 & 1 & 2 \\
+\hline
+P(X=x) & 0.25 & 0.50 & 0.25
+\end{array} ``]
+
+Does this represent a probability distribution table?
+
+[@ $rb_pd1->buttons() @]*
+
+---
+
+**Table 2**
+
+[`` \begin{array}{c|ccc}
+x & 0 & 1 & 2 \\
+\hline
+P(X=x) & 0.30 & 0.60 & 0.30
+\end{array} ``]
+
+Does this represent a probability distribution table?
+
+[@ $rb_pd2->buttons() @]*
+
+---
+
+**Table 3**
+
+[`` \begin{array}{c|ccc}
+x & 0 & 1 & 2 \\
+\hline
+P(X=x) & -0.10 & 0.60 & 0.50
+\end{array} ``]
+
+Does this represent a probability distribution table?
+
+[@ $rb_pd3->buttons() @]*
+
+---
+
+**Table 4**
+
+[`` \begin{array}{c|cc}
+x & 0 & 1 \\
+\hline
+P(X=x) & 0.25 & 0.75
+\end{array} ``]
+
+Assume [`` X ``] is supposed to have possible values [`` \{0,1,2\} ``].
+Does this represent a probability distribution table?
+
+[@ $rb_pd4->buttons() @]*
+END_PGML
+ANS($rb_pd1->cmp);
+ANS($rb_pd2->cmp);
+ANS($rb_pd3->cmp);
+ANS($rb_pd4->cmp);
+Section::End();
+
+Section::Begin("Outcomes and the random variable");
+BEGIN_PGML
+For two coin tosses, the outcomes are:
+[`` \{\text{HH},\text{HT},\text{TH},\text{TT}\} ``]
+and they are all equally likely.
+
+Now compute [`` X ``] (the number of heads) for each outcome:
+
+- [`` \text{HH} ``] gives [`` X=2 ``]
+- [`` \text{HT} ``] gives [`` X=1 ``]
+- [`` \text{TH} ``] gives [`` X=1 ``]
+- [`` \text{TT} ``] gives [`` X=0 ``]
+
+How many possible values can [`` X ``] take?
+
+[____]{$cmp_k}
+END_PGML
+Section::End();
+
+Section::Begin("Construct the probability distribution");
+BEGIN_PGML
+**How to build the table:**
+For each value of [`` X ``], count how many of the 4 equally likely outcomes produce that value, then divide by 4.
+
+Fill in the probability distribution table:
+END_PGML
+
+BEGIN_TEXT
+$BCENTER
+
+
+ | \(x\) |
+ \(P(X=x)\) |
+
+
+
+ | 2 |
+
+ \{ NAMED_ANS_RULE("p2",12) \}
+ |
+
+
+ | 3 |
+
+ \{ NAMED_ANS_RULE("p3",12) \}
+ |
+
+
+ | 4 |
+
+ \{ NAMED_ANS_RULE("p4",12) \}
+ |
+
+
+ | 5 |
+
+ \{ NAMED_ANS_RULE("p5",12) \}
+ |
+
+
+ | 6 |
+
+ \{ NAMED_ANS_RULE("p6",12) \}
+ |
+
+
+ | 7 |
+
+ \{ NAMED_ANS_RULE("p7",12) \}
+ |
+
+
+ | 8 |
+
+ \{ NAMED_ANS_RULE("p8",12) \}
+ |
+
+
+ | 9 |
+
+ \{ NAMED_ANS_RULE("p9",12) \}
+ |
+
+
+ | 10 |
+
+ \{ NAMED_ANS_RULE("p10",12) \}
+ |
+
+
+ | 11 |
+
+ \{ NAMED_ANS_RULE("p11",12) \}
+ |
+
+
+ | 12 |
+
+ \{ NAMED_ANS_RULE("p12",12) \}
+ |
+
+
+$ECENTER
+END_TEXT
+
+Section::End();
+
+# Register evaluators immediately after the table (top-to-bottom)
+NAMED_ANS("p2", $cmp_P2);
+NAMED_ANS("p3", $cmp_P3);
+NAMED_ANS("p4", $cmp_P4);
+NAMED_ANS("p5", $cmp_P5);
+NAMED_ANS("p6", $cmp_P6);
+NAMED_ANS("p7", $cmp_P7);
+NAMED_ANS("p8", $cmp_P8);
+NAMED_ANS("p9", $cmp_P9);
+NAMED_ANS("p10", $cmp_P10);
+NAMED_ANS("p11", $cmp_P11);
+NAMED_ANS("p12", $cmp_P12);
+
+Section::Begin("Check the probability distribution");
+BEGIN_PGML
+A valid probability distribution must have probabilities that add to 1.
+
+Compute:
+[`` P(X=2)+P(X=3)+\cdots+P(X=12)= ``] [____]{$cmp_sumP}
+END_PGML
+Section::End();
+
+Section::Begin("Compute P(X >= 9) (Part b)");
+BEGIN_PGML
+[`` X \ge 9 ``] means [`` X=9,10,11,12 ``].
+
+Compute:
+[`` P(X \ge 9)= ``] [____]{$cmp_Pge9}
+END_PGML
+Section::End();
+
+Section::Begin("Probability that X is even (Part c)");
+BEGIN_PGML
+[`` X ``] is even when [`` X \in \{2,4,6,8,10,12\} ``].
+
+Compute:
+[`` P(X \text{ is even})= ``] [____]{$cmp_Peven}
+END_PGML
+Section::End();
+
+# Feedback (hidden in LB)
+if ($ENABLE_GP_RATING && !$inLibraryBrowser) {
+ Section::Begin("Feedback");
+ BEGIN_PGML
+How useful was this guided problem for learning probability distributions?
+
+(1 = not useful, 2 = slightly useful, 3 = useful, 4 = very useful, 5 = extremely useful)
+Note: Consider using the Email Instructor button below to suggest improvements.
+
+Rating (enter 1-5): [___]{$cmp_rating}
+END_PGML
+ Section::End();
+}
+
+Scaffold::End();
+
+ENDDOCUMENT();
diff --git a/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/DiscreteRandomVariables/DiscreteRandomVariables_TypesOfRVs_GuidedProblem1.pg b/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/DiscreteRandomVariables/DiscreteRandomVariables_TypesOfRVs_GuidedProblem1.pg
new file mode 100644
index 0000000000..8d8944c6ed
--- /dev/null
+++ b/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/DiscreteRandomVariables/DiscreteRandomVariables_TypesOfRVs_GuidedProblem1.pg
@@ -0,0 +1,389 @@
+DOCUMENT();
+
+loadMacros(
+ "PGstandard.pl",
+ "PGanswermacros.pl",
+ "MathObjects.pl",
+ "PGML.pl",
+ "scaffold.pl",
+ "answerHints.pl",
+ "parserRadioButtons.pl",
+);
+
+TEXT(beginproblem());
+
+# ----------------------------------------------------------------
+# PILOT: usefulness rating toggle (ACTIVE)
+# ----------------------------------------------------------------
+$ENABLE_GP_RATING = 1;
+
+# =======================================================
+# Context
+# =======================================================
+Context("Numeric");
+Context()->flags->set(implicitMultiplication => 1);
+
+# =======================================================
+# Step 2 MC: Not a random variable
+# =======================================================
+$rb_not_rv = RadioButtons(
+ [
+ 'Toss a coin 10 times and let \(X\) be the number of heads.',
+ 'Roll a die once and let \(X\) be the number shown.',
+ 'Toss a coin once and record the result as H or T.',
+ 'Measure the time (in seconds) until a website loads and let \(X\) be that time.',
+ 'Choose 3 students and let \(X\) be the number of left-handed students.',
+ ],
+ 2, # NOT numeric
+ labels => "ABC",
+ displayLabels => 0,
+);
+
+# =======================================================
+# Practice: 8 NEW random variables (type + possible values)
+# NOTE: options are reordered so correct choices are NOT always A.
+# =======================================================
+
+# RV1: Die once (Discrete) --- values correct is B
+$rb_rv1_type = RadioButtons(
+ [ 'Discrete', 'Continuous' ],
+ 0,
+ labels => "ABC",
+ displayLabels => 0,
+);
+$rb_rv1_vals = RadioButtons(
+ [
+ 'Possible values: \( \{0,1,2,3,4,5,6\} \)',
+ 'Possible values: \( \{1,2,3,4,5,6\} \)', # correct (B)
+ 'Possible values: \( [1,6] \)',
+ ],
+ 1,
+ labels => "ABC",
+ displayLabels => 0,
+);
+
+# RV2: # of sixes in 8 rolls (Discrete) --- type order swapped, values correct is C
+$rb_rv2_type = RadioButtons(
+ [ 'Continuous', 'Discrete' ],
+ 1,
+ labels => "ABC",
+ displayLabels => 0,
+);
+$rb_rv2_vals = RadioButtons(
+ [
+ 'Possible values: \( [0,8] \)',
+ 'Possible values: \( \{1,2,3,4,5,6,7,8\} \)',
+ 'Possible values: \( \{0,1,2,3,4,5,6,7,8\} \)', # correct (C)
+ ],
+ 2,
+ labels => "ABC",
+ displayLabels => 0,
+);
+
+# RV3: toss coin until tails, X = number of tosses (Discrete) --- values correct is B
+$rb_rv3_type = RadioButtons(
+ [ 'Discrete', 'Continuous' ],
+ 0,
+ labels => "ABC",
+ displayLabels => 0,
+);
+$rb_rv3_vals = RadioButtons(
+ [
+ 'Possible values: \( \{0,1,2,3,\dots\} \)',
+ 'Possible values: \( \{1,2,3,4,\dots\} \)', # correct (B)
+ 'Possible values: \( (0,1) \)',
+ ],
+ 1,
+ labels => "ABC",
+ displayLabels => 0,
+);
+
+# RV4: customers in an hour (Discrete) --- type order swapped, values correct is C
+$rb_rv4_type = RadioButtons(
+ [ 'Continuous', 'Discrete' ],
+ 1,
+ labels => "ABC",
+ displayLabels => 0,
+);
+$rb_rv4_vals = RadioButtons(
+ [
+ 'Possible values: \( \{1,2,3,4,\dots\} \)',
+ 'Possible values: \( [0,\infty) \)',
+ 'Possible values: \( \{0,1,2,3,\dots\} \)', # correct (C)
+ ],
+ 2,
+ labels => "ABC",
+ displayLabels => 0,
+);
+
+# RV5: exam score out of 100 (Continuous) --- values correct is B
+$rb_rv5_type = RadioButtons(
+ [ 'Discrete', 'Continuous' ],
+ 1,
+ labels => "ABC",
+ displayLabels => 0,
+);
+$rb_rv5_vals = RadioButtons(
+ [
+ 'Possible values: \( \{0,1,2,3,\dots,100\} \)',
+ 'Possible values: \( [0,100] \)', # correct (B)
+ 'Possible values: \( (0,100) \)',
+ ],
+ 1,
+ labels => "ABC",
+ displayLabels => 0,
+);
+
+# RV6: point strictly between ends of a 1-meter stick (Continuous) --- type order swapped, values correct is C
+$rb_rv6_type = RadioButtons(
+ [ 'Continuous', 'Discrete' ],
+ 0,
+ labels => "ABC",
+ displayLabels => 0,
+);
+$rb_rv6_vals = RadioButtons(
+ [
+ 'Possible values: \( [0,1] \)',
+ 'Possible values: \( \{0,1\} \)',
+ 'Possible values: \( (0,1) \)', # correct (C)
+ ],
+ 2,
+ labels => "ABC",
+ displayLabels => 0,
+);
+
+# RV7: waiting time until next bus arrives (Continuous) --- values correct is B
+$rb_rv7_type = RadioButtons(
+ [ 'Discrete', 'Continuous' ],
+ 1,
+ labels => "ABC",
+ displayLabels => 0,
+);
+$rb_rv7_vals = RadioButtons(
+ [
+ 'Possible values: \( \{0,1,2,3,\dots\} \)',
+ 'Possible values: \( [0,\infty) \)', # correct (B)
+ 'Possible values: \( (0,\infty) \)',
+ ],
+ 1,
+ labels => "ABC",
+ displayLabels => 0,
+);
+
+# RV8: classroom temperature at noon, assume between 15 and 30 (Continuous) --- type order swapped, values correct is C
+$rb_rv8_type = RadioButtons(
+ [ 'Continuous', 'Discrete' ],
+ 0,
+ labels => "ABC",
+ displayLabels => 0,
+);
+$rb_rv8_vals = RadioButtons(
+ [
+ 'Possible values: \( (15,30) \)',
+ 'Possible values: \( [0,100] \)',
+ 'Possible values: \( [15,30] \)', # correct (C)
+ ],
+ 2,
+ labels => "ABC",
+ displayLabels => 0,
+);
+
+# =======================================================
+# Rating (1–5)
+# =======================================================
+$cmp_rating = Real(3)->cmp(
+ checker => sub {
+ my ($correct, $student, $ansHash) = @_;
+ my $v = eval { $student->value };
+
+ if (defined($v) && $v == int($v) && $v >= 1 && $v <= 5) {
+ $ansHash->{score} = 1;
+ $ansHash->{ans_message} = "Thanks!";
+ } else {
+ $ansHash->{score} = 0;
+ $ansHash->{ans_message} = "Please enter an integer from 1 to 5.";
+ }
+ return $ansHash->{score};
+ }
+);
+
+# =======================================================
+# Scaffold
+# =======================================================
+Scaffold::Begin();
+
+Section::Begin("Problem Statement");
+BEGIN_PGML
+In this guided problem, we learn what a **random variable** is and how to classify it as **discrete** or **continuous**.
+
+Then you will classify 8 random variables by:
+- the **type** (discrete or continuous)
+- the **set of possible values**
+END_PGML
+Section::End();
+
+Section::Begin("What is a random variable?");
+BEGIN_PGML
+A **random variable** is a **numerical** quantity that is generated by a random experiment.
+
+**Checkpoint:** Which one is **not** a random variable?
+
+[_]{$rb_not_rv}
+END_PGML
+Section::End();
+
+Section::Begin("Discrete vs Continuous");
+BEGIN_PGML
+A random variable is called **discrete** if it has either a **finite** or a **countable** number of possible values.
+
+A random variable is called **continuous** if its possible values contain a **whole interval** of numbers.
+
+Key ideas:
+
+- Discrete (finite): values can be listed and the list ends.
+- Discrete (countably infinite): values can be listed forever, like [`` \{1,2,3,4,\dots\} ``].
+- Continuous: values fill an interval like [`` [a,b] ``] or [`` (a,b) ``] or [`` [0,\infty) ``].
+END_PGML
+Section::End();
+
+Section::Begin("Examples (for learning the concept)");
+BEGIN_PGML
+Here are examples to model what we mean by “type” and “possible values.”
+
+**Example 1: Roll two fair dice**
+Let [`` X ``] be the sum of the dots on the top faces.
+Type: **Discrete (finite)**
+Possible values: [`` \{2,3,4,5,6,7,8,9,10,11,12\} ``]
+
+**Example 2: Flip a fair coin repeatedly**
+Let [`` X ``] be the number of tosses until the coin lands heads.
+Type: **Discrete (countably infinite)**
+Possible values: [`` \{1,2,3,4,\dots\} ``]
+
+**Example 3: Measure the voltage at an electrical outlet**
+Let [`` X ``] be the voltage measured.
+Type: **Continuous**
+Possible values (an interval): [`` [118,122] ``]
+
+**Example 4: Operate a light bulb until it burns out**
+Let [`` X ``] be the time until the bulb burns out.
+Type: **Continuous**
+Possible values: [`` [0,\infty) ``]
+END_PGML
+Section::End();
+
+Section::Begin("Practice 1 (Random Variables 1–2)");
+BEGIN_PGML
+**RV #1**
+Experiment: Roll a fair die once.
+Let [`` X ``] be the number rolled.
+
+Type:
+[_]{$rb_rv1_type}
+Possible values:
+[_]{$rb_rv1_vals}
+
+---
+
+**RV #2**
+Experiment: Roll a fair die 8 times.
+Let [`` X ``] be the number of sixes rolled.
+
+Type:
+[_]{$rb_rv2_type}
+Possible values:
+[_]{$rb_rv2_vals}
+END_PGML
+Section::End();
+
+Section::Begin("Practice 2 (Random Variables 3–4)");
+BEGIN_PGML
+**RV #3**
+Experiment: Toss a fair coin until you get tails.
+Let [`` X ``] be the number of tosses required.
+
+Type:
+[_]{$rb_rv3_type}
+Possible values:
+[_]{$rb_rv3_vals}
+
+---
+
+**RV #4**
+Experiment: Count the number of customers who enter a store between 12:00 and 1:00.
+Let [`` X ``] be the number of customers.
+
+Type:
+[_]{$rb_rv4_type}
+Possible values:
+[_]{$rb_rv4_vals}
+END_PGML
+Section::End();
+
+Section::Begin("Practice 3 (Random Variables 5–6)");
+BEGIN_PGML
+**RV #5**
+Experiment: Randomly pick a student and record their exam score out of 100 (as a percentage).
+Let [`` X ``] be the score.
+
+Type:
+[_]{$rb_rv5_type}
+Possible values:
+[_]{$rb_rv5_vals}
+
+---
+
+**RV #6**
+Experiment: Randomly choose a point strictly between the ends of a 1-meter stick.
+Let [`` X ``] be the distance from the left end (in meters).
+
+Type:
+[_]{$rb_rv6_type}
+Possible values:
+[_]{$rb_rv6_vals}
+END_PGML
+Section::End();
+
+Section::Begin("Practice 4 (Random Variables 7–8)");
+BEGIN_PGML
+**RV #7**
+Experiment: Measure the waiting time (in minutes) until the next bus arrives.
+Let [`` X ``] be the waiting time.
+
+Type:
+[_]{$rb_rv7_type}
+Possible values:
+[_]{$rb_rv7_vals}
+
+---
+
+**RV #8**
+Experiment: Measure the classroom temperature (in °C) at noon, assuming it stays between 15 and 30.
+Let [`` X ``] be the temperature.
+
+Type:
+[_]{$rb_rv8_type}
+Possible values:
+[_]{$rb_rv8_vals}
+END_PGML
+Section::End();
+
+# -------------------------------------------------------
+# Feedback (rating 1–5)
+# -------------------------------------------------------
+if ($ENABLE_GP_RATING) {
+ Section::Begin("Feedback");
+ BEGIN_PGML
+How useful was this guided problem for learning combinations?
+
+(1 = not useful, 2 = slightly useful, 3 = useful, 4 = very useful, 5 = extremely useful)
+Note: Consider using the Email Instructor button below to suggest improvements.
+
+Rating (enter 1-5): [___]{$cmp_rating}
+END_PGML
+ Section::End();
+}
+
+Scaffold::End();
+
+ENDDOCUMENT();
diff --git a/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/DiscreteRandomVariables/DiscreteRandomVariables_TypesOfRVs_GuidedProblem2.pg b/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/DiscreteRandomVariables/DiscreteRandomVariables_TypesOfRVs_GuidedProblem2.pg
new file mode 100644
index 0000000000..619ef61d0d
--- /dev/null
+++ b/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/DiscreteRandomVariables/DiscreteRandomVariables_TypesOfRVs_GuidedProblem2.pg
@@ -0,0 +1,402 @@
+## DESCRIPTION
+## WeBWorK Guided Problems Project — Random Variables: Discrete vs Continuous (Guided / Scaffolded)
+## ENDDESCRIPTION
+##
+## Institution(Dawson College)
+## Author(Mehdi Moodi)
+## DBsubject(Statistics)
+## DBchapter(Probability)
+## DBsection(Random Variables)
+## Level(2)
+## KEYWORDS('random variable','discrete','continuous','possible values','countable','interval')
+## Static(1)
+
+DOCUMENT();
+
+loadMacros(
+ "PGstandard.pl",
+ "PGanswermacros.pl",
+ "MathObjects.pl",
+ "PGML.pl",
+ "scaffold.pl",
+ "answerHints.pl",
+ "parserRadioButtons.pl",
+);
+
+TEXT(beginproblem());
+
+# ----------------------------------------------------------------
+# PILOT: usefulness rating toggle (ACTIVE)
+# ----------------------------------------------------------------
+$ENABLE_GP_RATING = 1;
+
+# =======================================================
+# Context
+# =======================================================
+Context("Numeric");
+Context()->flags->set(implicitMultiplication => 1);
+
+# =======================================================
+# Step 2 MC: Not a random variable
+# =======================================================
+$rb_not_rv = RadioButtons(
+ [
+ 'Toss a coin 10 times and let \(X\) be the number of heads.',
+ 'Roll a die once and let \(X\) be the number shown.',
+ 'Toss a coin once and record the result as H or T.',
+ 'Measure the time (in seconds) until a website loads and let \(X\) be that time.',
+ 'Choose 3 students and let \(X\) be the number of left-handed students.',
+ ],
+ 2, # NOT numeric
+ labels => "ABC",
+ displayLabels => 0,
+);
+
+# =======================================================
+# Practice: 8 NEW random variables (type + possible values)
+# NOTE: options are reordered so correct choices are NOT always A.
+# =======================================================
+
+# RV1: Roll two dice, X = absolute difference (Discrete) --- values correct is C
+$rb_rv1_type = RadioButtons(
+ [ 'Discrete', 'Continuous' ],
+ 0,
+ labels => "ABC",
+ displayLabels => 0,
+);
+$rb_rv1_vals = RadioButtons(
+ [
+ 'Possible values: \( \{1,2,3,4,5,6\} \)',
+ 'Possible values: \( [0,5] \)',
+ 'Possible values: \( \{0,1,2,3,4,5\} \)', # correct (C)
+ ],
+ 2,
+ labels => "ABC",
+ displayLabels => 0,
+);
+
+# RV2: Choose 5 items, X = number defective (Discrete) --- type swapped, values correct is B
+$rb_rv2_type = RadioButtons(
+ [ 'Continuous', 'Discrete' ],
+ 1,
+ labels => "ABC",
+ displayLabels => 0,
+);
+$rb_rv2_vals = RadioButtons(
+ [
+ 'Possible values: \( \{1,2,3,4,5\} \)',
+ 'Possible values: \( \{0,1,2,3,4,5\} \)', # correct (B)
+ 'Possible values: \( [0,5] \)',
+ ],
+ 1,
+ labels => "ABC",
+ displayLabels => 0,
+);
+
+# RV3: Flip a coin until first tails, X = number of flips (Discrete) --- values correct is B
+$rb_rv3_type = RadioButtons(
+ [ 'Discrete', 'Continuous' ],
+ 0,
+ labels => "ABC",
+ displayLabels => 0,
+);
+$rb_rv3_vals = RadioButtons(
+ [
+ 'Possible values: \( \{0,1,2,3,\dots\} \)',
+ 'Possible values: \( \{1,2,3,4,\dots\} \)', # correct (B)
+ 'Possible values: \( (0,1) \)',
+ ],
+ 1,
+ labels => "ABC",
+ displayLabels => 0,
+);
+
+# RV4: Number of emails received today (Discrete) --- values correct is C
+$rb_rv4_type = RadioButtons(
+ [ 'Discrete', 'Continuous' ],
+ 0,
+ labels => "ABC",
+ displayLabels => 0,
+);
+$rb_rv4_vals = RadioButtons(
+ [
+ 'Possible values: \( \{1,2,3,4,\dots\} \)',
+ 'Possible values: \( [0,\infty) \)',
+ 'Possible values: \( \{0,1,2,3,\dots\} \)', # correct (C)
+ ],
+ 2,
+ labels => "ABC",
+ displayLabels => 0,
+);
+
+# RV5: Amount of gasoline in a tank after filling (liters), assume between 0 and 60 (Continuous) --- values correct is C
+$rb_rv5_type = RadioButtons(
+ [ 'Discrete', 'Continuous' ],
+ 1,
+ labels => "ABC",
+ displayLabels => 0,
+);
+$rb_rv5_vals = RadioButtons(
+ [
+ 'Possible values: \( \{0,1,2,\dots,60\} \)',
+ 'Possible values: \( (0,60) \)',
+ 'Possible values: \( [0,60] \)', # correct (C)
+ ],
+ 2,
+ labels => "ABC",
+ displayLabels => 0,
+);
+
+# RV6: Choose a random time strictly during a 60-minute lecture, X = minutes after start (Continuous) --- values correct is B
+$rb_rv6_type = RadioButtons(
+ [ 'Continuous', 'Discrete' ],
+ 0,
+ labels => "ABC",
+ displayLabels => 0,
+);
+$rb_rv6_vals = RadioButtons(
+ [
+ 'Possible values: \( [0,60] \)',
+ 'Possible values: \( (0,60) \)', # correct (B)
+ 'Possible values: \( \{0,1,2,\dots,60\} \)',
+ ],
+ 1,
+ labels => "ABC",
+ displayLabels => 0,
+);
+
+# RV7: Distance from home to school (km) for a randomly selected student (Continuous) --- values correct is A
+$rb_rv7_type = RadioButtons(
+ [ 'Discrete', 'Continuous' ],
+ 1,
+ labels => "ABC",
+ displayLabels => 0,
+);
+$rb_rv7_vals = RadioButtons(
+ [
+ 'Possible values: \( [0,\infty) \)', # correct (A)
+ 'Possible values: \( (0,\infty) \)',
+ 'Possible values: \( \{0,1,2,3,\dots\} \)',
+ ],
+ 0,
+ labels => "ABC",
+ displayLabels => 0,
+);
+
+# RV8: Temperature of coffee right after pouring (°C), assume strictly between 0 and 100 (Continuous) --- values correct is C
+$rb_rv8_type = RadioButtons(
+ [ 'Continuous', 'Discrete' ],
+ 0,
+ labels => "ABC",
+ displayLabels => 0,
+);
+$rb_rv8_vals = RadioButtons(
+ [
+ 'Possible values: \( [0,100] \)',
+ 'Possible values: \( \{0,1,2,\dots,100\} \)',
+ 'Possible values: \( (0,100) \)', # correct (C)
+ ],
+ 2,
+ labels => "ABC",
+ displayLabels => 0,
+);
+
+# =======================================================
+# Rating (1–5)
+# =======================================================
+$cmp_rating = Real(3)->cmp(
+ checker => sub {
+ my ($correct, $student, $ansHash) = @_;
+ my $v = eval { $student->value };
+
+ if (defined($v) && $v == int($v) && $v >= 1 && $v <= 5) {
+ $ansHash->{score} = 1;
+ $ansHash->{ans_message} = "Thanks!";
+ } else {
+ $ansHash->{score} = 0;
+ $ansHash->{ans_message} = "Please enter an integer from 1 to 5.";
+ }
+ return $ansHash->{score};
+ }
+);
+
+# =======================================================
+# Scaffold
+# =======================================================
+Scaffold::Begin();
+
+Section::Begin("Problem Statement");
+BEGIN_PGML
+In this guided problem, we learn what a **random variable** is and how to classify it as **discrete** or **continuous**.
+
+Then you will classify 8 random variables by:
+- the **type** (discrete or continuous)
+- the **set of possible values**
+END_PGML
+Section::End();
+
+Section::Begin("What is a random variable?");
+BEGIN_PGML
+A **random variable** is a **numerical** quantity that is generated by a random experiment.
+
+**Checkpoint:** Which one is **not** a random variable?
+
+[_]{$rb_not_rv}
+END_PGML
+Section::End();
+
+Section::Begin("Discrete vs Continuous");
+BEGIN_PGML
+A random variable is called **discrete** if it has either a **finite** or a **countable** number of possible values.
+
+A random variable is called **continuous** if its possible values contain a **whole interval** of numbers.
+
+Key ideas:
+
+- Discrete (finite): values can be listed and the list ends.
+- Discrete (countably infinite): values can be listed forever, like [`` \{1,2,3,4,\dots\} ``].
+- Continuous: values fill an interval like [`` [a,b] ``] or [`` (a,b) ``] or [`` [0,\infty) ``].
+END_PGML
+Section::End();
+
+Section::Begin("Examples (for learning the concept)");
+BEGIN_PGML
+Here are examples to model what we mean by “type” and “possible values.”
+
+**Example 1: Spin a fair spinner labeled 1 through 8**
+Let [`` X ``] be the number you land on.
+Type: **Discrete (finite)**
+Possible values: [`` \{1,2,3,4,5,6,7,8\} ``]
+
+**Example 2: Roll a fair die repeatedly**
+Let [`` X ``] be the number of rolls until you get a 6.
+Type: **Discrete (countably infinite)**
+Possible values: [`` \{1,2,3,4,\dots\} ``]
+
+**Example 3: Measure the speed of a car in a school zone**
+Let [`` X ``] be the speed (km/h), assuming it stays between 30 and 50.
+Type: **Continuous**
+Possible values (a closed interval): [`` [30,50] ``]
+
+**Example 4: Choose a point strictly inside a circle of radius 1**
+Let [`` X ``] be the distance from the center.
+Type: **Continuous**
+Possible values (an open interval): [`` (0,1) ``]
+END_PGML
+Section::End();
+
+Section::Begin("Practice 1 (Random Variables 1–2)");
+BEGIN_PGML
+**RV #1**
+Experiment: Roll two fair dice and compute the absolute difference between the two numbers.
+Let [`` X ``] be the absolute difference.
+
+Type:
+[_]{$rb_rv1_type}
+Possible values:
+[_]{$rb_rv1_vals}
+
+---
+
+**RV #2**
+Experiment: Inspect 5 items and count how many are defective.
+Let [`` X ``] be the number of defective items.
+
+Type:
+[_]{$rb_rv2_type}
+Possible values:
+[_]{$rb_rv2_vals}
+END_PGML
+Section::End();
+
+Section::Begin("Practice 2 (Random Variables 3–4)");
+BEGIN_PGML
+**RV #3**
+Experiment: Flip a fair coin until you get tails.
+Let [`` X ``] be the number of flips required.
+
+Type:
+[_]{$rb_rv3_type}
+Possible values:
+[_]{$rb_rv3_vals}
+
+---
+
+**RV #4**
+Experiment: Count how many emails you receive in a day.
+Let [`` X ``] be the number of emails received.
+
+Type:
+[_]{$rb_rv4_type}
+Possible values:
+[_]{$rb_rv4_vals}
+END_PGML
+Section::End();
+
+Section::Begin("Practice 3 (Random Variables 5–6)");
+BEGIN_PGML
+**RV #5**
+Experiment: Measure the amount of gasoline in a car’s tank after filling (in liters), assuming it stays between 0 and 60.
+Let [`` X ``] be the amount of gasoline.
+
+Type:
+[_]{$rb_rv5_type}
+Possible values:
+[_]{$rb_rv5_vals}
+
+---
+
+**RV #6**
+Experiment: Choose a random time strictly during a 60-minute lecture.
+Let [`` X ``] be the number of minutes after the lecture starts.
+
+Type:
+[_]{$rb_rv6_type}
+Possible values:
+[_]{$rb_rv6_vals}
+END_PGML
+Section::End();
+
+Section::Begin("Practice 4 (Random Variables 7–8)");
+BEGIN_PGML
+**RV #7**
+Experiment: Randomly select a student and record the distance (in km) from their home to school.
+Let [`` X ``] be the distance.
+
+Type:
+[_]{$rb_rv7_type}
+Possible values:
+[_]{$rb_rv7_vals}
+
+---
+
+**RV #8**
+Experiment: Measure the temperature of coffee (in °C) right after it is poured, assuming it is strictly between 0 and 100.
+Let [`` X ``] be the temperature.
+
+Type:
+[_]{$rb_rv8_type}
+Possible values:
+[_]{$rb_rv8_vals}
+END_PGML
+Section::End();
+
+# -------------------------------------------------------
+# Feedback (rating 1–5)
+# -------------------------------------------------------
+if ($ENABLE_GP_RATING) {
+ Section::Begin("Feedback");
+ BEGIN_PGML
+How useful was this guided problem for learning combinations?
+
+(1 = not useful, 2 = slightly useful, 3 = useful, 4 = very useful, 5 = extremely useful)
+Note: Consider using the Email Instructor button below to suggest improvements.
+
+Rating (enter 1-5): [___]{$cmp_rating}
+END_PGML
+ Section::End();
+}
+
+Scaffold::End();
+
+ENDDOCUMENT();
diff --git a/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/Estimation/ConfidenceInterval_LargeSample_GuidedProblem1.pg b/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/Estimation/ConfidenceInterval_LargeSample_GuidedProblem1.pg
new file mode 100644
index 0000000000..2c1893b5cb
--- /dev/null
+++ b/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/Estimation/ConfidenceInterval_LargeSample_GuidedProblem1.pg
@@ -0,0 +1,435 @@
+## DESCRIPTION
+## WeBWorK Guided Problems Project — Large-Sample Confidence Intervals for the Mean (Guided / Scaffolded)
+## ENDDESCRIPTION
+##
+## Institution(Dawson College)
+## Author(Mehdi Moodi)
+## DBsubject(Statistics)
+## DBchapter(Confidence Intervals)
+## DBsection(Confidence Interval for the Mean)
+## Level(2)
+## KEYWORDS('confidence interval','large sample','mean','known sigma','unknown sigma','confidence level','critical value','z alpha over 2','sample standard deviation')
+
+DOCUMENT();
+
+loadMacros(
+ "PGstandard.pl",
+ "PGanswermacros.pl",
+ "MathObjects.pl",
+ "PGML.pl",
+ "scaffold.pl",
+ "answerHints.pl",
+ "parserRadioButtons.pl",
+ "PGcourse.pl",
+);
+
+TEXT(beginproblem());
+
+# ----------------------------------------------------------------
+# PILOT: usefulness rating toggle (ACTIVE)
+# ----------------------------------------------------------------
+$ENABLE_GP_RATING = 1;
+
+# ----------------------------------------------------------------
+# Library Browser detection (LB stability)
+# ----------------------------------------------------------------
+$inLibraryBrowser = ($envir{isLibraryBrowser} || $envir{isLibrary} || 0);
+
+Context("Numeric");
+
+# =======================================================
+# Postfilter: add a hint ONLY when the answer is wrong
+# =======================================================
+sub HintIfWrong {
+ my $msg = shift;
+ return sub {
+ my $ans = shift;
+ if (defined($ans->{score}) && $ans->{score} < 1) {
+ $ans->{ans_message} = "" unless defined($ans->{ans_message});
+ $ans->{ans_message} .= " $msg";
+ }
+ return $ans;
+ };
+}
+
+# =======================================================
+# Data for sample standard deviation checkpoint
+# =======================================================
+$x1 = 8;
+$x2 = 10;
+$x3 = 12;
+$x4 = 14;
+$x5 = 16;
+
+# sample standard deviation = sqrt(40/4) = sqrt(10)
+$s_exact = 3.16227766016838;
+
+# =======================================================
+# Correct targets
+# =======================================================
+$alpha_98 = Real(0.02);
+
+$z_010 = Real(1.645);
+$z_005 = Real(1.960);
+$z_002 = Real(2.325);
+$z_001 = Real(2.575);
+
+# =======================================================
+# Tolerances
+# =======================================================
+$tol_alpha = 0.0005;
+$tol_s = 0.02;
+$tol_z = 0.01;
+
+# =======================================================
+# Comparators
+# =======================================================
+
+# Section B MC: correct choice = 2 (xbar)
+$cmp_point_est = Real(2)->cmp(
+ checker => sub {
+ my ($correct, $student, $ansHash) = @_;
+ return 0 unless defined $student;
+ my $v = $student->value;
+ return 0 unless defined $v;
+ return 0 unless $v == int($v);
+ return 0 unless ($v >= 1 && $v <= 5);
+ return ($v == 2) ? 1 : 0;
+ }
+)->withPostFilter(
+ HintIfWrong("The appropriate point estimate for mu is the sample mean xbar.")
+);
+
+$cmp_alpha_98 = $alpha_98->cmp(
+ tolType => "absolute",
+ tolerance => $tol_alpha
+)->withPostFilter(
+ HintIfWrong("For a 98% confidence level, 1 - alpha = 0.98, so alpha = 0.02.")
+);
+
+# Section C MC: correct choice = 3
+$cmp_conf_meaning = Real(3)->cmp(
+ checker => sub {
+ my ($correct, $student, $ansHash) = @_;
+ return 0 unless defined $student;
+ my $v = $student->value;
+ return 0 unless defined $v;
+ return 0 unless $v == int($v);
+ return 0 unless ($v >= 1 && $v <= 5);
+ return ($v == 3) ? 1 : 0;
+ }
+)->withPostFilter(
+ HintIfWrong("The confidence level describes the long-run success rate of the method over many random samples.")
+);
+
+$cmp_s = Real($s_exact)->cmp(
+ tolType => "absolute",
+ tolerance => $tol_s
+)->withPostFilter(
+ HintIfWrong("Use the sample standard deviation on your calculator, not the population standard deviation.")
+);
+
+$cmp_z_010 = $z_010->cmp(
+ tolType => "absolute",
+ tolerance => $tol_z
+)->withPostFilter(
+ HintIfWrong("For alpha = 0.10, alpha/2 = 0.05, so the cutoff is z_(0.05) = 1.645.")
+);
+
+$cmp_z_005 = $z_005->cmp(
+ tolType => "absolute",
+ tolerance => $tol_z
+)->withPostFilter(
+ HintIfWrong("For alpha = 0.05, alpha/2 = 0.025, so the cutoff is z_(0.025) = 1.960.")
+);
+
+$cmp_z_002 = $z_002->cmp(
+ tolType => "absolute",
+ tolerance => $tol_z
+)->withPostFilter(
+ HintIfWrong("For alpha = 0.02, alpha/2 = 0.01, so the cutoff is z_(0.01) = 2.326.")
+);
+
+$cmp_z_001 = $z_001->cmp(
+ tolType => "absolute",
+ tolerance => $tol_z
+)->withPostFilter(
+ HintIfWrong("For alpha = 0.01, alpha/2 = 0.005, so the cutoff is z_(0.005) = 2.576.")
+);
+
+# Rating checker
+$cmp_rating = Real(3)->cmp(
+ checker => sub {
+ my ($correct, $student, $ansHash) = @_;
+ return 0 unless defined $student;
+ my $v = $student->value;
+ return 0 unless defined $v;
+ return 0 unless $v == int($v);
+ return 0 unless ($v >= 1 && $v <= 5);
+ return 1;
+ }
+);
+
+# =======================================================
+# Scaffold setup
+# =======================================================
+if ($inLibraryBrowser) {
+ Scaffold::Begin(
+ numbered => 1,
+ can_open => "always",
+ is_open => "always"
+ );
+} else {
+ Scaffold::Begin(
+ numbered => 1,
+ can_open => "when_previous_correct",
+ is_open => "correct_or_first_incorrect"
+ );
+}
+
+# -------------------------------------------------------
+# Step 1 — Problem Statement
+# -------------------------------------------------------
+Section::Begin("Problem Statement");
+
+BEGIN_PGML
+In this Guided Problem, we introduce the idea of a large-sample confidence interval for the population mean [``\mu``].
+
+Assume a random sample of size [``n \ge 30``] is taken from a population. The goal is to estimate [``\mu``] using the sample data.
+
+In this GP, you will learn:
+
+- what a [``100(1-\alpha)\%``] confidence interval means,
+- what the confidence level represents,
+- the two large-sample formulas for a confidence interval for [``\mu``],
+- and the normal cutoffs [``z_{\alpha/2}``] for common values of [``\alpha``].
+
+Rounding: Round cutoff values to 3 decimals.
+END_PGML
+
+Section::End();
+
+# -------------------------------------------------------
+# Step 2 — Section B
+# -------------------------------------------------------
+Section::Begin("Section B — Appropriate point estimate for mu");
+
+BEGIN_PGML
+A confidence interval for [``\mu``] has the form
+
+[``\text{estimate} \pm \text{margin of error}``]
+
+What is the appropriate point estimate for the population mean [``\mu``]?
+
+1) [``\mu``]
+2) [``\bar x``]
+3) [``\sigma``]
+4) [``s``]
+5) [``\alpha``]
+
+Enter the number (1-5): [____]
+END_PGML
+
+ANS($cmp_point_est);
+
+Section::End();
+
+# -------------------------------------------------------
+# Step 3 — Section C
+# -------------------------------------------------------
+Section::Begin("Section C — Level of confidence");
+
+BEGIN_PGML
+When we build a confidence interval, we must consider the required level of confidence.
+
+Common values are usually [``90\%``] ([``\alpha = 0.10``]), [``95\%``] ([``\alpha = 0.05``]), and [``99\%``] ([``\alpha = 0.01``]).
+
+The confidence level describes how the method behaves over many random samples. For example, a [``95\%``] level of confidence means [``95\%``] of intervals built using this method, by repeatedly sampling the population, will capture the actual [``\mu``].
+
+1) If the confidence level is [``98\%``], enter the value of [``\alpha``]:
+
+[____]
+
+2) Choose the best interpretation of a [``100(1-\alpha)\%``] confidence interval:
+
+1) [``100(1-\alpha)\%``] of all sample means are equal to [``\mu``].
+2) There is a [``100(1-\alpha)\%``] chance that [``\mu``] changes from sample to sample.
+3) If we repeatedly took many random samples and built intervals the same way each time, about [``100(1-\alpha)\%``] of those intervals would contain [``\mu``].
+4) A [``100(1-\alpha)\%``] confidence interval always contains [``\mu``].
+5) The confidence level tells us that this particular sample must be correct.
+
+Enter the number (1-5): [____]
+END_PGML
+
+ANS($cmp_alpha_98, $cmp_conf_meaning);
+
+Section::End();
+
+# -------------------------------------------------------
+# Step 4 — Section D
+# -------------------------------------------------------
+Section::Begin("Section D — Large-sample formulas and sample standard deviation");
+
+BEGIN_PGML
+For large samples, a confidence interval for [``\mu``] has the form
+
+[``\bar x \pm z_{\alpha/2}(\text{standard error})``]
+
+There are two common large-sample formulas:
+
+**Case 1: sigma known**
+[``\bar x \pm z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}``]
+
+**Case 2: sigma unknown, but n is large**
+[``\bar x \pm z_{\alpha/2}\dfrac{s}{\sqrt{n}}``]
+
+So when [``\sigma``] is unknown, we replace it with the sample standard deviation [``s``].
+
+Use your calculator to find the **sample standard deviation** of the following data:
+
+[``8,\ 10,\ 12,\ 14,\ 16``]
+
+Enter [``s``]: [____]
+END_PGML
+
+ANS($cmp_s);
+
+Section::End();
+
+# -------------------------------------------------------
+# Step 5 — Sections E
+# -------------------------------------------------------
+Section::Begin("Sections E — Cutoff values from the normal table");
+
+BEGIN_PGML
+For a [``100(1-\alpha)\%``] confidence interval:
+
+- the middle area is [``1-\alpha``]
+- the total area in the two tails is [``\alpha``]
+- each tail has area [``\alpha/2``]
+- the positive cutoff is [``z_{\alpha/2}``]
+
+[@ image("cutoff.jpg", width=>520) @]*
+
+Now use the normal table to find the positive cutoff [``z_{\alpha/2}``] for each value of [``\alpha``].
+
+For [``\alpha = 0.10``], enter [``z_{\alpha/2}``]: [____]
+
+For [``\alpha = 0.05``], enter [``z_{\alpha/2}``]: [____]
+
+For [``\alpha = 0.02``], enter [``z_{\alpha/2}``]: [____]
+
+For [``\alpha = 0.01``], enter [``z_{\alpha/2}``]: [____]
+
+Note that smaller [``\alpha``] means higher confidence level, which is desirable. However, smaller [``\alpha``] also means a bigger cutoff value that will result in a bigger margin of error for the confidence interval. This is a trade-off. Higher confidence level comes at the expense of a wider, and thus less precise, confidence interval.
+END_PGML
+
+ANS($cmp_z_010, $cmp_z_005, $cmp_z_002, $cmp_z_001);
+
+Section::End();
+
+# -------------------------------------------------------
+# Feedback (rating) — hidden in Library Browser
+# -------------------------------------------------------
+if ($ENABLE_GP_RATING && !$inLibraryBrowser) {
+ Section::Begin("Feedback");
+
+ BEGIN_PGML
+How useful was this Guided Problem for you?
+
+(1 = not useful, 2 = slightly useful, 3 = useful, 4 = very useful, 5 = extremely useful)
+
+Note: Consider using the Email Instructor button below to suggest improvements.
+
+Rating (1-5): [____]
+END_PGML
+
+ ANS($cmp_rating);
+
+ Section::End();
+}
+
+Scaffold::End();
+
+BEGIN_PGML_SOLUTION
+**1. Appropriate point estimate**
+
+A confidence interval for [``\mu``] is centered at the sample mean.
+
+So the appropriate point estimate for the population mean [``\mu``] is
+
+[``\bar x``]
+
+**2. Confidence level and alpha**
+
+If the confidence level is [``98\%``], then
+
+[``1-\alpha = 0.98``]
+
+so
+
+[``\alpha = 0.02``]
+
+The confidence level describes the long-run success rate of the method. For example, a [``95\%``] confidence level means that if we repeatedly sample the population and build intervals the same way each time, about [``95\%``] of those intervals will capture the true [``\mu``].
+
+**3. Large-sample formulas**
+
+For large samples, the confidence interval for [``\mu``] has the form
+
+[``\bar x \pm z_{\alpha/2}(\text{standard error})``]
+
+If [``\sigma``] is known:
+
+[``\bar x \pm z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}``]
+
+If [``\sigma``] is unknown but the sample is large:
+
+[``\bar x \pm z_{\alpha/2}\dfrac{s}{\sqrt{n}}``]
+
+So when [``\sigma``] is unknown, we use the sample standard deviation [``s``].
+
+**4. Compute the sample standard deviation**
+
+For the data
+
+[``8,\ 10,\ 12,\ 14,\ 16``]
+
+the sample mean is
+
+[``\bar x = 12``]
+
+The deviations from the mean are
+
+[``-4,\ -2,\ 0,\ 2,\ 4``]
+
+Their squares are
+
+[``16,\ 4,\ 0,\ 4,\ 16``]
+
+The sum of squared deviations is
+
+[``40``]
+
+So the sample variance is
+
+[``s^2 = \dfrac{40}{5-1} = 10``]
+
+and the sample standard deviation is
+
+[``s = \sqrt{10} \approx 3.162``]
+
+**5. Cutoff values**
+
+For a [``100(1-\alpha)\%``] confidence interval, each tail has area [``\alpha/2``], and the positive cutoff is [``z_{\alpha/2}``].
+
+The common cutoff values are:
+
+- for [``\alpha = 0.10``]: [``z_{0.05} = 1.645``]
+- for [``\alpha = 0.05``]: [``z_{0.025} = 1.960``]
+- for [``\alpha = 0.02``]: [``z_{0.01} = 2.325``]
+- for [``\alpha = 0.01``]: [``z_{0.005} = 2.575``]
+
+These are the normal-table cutoffs commonly used for large-sample confidence intervals for the mean.
+END_PGML_SOLUTION
+
+ENDDOCUMENT();
\ No newline at end of file
diff --git a/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/Estimation/ConfidenceInterval_LargeSample_GuidedProblem2.pg b/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/Estimation/ConfidenceInterval_LargeSample_GuidedProblem2.pg
new file mode 100644
index 0000000000..58b343e75b
--- /dev/null
+++ b/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/Estimation/ConfidenceInterval_LargeSample_GuidedProblem2.pg
@@ -0,0 +1,491 @@
+## DESCRIPTION
+## WeBWorK Guided Problems Project — Confidence Interval for the Mean:
+## Large Sample, Sigma Known (Guided / Scaffolded)
+## ENDDESCRIPTION
+##
+## Institution(Dawson College)
+## Author(Mehdi Moodi)
+## DBsubject(Statistics)
+## DBchapter(Confidence Intervals)
+## DBsection(Confidence Interval for the Mean)
+## Level(2)
+## KEYWORDS(
+## 'confidence interval',
+## 'population mean',
+## 'large sample',
+## 'known sigma',
+## 'z critical value',
+## 'standard error',
+## 'margin of error'
+## )
+
+DOCUMENT();
+
+loadMacros(
+ "PGstandard.pl",
+ "PGanswermacros.pl",
+ "MathObjects.pl",
+ "PGML.pl",
+ "scaffold.pl",
+ "answerHints.pl",
+ "parserRadioButtons.pl",
+ "PGcourse.pl",
+);
+
+TEXT(beginproblem());
+
+# ============================================================
+# Settings
+# ============================================================
+
+# Pilot: usefulness rating toggle
+$ENABLE_GP_RATING = 1;
+
+# Library Browser detection
+$inLibraryBrowser = ($envir{isLibraryBrowser} || $envir{isLibrary} || 0);
+
+Context("Numeric");
+
+# ============================================================
+# Helper: show a hint only when the student's answer is wrong
+# ============================================================
+sub HintIfWrong {
+ my $msg = shift;
+
+ return sub {
+ my $ans = shift;
+
+ if (defined($ans->{score}) && $ans->{score} < 1) {
+ $ans->{ans_message} = "" unless defined($ans->{ans_message});
+ $ans->{ans_message} .= " $msg";
+ }
+
+ return $ans;
+ };
+}
+
+# ============================================================
+# Data
+# ============================================================
+$xbar = 483;
+$sigma = 12;
+$n = 64;
+$conf = 0.95;
+
+$alpha = Real(0.05);
+$z_cutoff = Real(1.96);
+$se = Real(1.5);
+$me = Real(2.94);
+$lcl = Real(480.06);
+$ucl = Real(485.94);
+
+# ============================================================
+# Tolerances
+# ============================================================
+$tol_alpha = 0.0005;
+$tol_z = 0.005;
+$tol_se = 0.005;
+$tol_me = 0.01;
+$tol_end = 0.02;
+
+# ============================================================
+# Comparators
+# ============================================================
+
+# ------------------------------------------------------------
+# Formula-selection multiple choice
+# Correct answer: 4
+# ------------------------------------------------------------
+$cmp_formula = Real(4)->cmp(
+ checker => sub {
+ my ($correct, $student, $ansHash) = @_;
+
+ return 0 unless defined $student;
+
+ my $v = $student->value;
+ return 0 unless defined $v;
+ return 0 unless $v == int($v);
+ return 0 unless ($v >= 1 && $v <= 5);
+
+ return ($v == 4) ? 1 : 0;
+ }
+)->withPostFilter(
+ HintIfWrong(
+ "Because the sample is large and sigma is known, use xbar +/- z_(alpha/2)(sigma/sqrt(n))."
+ )
+);
+
+# ------------------------------------------------------------
+# Alpha
+# ------------------------------------------------------------
+$cmp_alpha = $alpha->cmp(
+ tolType => "absolute",
+ tolerance => $tol_alpha
+)->withPostFilter(
+ HintIfWrong("For a 95% confidence level, 1 - alpha = 0.95, so alpha = 0.05.")
+);
+
+# ------------------------------------------------------------
+# z cutoff
+# ------------------------------------------------------------
+$cmp_z = $z_cutoff->cmp(
+ tolType => "absolute",
+ tolerance => $tol_z
+)->withPostFilter(
+ HintIfWrong(
+ "For a 95% confidence interval, alpha = 0.05, so alpha/2 = 0.025 and z_(alpha/2) = 1.960."
+ )
+);
+
+# ------------------------------------------------------------
+# Standard error
+# ------------------------------------------------------------
+$cmp_se = $se->cmp(
+ tolType => "absolute",
+ tolerance => $tol_se
+)->withPostFilter(
+ HintIfWrong("The standard error here is sigma/sqrt(n).")
+);
+
+# ------------------------------------------------------------
+# Margin of error
+# ------------------------------------------------------------
+$cmp_me = $me->cmp(
+ tolType => "absolute",
+ tolerance => $tol_me
+)->withPostFilter(
+ HintIfWrong("The margin of error is z_(alpha/2)(sigma/sqrt(n)).")
+);
+
+# ------------------------------------------------------------
+# Lower endpoint
+# ------------------------------------------------------------
+$cmp_lcl = $lcl->cmp(
+ tolType => "absolute",
+ tolerance => $tol_end
+)->withPostFilter(
+ HintIfWrong("The lower endpoint is xbar minus the margin of error.")
+);
+
+# ------------------------------------------------------------
+# Upper endpoint
+# ------------------------------------------------------------
+$cmp_ucl = $ucl->cmp(
+ tolType => "absolute",
+ tolerance => $tol_end
+)->withPostFilter(
+ HintIfWrong("The upper endpoint is xbar plus the margin of error.")
+);
+
+# ------------------------------------------------------------
+# Interpretation multiple choice
+# Correct answer: 2
+# ------------------------------------------------------------
+$cmp_interp = Real(2)->cmp(
+ checker => sub {
+ my ($correct, $student, $ansHash) = @_;
+
+ return 0 unless defined $student;
+
+ my $v = $student->value;
+ return 0 unless defined $v;
+ return 0 unless $v == int($v);
+ return 0 unless ($v >= 1 && $v <= 5);
+
+ return ($v == 2) ? 1 : 0;
+ }
+)->withPostFilter(
+ HintIfWrong(
+ "A confidence interval for mu is interpreted as a plausible range for the true population mean."
+ )
+);
+
+# ------------------------------------------------------------
+# Rating checker
+# Any integer 1 to 5 is accepted
+# ------------------------------------------------------------
+$cmp_rating = Real(3)->cmp(
+ checker => sub {
+ my ($correct, $student, $ansHash) = @_;
+
+ return 0 unless defined $student;
+
+ my $v = $student->value;
+ return 0 unless defined $v;
+ return 0 unless $v == int($v);
+ return 0 unless ($v >= 1 && $v <= 5);
+
+ return 1;
+ }
+);
+
+# ============================================================
+# Scaffold setup
+# ============================================================
+if ($inLibraryBrowser) {
+ Scaffold::Begin(
+ numbered => 1,
+ can_open => "always",
+ is_open => "always"
+ );
+}
+else {
+ Scaffold::Begin(
+ numbered => 1,
+ can_open => "when_previous_correct",
+ is_open => "correct_or_first_incorrect"
+ );
+}
+
+# ============================================================
+# Step 1 — Problem Statement
+# ============================================================
+Section::Begin("Problem Statement");
+
+BEGIN_PGML
+A beverage company wants to estimate the true mean amount of juice in its bottles.
+
+A random sample of [``n=64``] bottles is selected. The sample mean is [``\bar x = 483``] mL.
+The population standard deviation is known to be [``\sigma = 12``] mL.
+
+Construct a [``95\%``] confidence interval for the true population mean [``\mu``].
+
+Because the sample is large, according to the CLT, the sampling distribution of [``\bar X``] is approximately normal.
+
+Round the cutoff, standard error, margin of error, and interval endpoints to 3 decimals.
+END_PGML
+
+Section::End();
+
+# ============================================================
+# Step 2 — Choose the correct formula
+# ============================================================
+Section::Begin("Section B — Choose the correct formula");
+
+BEGIN_PGML
+Choose the correct confidence-interval formula for this problem, considering that the sample is large
+and the population standard deviation [``\sigma``] is known.
+
+1) [``\bar x \pm t_{\alpha/2}\dfrac{s}{\sqrt{n}}``]
+
+2) [``\hat p \pm z_{\alpha/2}\sqrt{\dfrac{\hat p(1-\hat p)}{n}}``]
+
+3) [``\bar x \pm z_{\alpha/2}\dfrac{s}{\sqrt{n}}``]
+
+4) [``\bar x \pm z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}``]
+
+5) [``\bar x \pm t_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}``]
+
+Enter the number (1-5): [____]
+END_PGML
+
+ANS($cmp_formula);
+
+Section::End();
+
+# ============================================================
+# Step 3 — Find alpha and the cutoff
+# ============================================================
+Section::Begin("Section C — Find alpha and the cutoff");
+
+BEGIN_PGML
+For a [``95\%``] confidence interval:
+
+- [``1-\alpha = 0.95``]
+- each tail has area [``\alpha/2``]
+
+1) Enter the value of [``\alpha``]:
+
+[____]
+
+[@ image("cutoff.jpg", width=>520) @]*
+
+2) Use the normal table to find the positive cutoff [``z_{\alpha/2}``].
+
+Enter [``z_{\alpha/2}``]: [____]
+END_PGML
+
+ANS($cmp_alpha, $cmp_z);
+
+Section::End();
+
+# ============================================================
+# Step 4 — Compute the standard error and margin of error
+# ============================================================
+Section::Begin("Section D — Compute the standard error and margin of error");
+
+BEGIN_PGML
+Compute the standard error:
+
+[``\dfrac{\sigma}{\sqrt{n}}``]
+
+For this problem,
+
+[``\dfrac{12}{\sqrt{64}} =``] [____]
+
+Now compute the margin of error:
+
+[``E = z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}``]
+
+For this problem,
+
+[``E = (1.960)\left(\dfrac{12}{\sqrt{64}}\right) =``] [____]
+END_PGML
+
+ANS($cmp_se, $cmp_me);
+
+Section::End();
+
+# ============================================================
+# Step 5 — Construct the confidence interval
+# ============================================================
+Section::Begin("Section E — Construct the confidence interval");
+
+BEGIN_PGML
+Now construct the confidence interval:
+
+[``\bar x \pm E``]
+
+Using [``\bar x = 483``] and your margin of error, enter the two endpoints.
+
+Lower endpoint: [____]
+
+Upper endpoint: [____]
+END_PGML
+
+ANS($cmp_lcl, $cmp_ucl);
+
+Section::End();
+
+# ============================================================
+# Step 6 — Interpretation
+# ============================================================
+Section::Begin("Section F — Interpret the confidence interval");
+
+BEGIN_PGML
+The [``95\%``] confidence interval is
+
+[``(480.060,\ 485.940)``]
+
+Choose the best interpretation.
+
+1) [``95\%``] of all bottles contain between [``480.060``] mL and [``485.940``] mL.
+
+2) We are [``95\%``] confident that the true mean amount of juice in all bottles is between
+[``480.060``] mL and [``485.940``] mL.
+
+3) There is a [``95\%``] chance that the sample mean [``\bar x``] is between
+[``480.060``] and [``485.940``].
+
+4) [``95\%``] of all possible sample means are exactly equal to [``\mu``].
+
+5) The interval proves that [``\mu = 483``] exactly.
+
+Enter the number (1-5): [____]
+END_PGML
+
+ANS($cmp_interp);
+
+Section::End();
+
+# ============================================================
+# Feedback — hidden in Library Browser
+# ============================================================
+if ($ENABLE_GP_RATING && !$inLibraryBrowser) {
+ Section::Begin("Feedback");
+
+ BEGIN_PGML
+How useful was this Guided Problem for you?
+
+(1 = not useful, 2 = slightly useful, 3 = useful, 4 = very useful, 5 = extremely useful)
+
+Note: Consider using the Email Instructor button below to suggest improvements.
+
+Rating (1-5): [____]
+END_PGML
+
+ ANS($cmp_rating);
+
+ Section::End();
+}
+
+Scaffold::End();
+
+# ============================================================
+# Solution
+# ============================================================
+BEGIN_PGML_SOLUTION
+**1. Choose the correct formula**
+
+Because the sample is large and the population standard deviation [``\sigma``] is known,
+the correct formula is
+
+[``\bar x \pm z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}``]
+
+**2. Find alpha and the cutoff**
+
+For a [``95\%``] confidence interval,
+
+[``1-\alpha = 0.95``]
+
+so
+
+[``\alpha = 0.05``]
+
+Then
+
+[``\alpha/2 = 0.025``]
+
+From the normal table,
+
+[``z_{\alpha/2} = z_{0.025} = 1.960``]
+
+**3. Compute the standard error**
+
+[``\dfrac{\sigma}{\sqrt{n}} = \dfrac{12}{\sqrt{64}} = \dfrac{12}{8} = 1.500``]
+
+So the standard error is [``1.500``].
+
+**4. Compute the margin of error**
+
+[``E = z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}``]
+
+So
+
+[``E = (1.960)(1.500) = 2.940``]
+
+**5. Construct the confidence interval**
+
+Use
+
+[``\bar x \pm E``]
+
+So
+
+[``483 \pm 2.940``]
+
+Lower endpoint:
+
+[``483 - 2.940 = 480.060``]
+
+Upper endpoint:
+
+[``483 + 2.940 = 485.940``]
+
+Therefore, the [``95\%``] confidence interval for [``\mu``] is
+
+[``(480.060,\ 485.940)``]
+
+**6. Interpret the confidence interval**
+
+The correct interpretation is:
+
+We are [``95\%``] confident that the true mean amount of juice in all bottles is between
+[``480.060``] mL and [``485.940``] mL.
+
+This interval gives a plausible range for the population mean [``\mu``].
+It does **not** mean that [``95\%``] of individual bottles must fall in this interval,
+and it does **not** mean that [``\bar x``] has a [``95\%``] chance of being in that range.
+END_PGML_SOLUTION
+
+ENDDOCUMENT();
\ No newline at end of file
diff --git a/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/Estimation/ConfidenceInterval_LargeSample_GuidedProblem3.pg b/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/Estimation/ConfidenceInterval_LargeSample_GuidedProblem3.pg
new file mode 100644
index 0000000000..dcd02ff8bb
--- /dev/null
+++ b/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/Estimation/ConfidenceInterval_LargeSample_GuidedProblem3.pg
@@ -0,0 +1,425 @@
+## DESCRIPTION
+## WeBWorK Guided Problems Project — Confidence Interval for the Mean: Large Sample, Sigma Unknown (Guided / Scaffolded)
+## ENDDESCRIPTION
+##
+## Institution(Dawson College)
+## Author(Mehdi Moodi)
+## DBsubject(Statistics)
+## DBchapter(Confidence Intervals)
+## DBsection(Confidence Interval for the Mean)
+## Level(2)
+## KEYWORDS('confidence interval','population mean','large sample','sigma unknown','sample standard deviation','z critical value','standard error','margin of error')
+
+DOCUMENT();
+
+loadMacros(
+ "PGstandard.pl",
+ "PGanswermacros.pl",
+ "MathObjects.pl",
+ "PGML.pl",
+ "scaffold.pl",
+ "answerHints.pl",
+ "parserRadioButtons.pl",
+ "PGcourse.pl",
+);
+
+TEXT(beginproblem());
+
+# ----------------------------------------------------------------
+# PILOT: usefulness rating toggle
+# ----------------------------------------------------------------
+$ENABLE_GP_RATING = 1;
+
+# ----------------------------------------------------------------
+# Library Browser detection (LB stability)
+# ----------------------------------------------------------------
+$inLibraryBrowser = ($envir{isLibraryBrowser} || $envir{isLibrary} || 0);
+
+Context("Numeric");
+
+# =======================================================
+# Postfilter: add a hint ONLY when the answer is wrong
+# =======================================================
+sub HintIfWrong {
+ my $msg = shift;
+ return sub {
+ my $ans = shift;
+ if (defined($ans->{score}) && $ans->{score} < 1) {
+ $ans->{ans_message} = "" unless defined($ans->{ans_message});
+ $ans->{ans_message} .= " $msg";
+ }
+ return $ans;
+ };
+}
+
+# =======================================================
+# Data
+# =======================================================
+$xbar = 72;
+$s = 15;
+$n = 100;
+$conf = 0.98;
+
+$alpha = Real(0.02);
+$z_cutoff = Real(2.325);
+$se = Real(1.5);
+$me = Real(3.4875);
+$lcl = Real(68.5125);
+$ucl = Real(75.4875);
+
+# =======================================================
+# Tolerances
+# =======================================================
+$tol_alpha = 0.0005;
+$tol_z = 0.005;
+$tol_se = 0.005;
+$tol_me = 0.01;
+$tol_end = 0.02;
+
+# =======================================================
+# Comparators
+# =======================================================
+
+# Formula-selection MC: correct choice = 3
+$cmp_formula = Real(3)->cmp(
+ checker => sub {
+ my ($correct, $student, $ansHash) = @_;
+ return 0 unless defined $student;
+ my $v = $student->value;
+ return 0 unless defined $v;
+ return 0 unless $v == int($v);
+ return 0 unless ($v >= 1 && $v <= 5);
+ return ($v == 3) ? 1 : 0;
+ }
+)->withPostFilter(
+ HintIfWrong("Because the sample is large and sigma is unknown, we use xbar +/- z_(alpha/2)(s/sqrt(n)).")
+);
+
+$cmp_alpha = $alpha->cmp(
+ tolType => "absolute",
+ tolerance => $tol_alpha
+)->withPostFilter(
+ HintIfWrong("For a 98% confidence level, 1 - alpha = 0.98, so alpha = 0.02.")
+);
+
+$cmp_z = $z_cutoff->cmp(
+ tolType => "absolute",
+ tolerance => $tol_z
+)->withPostFilter(
+ HintIfWrong("For a 98% confidence interval, alpha = 0.02, so alpha/2 = 0.01 and the table cutoff is 2.325.")
+);
+
+$cmp_se = $se->cmp(
+ tolType => "absolute",
+ tolerance => $tol_se
+)->withPostFilter(
+ HintIfWrong("The standard error here is s/sqrt(n).")
+);
+
+$cmp_me = $me->cmp(
+ tolType => "absolute",
+ tolerance => $tol_me
+)->withPostFilter(
+ HintIfWrong("The margin of error is z_(alpha/2)(s/sqrt(n)).")
+);
+
+$cmp_lcl = $lcl->cmp(
+ tolType => "absolute",
+ tolerance => $tol_end
+)->withPostFilter(
+ HintIfWrong("The lower endpoint is xbar minus the margin of error.")
+);
+
+$cmp_ucl = $ucl->cmp(
+ tolType => "absolute",
+ tolerance => $tol_end
+)->withPostFilter(
+ HintIfWrong("The upper endpoint is xbar plus the margin of error.")
+);
+
+# Interpretation MC: correct choice = 4
+$cmp_interp = Real(4)->cmp(
+ checker => sub {
+ my ($correct, $student, $ansHash) = @_;
+ return 0 unless defined $student;
+ my $v = $student->value;
+ return 0 unless defined $v;
+ return 0 unless $v == int($v);
+ return 0 unless ($v >= 1 && $v <= 5);
+ return ($v == 4) ? 1 : 0;
+ }
+)->withPostFilter(
+ HintIfWrong("A confidence interval for mu is interpreted as a plausible range for the true population mean.")
+);
+
+# Rating checker
+$cmp_rating = Real(3)->cmp(
+ checker => sub {
+ my ($correct, $student, $ansHash) = @_;
+ return 0 unless defined $student;
+ my $v = $student->value;
+ return 0 unless defined $v;
+ return 0 unless $v == int($v);
+ return 0 unless ($v >= 1 && $v <= 5);
+ return 1;
+ }
+);
+
+# =======================================================
+# Scaffold setup
+# =======================================================
+if ($inLibraryBrowser) {
+ Scaffold::Begin(
+ numbered => 1,
+ can_open => "always",
+ is_open => "always"
+ );
+} else {
+ Scaffold::Begin(
+ numbered => 1,
+ can_open => "when_previous_correct",
+ is_open => "correct_or_first_incorrect"
+ );
+}
+
+# -------------------------------------------------------
+# Step 1 — Problem Statement
+# -------------------------------------------------------
+Section::Begin("Problem Statement");
+
+BEGIN_PGML
+A streaming platform wants to estimate the true mean number of minutes users spend watching content in a session.
+
+A random sample of [``n=100``] users is selected. The sample mean is [``\bar x = 72``] minutes. The population standard deviation is unknown, but the sample standard deviation is [``s = 15``] minutes.
+
+Construct a [``98\%``] confidence interval for the true population mean [``\mu``].
+
+Because the sample is large, according to the CLT, the sampling distribution of [``\bar X``] is approximately normal.
+
+Round the cutoff, standard error, margin of error, and interval endpoints to 3 decimals.
+END_PGML
+
+Section::End();
+
+# -------------------------------------------------------
+# Step 2 — Choose the correct formula
+# -------------------------------------------------------
+Section::Begin("Section B — Choose the correct formula");
+
+BEGIN_PGML
+Choose the correct confidence-interval formula for this problem, considering that the sample is large and the population standard deviation [``\sigma``] is unknown.
+
+1) [``\bar x \pm t_{\alpha/2}\dfrac{s}{\sqrt{n}}``]
+
+2) [``\hat p \pm z_{\alpha/2}\sqrt{\dfrac{\hat p(1-\hat p)}{n}}``]
+
+3) [``\bar x \pm z_{\alpha/2}\dfrac{s}{\sqrt{n}}``]
+
+4) [``\bar x \pm z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}``]
+
+5) [``\bar x \pm t_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}``]
+
+Enter the number (1-5): [____]
+END_PGML
+
+ANS($cmp_formula);
+
+Section::End();
+
+# -------------------------------------------------------
+# Step 3 — Find alpha and the cutoff
+# -------------------------------------------------------
+Section::Begin("Section C — Find alpha and the cutoff");
+
+BEGIN_PGML
+For a [``98\%``] confidence interval:
+
+- [``1-\alpha = 0.98``]
+- each tail has area [``\alpha/2``]
+
+1) Enter the value of [``\alpha``]:
+
+[____]
+
+[@ image("cutoff.jpg", width=>520) @]*
+
+2) Use the normal table to find the positive cutoff [``z_{\alpha/2}``].
+
+Enter [``z_{\alpha/2}``]: [____]
+END_PGML
+
+ANS($cmp_alpha, $cmp_z);
+
+Section::End();
+
+# -------------------------------------------------------
+# Step 4 — Compute the standard error and margin of error
+# -------------------------------------------------------
+Section::Begin("Section D — Compute the standard error and margin of error");
+
+BEGIN_PGML
+Compute the standard error:
+
+[``\dfrac{s}{\sqrt{n}}``]
+
+For this problem,
+
+[``\dfrac{15}{\sqrt{100}} =``] [____]
+
+Now compute the margin of error:
+
+[``E = z_{\alpha/2}\dfrac{s}{\sqrt{n}}``]
+
+For this problem,
+
+[``E = (2.325)\left(\dfrac{15}{\sqrt{100}}\right) =``] [____]
+END_PGML
+
+ANS($cmp_se, $cmp_me);
+
+Section::End();
+
+# -------------------------------------------------------
+# Step 5 — Construct the confidence interval
+# -------------------------------------------------------
+Section::Begin("Section E — Construct the confidence interval");
+
+BEGIN_PGML
+Now construct the confidence interval:
+
+[``\bar x \pm E``]
+
+Using [``\bar x = 72``] and your margin of error, enter the two endpoints.
+
+Lower endpoint: [____]
+
+Upper endpoint: [____]
+END_PGML
+
+ANS($cmp_lcl, $cmp_ucl);
+
+Section::End();
+
+# -------------------------------------------------------
+# Step 6 — Interpretation
+# -------------------------------------------------------
+Section::Begin("Section F — Interpret the confidence interval");
+
+BEGIN_PGML
+The [``98\%``] confidence interval is
+
+[``(68.513,\ 75.488)``]
+
+Choose the best interpretation.
+
+1) [``98\%``] of all user sessions last exactly between [``68.513``] and [``75.488``] minutes.
+
+2) There is a [``98\%``] chance that the sample mean [``\bar x``] is between [``68.513``] and [``75.488``].
+
+3) The interval proves that the true population mean is exactly [``72``] minutes.
+
+4) We are [``98\%``] confident that the true mean session time for all users is between [``68.513``] and [``75.488``] minutes.
+
+5) [``98\%``] of all possible sample means are exactly equal to [``\mu``].
+
+Enter the number (1-5): [____]
+END_PGML
+
+ANS($cmp_interp);
+
+Section::End();
+
+# -------------------------------------------------------
+# Feedback (rating) — hidden in Library Browser
+# -------------------------------------------------------
+if ($ENABLE_GP_RATING && !$inLibraryBrowser) {
+ Section::Begin("Feedback");
+
+ BEGIN_PGML
+How useful was this Guided Problem for you?
+
+(1 = not useful, 2 = slightly useful, 3 = useful, 4 = very useful, 5 = extremely useful)
+
+Note: Consider using the Email Instructor button below to suggest improvements.
+
+Rating (1-5): [____]
+END_PGML
+
+ ANS($cmp_rating);
+
+ Section::End();
+}
+
+Scaffold::End();
+
+BEGIN_PGML_SOLUTION
+**1. Choose the correct formula**
+
+Because the sample is large and the population standard deviation [``\sigma``] is unknown, we use the sample standard deviation [``s``]. The correct formula is
+
+[``\bar x \pm z_{\alpha/2}\dfrac{s}{\sqrt{n}}``]
+
+**2. Find alpha and the cutoff**
+
+For a [``98\%``] confidence interval,
+
+[``1-\alpha = 0.98``]
+
+so
+
+[``\alpha = 0.02``]
+
+Then
+
+[``\alpha/2 = 0.01``]
+
+From the normal table,
+
+[``z_{\alpha/2} = z_{0.01} = 2.325``]
+
+**3. Compute the standard error**
+
+[``\dfrac{s}{\sqrt{n}} = \dfrac{15}{\sqrt{100}} = \dfrac{15}{10} = 1.500``]
+
+So the standard error is [``1.500``].
+
+**4. Compute the margin of error**
+
+[``E = z_{\alpha/2}\dfrac{s}{\sqrt{n}}``]
+
+So
+
+[``E = (2.325)(1.500) = 3.488``]
+
+**5. Construct the confidence interval**
+
+Use
+
+[``\bar x \pm E``]
+
+So
+
+[``72 \pm 3.488``]
+
+Lower endpoint:
+
+[``72 - 3.488 = 68.513``]
+
+Upper endpoint:
+
+[``72 + 3.488 = 75.488``]
+
+Therefore, the [``98\%``] confidence interval for [``\mu``] is
+
+[``(68.513,\ 75.488)``]
+
+**6. Interpret the confidence interval**
+
+The correct interpretation is:
+
+We are [``98\%``] confident that the true mean session time for all users is between [``68.513``] and [``75.488``] minutes.
+
+This interval gives a plausible range for the population mean [``\mu``]. It does **not** describe where [``98\%``] of individual session times fall, and it does **not** mean that the sample mean has a [``98\%``] chance of being in that interval.
+END_PGML_SOLUTION
+
+ENDDOCUMENT();
diff --git a/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/Estimation/ConfidenceInterval_SampleProportion_GuidedProblem1.pg b/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/Estimation/ConfidenceInterval_SampleProportion_GuidedProblem1.pg
new file mode 100644
index 0000000000..83c1d9a3ae
--- /dev/null
+++ b/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/Estimation/ConfidenceInterval_SampleProportion_GuidedProblem1.pg
@@ -0,0 +1,508 @@
+## DESCRIPTION
+## WeBWorK Guided Problems Project — Confidence Intervals for a Population Proportion (Guided / Scaffolded)
+## ENDDESCRIPTION
+##
+## Institution(Dawson College)
+## Author(Mehdi Moodi)
+## DBsubject(Statistics)
+## DBchapter(Confidence Intervals)
+## DBsection(Confidence Interval for a Population Proportion)
+## Level(2)
+## KEYWORDS('confidence interval','population proportion','sample proportion','normal approximation','success-failure condition','z critical value','confidence level')
+
+DOCUMENT();
+
+loadMacros(
+ "PGstandard.pl",
+ "PGanswermacros.pl",
+ "MathObjects.pl",
+ "PGML.pl",
+ "scaffold.pl",
+ "answerHints.pl",
+ "parserRadioButtons.pl",
+ "PGcourse.pl",
+);
+
+TEXT(beginproblem());
+
+# ----------------------------------------------------------------
+# PILOT: usefulness rating toggle (ACTIVE)
+# ----------------------------------------------------------------
+$ENABLE_GP_RATING = 1;
+
+# ----------------------------------------------------------------
+# Library Browser detection (LB stability)
+# ----------------------------------------------------------------
+$inLibraryBrowser = ($envir{isLibraryBrowser} || $envir{isLibrary} || 0);
+
+Context("Numeric");
+
+# =======================================================
+# Postfilter: add a hint ONLY when the answer is wrong
+# =======================================================
+sub HintIfWrong {
+ my $msg = shift;
+ return sub {
+ my $ans = shift;
+ if (defined($ans->{score}) && $ans->{score} < 1) {
+ $ans->{ans_message} = "" unless defined($ans->{ans_message});
+ $ans->{ans_message} .= " $msg";
+ }
+ return $ans;
+ };
+}
+
+# =======================================================
+# Data
+# =======================================================
+$n_sample = 80;
+$x_success = 52;
+$p_hat_val = 0.65;
+$q_hat_val = 0.35;
+
+$np_hat_val = 52;
+$nq_hat_val = 28;
+
+$alpha_98 = Real(0.02);
+
+$z_010 = Real(1.645);
+$z_005 = Real(1.960);
+$z_002 = Real(2.325);
+$z_001 = Real(2.575);
+
+# Approximate 95% CI for the example
+$se_95 = 0.053326; # sqrt(0.65*0.35/80)
+$me_95 = 0.104519; # 1.96 * se
+$lcl_95 = 0.545;
+$ucl_95 = 0.755;
+
+# =======================================================
+# Tolerances
+# =======================================================
+$tol_alpha = 0.0005;
+$tol_phat = 0.0005;
+$tol_count = 0.0005;
+$tol_z = 0.01;
+
+# =======================================================
+# Comparators
+# =======================================================
+
+# Section B MC: correct choice = 2 (phat)
+$cmp_point_est = Real(2)->cmp(
+ checker => sub {
+ my ($correct, $student, $ansHash) = @_;
+ return 0 unless defined $student;
+ my $v = $student->value;
+ return 0 unless defined $v;
+ return 0 unless $v == int($v);
+ return 0 unless ($v >= 1 && $v <= 5);
+ return ($v == 2) ? 1 : 0;
+ }
+)->withPostFilter(
+ HintIfWrong("The appropriate point estimate for p is the sample proportion phat.")
+);
+
+$cmp_alpha_98 = $alpha_98->cmp(
+ tolType => "absolute",
+ tolerance => $tol_alpha
+)->withPostFilter(
+ HintIfWrong("For a 98% confidence level, 1 - alpha = 0.98, so alpha = 0.02.")
+);
+
+$cmp_phat = Real($p_hat_val)->cmp(
+ tolType => "absolute",
+ tolerance => $tol_phat
+)->withPostFilter(
+ HintIfWrong("Use phat = x/n = number of successes divided by the sample size.")
+);
+
+$cmp_np_hat = Real($np_hat_val)->cmp(
+ tolType => "absolute",
+ tolerance => $tol_count
+)->withPostFilter(
+ HintIfWrong("Compute n(phat) using your value of phat.")
+);
+
+$cmp_nq_hat = Real($nq_hat_val)->cmp(
+ tolType => "absolute",
+ tolerance => $tol_count
+)->withPostFilter(
+ HintIfWrong("Compute n(1-phat). Since phat = 0.65, 1-phat = 0.35.")
+);
+
+$cmp_z_010 = $z_010->cmp(
+ tolType => "absolute",
+ tolerance => $tol_z
+)->withPostFilter(
+ HintIfWrong("For alpha = 0.10, alpha/2 = 0.05, so the cutoff is z_(0.05) = 1.645.")
+);
+
+$cmp_z_005 = $z_005->cmp(
+ tolType => "absolute",
+ tolerance => $tol_z
+)->withPostFilter(
+ HintIfWrong("For alpha = 0.05, alpha/2 = 0.025, so the cutoff is z_(0.025) = 1.960.")
+);
+
+$cmp_z_002 = $z_002->cmp(
+ tolType => "absolute",
+ tolerance => $tol_z
+)->withPostFilter(
+ HintIfWrong("For alpha = 0.02, alpha/2 = 0.01, so the cutoff is z_(0.01) = 2.326.")
+);
+
+$cmp_z_001 = $z_001->cmp(
+ tolType => "absolute",
+ tolerance => $tol_z
+)->withPostFilter(
+ HintIfWrong("For alpha = 0.01, alpha/2 = 0.005, so the cutoff is z_(0.005) = 2.576.")
+);
+
+# Section G MC: correct choice = 3
+$cmp_interp = Real(3)->cmp(
+ checker => sub {
+ my ($correct, $student, $ansHash) = @_;
+ return 0 unless defined $student;
+ my $v = $student->value;
+ return 0 unless defined $v;
+ return 0 unless $v == int($v);
+ return 0 unless ($v >= 1 && $v <= 5);
+ return ($v == 3) ? 1 : 0;
+ }
+)->withPostFilter(
+ HintIfWrong("A confidence interval for p is interpreted as a plausible range for the true population proportion.")
+);
+
+# Rating checker
+$cmp_rating = Real(3)->cmp(
+ checker => sub {
+ my ($correct, $student, $ansHash) = @_;
+ return 0 unless defined $student;
+ my $v = $student->value;
+ return 0 unless defined $v;
+ return 0 unless $v == int($v);
+ return 0 unless ($v >= 1 && $v <= 5);
+ return 1;
+ }
+);
+
+# =======================================================
+# Scaffold setup
+# =======================================================
+if ($inLibraryBrowser) {
+ Scaffold::Begin(
+ numbered => 1,
+ can_open => "always",
+ is_open => "always"
+ );
+} else {
+ Scaffold::Begin(
+ numbered => 1,
+ can_open => "when_previous_correct",
+ is_open => "correct_or_first_incorrect"
+ );
+}
+
+# -------------------------------------------------------
+# Step 1 — Problem Statement
+# -------------------------------------------------------
+Section::Begin("Problem Statement");
+
+BEGIN_PGML
+In this Guided Problem, we introduce the idea of a confidence interval for the population proportion [``p``].
+
+Suppose a random sample of size [``n``] is taken from a population, and each individual answers a yes/no question. For example, do you prefer this new product to the old one?
+
+The goal is to estimate the true population proportion [``p``] of the individuals who prefer the new product, using the sample proportion [``\hat p``].
+
+In this GP, you will learn:
+
+- what a [``100(1-\alpha)\%``] confidence interval means,
+- what the appropriate point estimate for [``p``] is,
+- the formula for a confidence interval for [``p``],
+- the expected success-failure conditions needed for the normal approximation,
+- the common cutoff values [``z_{\alpha/2}``],
+- and how to interpret a confidence interval in context.
+
+For confidence intervals for [``p``], the cutoff comes from the **normal table**.
+
+Rounding: Round cutoff values to 3 decimals.
+END_PGML
+
+Section::End();
+
+# -------------------------------------------------------
+# Step 2 — Section B
+# -------------------------------------------------------
+Section::Begin("Section B — Appropriate point estimate for p");
+
+BEGIN_PGML
+A confidence interval for [``p``] has the form
+
+[``\text{estimate} \pm \text{margin of error}``]
+
+What is the appropriate point estimate for the population proportion [``p``]?
+
+1) [``p``]
+2) [``\hat p``]
+3) [``\sigma``]
+4) [``s``]
+5) [``\alpha``]
+
+Enter the number (1-5): [____]
+END_PGML
+
+ANS($cmp_point_est);
+
+Section::End();
+
+# -------------------------------------------------------
+# Step 3 — Section C
+# -------------------------------------------------------
+Section::Begin("Section C — Level of confidence");
+
+BEGIN_PGML
+When we build a confidence interval, we must consider the required level of confidence.
+
+Common values are usually [``90\%``] ([``\alpha = 0.10``]), [``95\%``] ([``\alpha = 0.05``]), and [``99\%``] ([``\alpha = 0.01``]).
+
+The confidence level describes how the method behaves over many random samples. For example, a [``95\%``] level of confidence means [``95\%``] of intervals built using this method, by repeatedly sampling the population, will capture the actual [``p``].
+
+If the confidence level is [``98\%``], enter the value of [``\alpha``]:
+
+[____]
+END_PGML
+
+ANS($cmp_alpha_98);
+
+Section::End();
+
+# -------------------------------------------------------
+# Step 4 — Section D
+# -------------------------------------------------------
+Section::Begin("Section D — Formula and sample proportion");
+
+BEGIN_PGML
+The confidence interval formula for a population proportion is
+
+[``\hat p \pm z_{\alpha/2}\sqrt{\dfrac{\hat p(1-\hat p)}{n}}``]
+
+Here:
+
+- [``\hat p``], called the sample proportion, is the center of the interval,
+- [``z_{\alpha/2}``] is the cutoff from the **normal table**,
+- and [``\sqrt{\dfrac{\hat p(1-\hat p)}{n}}``] is the standard error.
+
+A survey of [``80``] customers found that [``52``] prefer a new product.
+
+Enter the sample proportion [``\hat p``]:
+
+[____]
+END_PGML
+
+ANS($cmp_phat);
+
+Section::End();
+
+# -------------------------------------------------------
+# Step 5 — Section E
+# -------------------------------------------------------
+Section::Begin("Section E — Expected success-failure conditions");
+
+BEGIN_PGML
+For the normal approximation to be reasonable when building a confidence interval for [``p``], we check the expected success-failure conditions:
+
+[``n\hat p \ge 5 \qquad \text{and} \qquad n(1-\hat p)\ge 5``]
+
+Using the same sample with [``n=80``] and your value of [``\hat p``], compute:
+
+1) [``n\hat p``] = [____]
+
+2) [``n(1-\hat p)``] = [____]
+
+If both values are at least [``5``], then the normal approximation is reasonable.
+END_PGML
+
+ANS($cmp_np_hat, $cmp_nq_hat);
+
+Section::End();
+
+# -------------------------------------------------------
+# Step 6 — Section F
+# -------------------------------------------------------
+Section::Begin("Section F — Cutoff values from the normal table");
+
+BEGIN_PGML
+For a [``100(1-\alpha)\%``] confidence interval:
+
+- the middle area is [``1-\alpha``]
+- the total area in the two tails is [``\alpha``]
+- each tail has area [``\alpha/2``]
+- the positive cutoff is [``z_{\alpha/2}``]
+
+[@ image("cutoff.jpg", width=>520) @]*
+
+Now use the normal table to find the positive cutoff [``z_{\alpha/2}``] for each value of [``\alpha``].
+
+For [``\alpha = 0.10``], enter [``z_{\alpha/2}``]: [____]
+
+For [``\alpha = 0.05``], enter [``z_{\alpha/2}``]: [____]
+
+For [``\alpha = 0.02``], enter [``z_{\alpha/2}``]: [____]
+
+For [``\alpha = 0.01``], enter [``z_{\alpha/2}``]: [____]
+
+Note that smaller [``\alpha``] means higher confidence level, which is desirable. However, smaller [``\alpha``] also means a bigger cutoff value that will result in a bigger margin of error for the confidence interval. This is a trade-off. Higher confidence level comes at the expense of a wider, and thus less precise, confidence interval.
+END_PGML
+
+ANS($cmp_z_010, $cmp_z_005, $cmp_z_002, $cmp_z_001);
+
+Section::End();
+
+# -------------------------------------------------------
+# Step 7 — Section G
+# -------------------------------------------------------
+Section::Begin("Section G — Interpretation in context");
+
+BEGIN_PGML
+Using the sample of [``80``] customers with [``52``] successes, a [``95\%``] confidence interval for the true population proportion [``p``] is approximately
+
+[``(0.545,\ 0.755)``]
+
+Choose the best interpretation.
+
+1) [``95\%``] of the customers in this sample prefer the new product.
+2) There is a [``95\%``] chance that [``p``] is exactly [``0.65``].
+3) We are [``95\%``] confident that the true proportion of all customers who prefer the new product is between [``0.545``] and [``0.755``].
+4) [``95\%``] of all possible sample proportions must lie between [``0.545``] and [``0.755``].
+5) The interval proves that exactly [``65\%``] of all customers prefer the new product.
+
+Enter the number (1-5): [____]
+END_PGML
+
+ANS($cmp_interp);
+
+Section::End();
+
+# -------------------------------------------------------
+# Feedback (rating) — hidden in Library Browser
+# -------------------------------------------------------
+if ($ENABLE_GP_RATING && !$inLibraryBrowser) {
+ Section::Begin("Feedback");
+
+ BEGIN_PGML
+How useful was this Guided Problem for you?
+
+(1 = not useful, 2 = slightly useful, 3 = useful, 4 = very useful, 5 = extremely useful)
+
+Note: Consider using the Email Instructor button below to suggest improvements.
+
+Rating (1-5): [____]
+END_PGML
+
+ ANS($cmp_rating);
+
+ Section::End();
+}
+
+Scaffold::End();
+
+BEGIN_PGML_SOLUTION
+**1. Appropriate point estimate**
+
+A confidence interval for [``p``] is centered at the sample proportion.
+
+So the appropriate point estimate for the population proportion [``p``] is
+
+[``\hat p``]
+
+**2. Confidence level and alpha**
+
+If the confidence level is [``98\%``], then
+
+[``1-\alpha = 0.98``]
+
+so
+
+[``\alpha = 0.02``]
+
+The confidence level describes the long-run success rate of the method. For example, a [``95\%``] confidence level means that if we repeatedly sample the population and build intervals the same way each time, about [``95\%``] of those intervals will capture the true [``p``].
+
+**3. Formula for a confidence interval for p**
+
+The formula is
+
+[``\hat p \pm z_{\alpha/2}\sqrt{\dfrac{\hat p(1-\hat p)}{n}}``]
+
+Here:
+
+- [``\hat p``] is the center,
+- [``z_{\alpha/2}``] is the cutoff from the normal table,
+- and [``\sqrt{\dfrac{\hat p(1-\hat p)}{n}}``] is the standard error.
+
+**4. Compute the sample proportion**
+
+In the sample, [``52``] out of [``80``] customers are successes.
+
+So
+
+[``\hat p = \dfrac{52}{80} = 0.65``]
+
+**5. Expected success-failure conditions**
+
+We check:
+
+[``n\hat p \ge 5 \qquad \text{and} \qquad n(1-\hat p)\ge 5``]
+
+Since [``\hat p=0.65``], we have
+
+[``n\hat p = 80(0.65)=52``]
+
+and
+
+[``n(1-\hat p)=80(0.35)=28``]
+
+Both values are at least [``5``], so the normal approximation is reasonable.
+
+**6. Cutoff values**
+
+For a [``100(1-\alpha)\%``] confidence interval, each tail has area [``\alpha/2``], and the positive cutoff is [``z_{\alpha/2}``].
+
+The common cutoff values are:
+
+- for [``\alpha = 0.10``]: [``z_{0.05} = 1.645``]
+- for [``\alpha = 0.05``]: [``z_{0.025} = 1.960``]
+- for [``\alpha = 0.02``]: [``z_{0.01} = 2.325``]
+- for [``\alpha = 0.01``]: [``z_{0.005} = 2.575``]
+
+**7. Approximate 95% confidence interval**
+
+Using [``\hat p=0.65``], [``n=80``], and [``z_{0.025}=1.96``]:
+
+First compute the standard error:
+
+[``\sqrt{\dfrac{(0.65)(0.35)}{80}} \approx 0.0533``]
+
+Then the margin of error is
+
+[``1.96(0.0533)\approx 0.1045``]
+
+So the confidence interval is
+
+[``0.65 \pm 0.1045``]
+
+which gives
+
+[``(0.545,\ 0.755)``]
+
+**8. Interpretation in context**
+
+The correct interpretation is:
+
+We are [``95\%``] confident that the true proportion of all customers who prefer the new product is between [``0.545``] and [``0.755``].
+
+That means this interval gives a plausible range for the population proportion [``p``]. It does **not** mean that [``95\%``] of the sample had to be successes, and it does **not** mean that [``p``] has a [``95\%``] chance of being a single exact value.
+END_PGML_SOLUTION
+
+ENDDOCUMENT();
\ No newline at end of file
diff --git a/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/Estimation/ConfidenceInterval_SampleProportion_GuidedProblem2.pg b/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/Estimation/ConfidenceInterval_SampleProportion_GuidedProblem2.pg
new file mode 100644
index 0000000000..3a70bf1d73
--- /dev/null
+++ b/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/Estimation/ConfidenceInterval_SampleProportion_GuidedProblem2.pg
@@ -0,0 +1,585 @@
+## DESCRIPTION
+## WeBWorK Guided Problems Project — Confidence Interval for a Population Proportion:
+## Raw Data, Normal Approximation Check (Guided / Scaffolded)
+## ENDDESCRIPTION
+##
+## Institution(Dawson College)
+## Author(Mehdi Moodi)
+## DBsubject(Statistics)
+## DBchapter(Confidence Intervals)
+## DBsection(Confidence Interval for a Population Proportion)
+## Level(2)
+## KEYWORDS(
+## 'confidence interval',
+## 'population proportion',
+## 'sample proportion',
+## 'normal approximation',
+## 'z critical value',
+## 'margin of error'
+## )
+
+DOCUMENT();
+
+loadMacros(
+ "PGstandard.pl",
+ "PGanswermacros.pl",
+ "MathObjects.pl",
+ "PGML.pl",
+ "scaffold.pl",
+ "answerHints.pl",
+ "parserRadioButtons.pl",
+ "PGcourse.pl",
+);
+
+TEXT(beginproblem());
+
+# ============================================================
+# Settings
+# ============================================================
+
+$ENABLE_GP_RATING = 1;
+
+$inLibraryBrowser = 0;
+if ((defined($envir{isLibraryBrowser}) && $envir{isLibraryBrowser})
+ || (defined($envir{isLibrary}) && $envir{isLibrary})) {
+ $inLibraryBrowser = 1;
+}
+
+Context("Numeric");
+
+# ============================================================
+# Helper: show a hint only when the student's answer is wrong
+# ============================================================
+sub HintIfWrong {
+ my $msg = shift;
+
+ return sub {
+ my $ans = shift;
+
+ if (defined($ans->{score}) && $ans->{score} < 1) {
+ $ans->{ans_message} = "" unless defined($ans->{ans_message});
+ $ans->{ans_message} .= " $msg";
+ }
+
+ return $ans;
+ };
+}
+
+# ============================================================
+# Data
+# Context: library self-checkout use
+# ============================================================
+
+$n = 120;
+$x = 78;
+$not_x = 42;
+$conf = 0.97;
+
+$phat = Real(0.65);
+$alpha = Real(0.03);
+$z_cut = Real(2.170);
+$np = Real(78);
+$nq = Real(42);
+$se = Real(0.04354116896);
+$me = Real(0.09448433665);
+$lcl = Real(0.55551566335);
+$ucl = Real(0.74448433665);
+
+# ============================================================
+# Tolerances
+# ============================================================
+
+$tol_phat = 0.005;
+$tol_cond = 0.01;
+$tol_alpha = 0.0005;
+$tol_z = 0.005;
+$tol_se = 0.005;
+$tol_me = 0.01;
+$tol_end = 0.01;
+
+# ============================================================
+# Comparators
+# ============================================================
+
+# ------------------------------------------------------------
+# Formula-selection multiple choice
+# Correct answer: 2
+# ------------------------------------------------------------
+$cmp_formula = Real(2)->cmp(
+ checker => sub {
+ my ($correct, $student, $ansHash) = @_;
+
+ return 0 unless defined $student;
+
+ my $v = $student->value;
+ return 0 unless defined $v;
+ return 0 unless $v == int($v);
+ return 0 unless ($v >= 1 && $v <= 5);
+
+ return ($v == 2) ? 1 : 0;
+ }
+)->withPostFilter(
+ HintIfWrong(
+ "For a confidence interval for a population proportion, use p-hat +/- z_(alpha/2) sqrt((p-hat(1-p-hat))/n)."
+ )
+);
+
+# ------------------------------------------------------------
+# Sample proportion
+# ------------------------------------------------------------
+$cmp_phat = $phat->cmp(
+ tolType => "absolute",
+ tolerance => $tol_phat
+)->withPostFilter(
+ HintIfWrong("Compute p-hat using x/n, where x is the number of successes and n is the sample size.")
+);
+
+# ------------------------------------------------------------
+# Normal approximation conditions
+# ------------------------------------------------------------
+$cmp_np = $np->cmp(
+ tolType => "absolute",
+ tolerance => $tol_cond
+)->withPostFilter(
+ HintIfWrong("Compute n(p-hat) using the sample size and your value of p-hat.")
+);
+
+$cmp_nq = $nq->cmp(
+ tolType => "absolute",
+ tolerance => $tol_cond
+)->withPostFilter(
+ HintIfWrong("Compute n(1-p-hat) using the sample size and your value of p-hat.")
+);
+
+# ------------------------------------------------------------
+# Alpha
+# ------------------------------------------------------------
+$cmp_alpha = $alpha->cmp(
+ tolType => "absolute",
+ tolerance => $tol_alpha
+)->withPostFilter(
+ HintIfWrong("For a 97% confidence level, 1 - alpha = 0.97, so alpha = 0.03.")
+);
+
+# ------------------------------------------------------------
+# z cutoff
+# ------------------------------------------------------------
+$cmp_z = $z_cut->cmp(
+ tolType => "absolute",
+ tolerance => $tol_z
+)->withPostFilter(
+ HintIfWrong(
+ "For a 97% confidence interval, alpha = 0.03, so alpha/2 = 0.015. Use the normal table to find z_(alpha/2)."
+ )
+);
+
+# ------------------------------------------------------------
+# Standard error
+# ------------------------------------------------------------
+$cmp_se = $se->cmp(
+ tolType => "absolute",
+ tolerance => $tol_se
+)->withPostFilter(
+ HintIfWrong("For a proportion, the standard error is sqrt((p-hat(1-p-hat))/n).")
+);
+
+# ------------------------------------------------------------
+# Margin of error
+# ------------------------------------------------------------
+$cmp_me = $me->cmp(
+ tolType => "absolute",
+ tolerance => $tol_me
+)->withPostFilter(
+ HintIfWrong("The margin of error is z_(alpha/2) times the standard error.")
+);
+
+# ------------------------------------------------------------
+# Lower endpoint
+# ------------------------------------------------------------
+$cmp_lcl = $lcl->cmp(
+ tolType => "absolute",
+ tolerance => $tol_end
+)->withPostFilter(
+ HintIfWrong("The lower endpoint is p-hat minus the margin of error.")
+);
+
+# ------------------------------------------------------------
+# Upper endpoint
+# ------------------------------------------------------------
+$cmp_ucl = $ucl->cmp(
+ tolType => "absolute",
+ tolerance => $tol_end
+)->withPostFilter(
+ HintIfWrong("The upper endpoint is p-hat plus the margin of error.")
+);
+
+# ------------------------------------------------------------
+# Interpretation multiple choice
+# Correct answer: 4
+# ------------------------------------------------------------
+$cmp_interp = Real(4)->cmp(
+ checker => sub {
+ my ($correct, $student, $ansHash) = @_;
+
+ return 0 unless defined $student;
+
+ my $v = $student->value;
+ return 0 unless defined $v;
+ return 0 unless $v == int($v);
+ return 0 unless ($v >= 1 && $v <= 5);
+
+ return ($v == 4) ? 1 : 0;
+ }
+)->withPostFilter(
+ HintIfWrong(
+ "A confidence interval for p is interpreted as a plausible range for the true population proportion."
+ )
+);
+
+# ------------------------------------------------------------
+# Rating checker
+# Any integer 1 to 5 is accepted
+# ------------------------------------------------------------
+$cmp_rating = Real(3)->cmp(
+ checker => sub {
+ my ($correct, $student, $ansHash) = @_;
+
+ return 0 unless defined $student;
+
+ my $v = $student->value;
+ return 0 unless defined $v;
+ return 0 unless $v == int($v);
+ return 0 unless ($v >= 1 && $v <= 5);
+
+ return 1;
+ }
+);
+
+# ============================================================
+# Scaffold setup
+# ============================================================
+
+if ($inLibraryBrowser) {
+ Scaffold::Begin(
+ numbered => 1,
+ can_open => "always",
+ is_open => "always"
+ );
+}
+else {
+ Scaffold::Begin(
+ numbered => 1,
+ can_open => "when_previous_correct",
+ is_open => "correct_or_first_incorrect"
+ );
+}
+
+# ============================================================
+# Step 1 — Problem Statement
+# ============================================================
+Section::Begin("Problem Statement");
+
+BEGIN_PGML
+A public library wants to estimate the true proportion of its members who regularly use the self-checkout kiosk.
+
+A random sample of [``120``] members is selected. The results are:
+
+- [``78``] members said they regularly use the self-checkout kiosk.
+- [``42``] members said they do not.
+
+Construct a [``97\%``] confidence interval for the true population proportion [``p``].
+
+Round [``\hat p``], the cutoff, the standard error, the margin of error, and the interval endpoints to 3 decimals.
+END_PGML
+
+Section::End();
+
+# ============================================================
+# Step 2 — Choose the correct formula
+# ============================================================
+Section::Begin("Section B — Choose the correct formula");
+
+BEGIN_PGML
+Choose the correct confidence-interval formula for this problem.
+
+1) [``\bar x \pm z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}``]
+
+2) [``\hat p \pm z_{\alpha/2}\sqrt{\dfrac{\hat p(1-\hat p)}{n}}``]
+
+3) [``\hat p \pm t_{\alpha/2}\sqrt{\dfrac{\hat p(1-\hat p)}{n}}``]
+
+4) [``\bar x \pm t_{\alpha/2}\dfrac{s}{\sqrt{n}}``]
+
+5) [``\hat p \pm z_{\alpha/2}\dfrac{\hat p(1-\hat p)}{n}``]
+
+Enter the number (1-5): [____]
+END_PGML
+
+ANS($cmp_formula);
+
+Section::End();
+
+# ============================================================
+# Step 3 — Compute p-hat from the sample data
+# ============================================================
+Section::Begin("Section C — Compute the sample proportion");
+
+BEGIN_PGML
+For this problem, a "success" means a sampled member regularly uses the self-checkout kiosk.
+
+Compute the sample proportion:
+
+[``\hat p = \dfrac{x}{n}``]
+
+For this sample,
+
+[``\hat p = \dfrac{78}{120} =``] [____]
+END_PGML
+
+ANS($cmp_phat);
+
+Section::End();
+
+# ============================================================
+# Step 4 — Check the normal approximation conditions
+# ============================================================
+Section::Begin("Section D — Check the normal approximation conditions");
+
+BEGIN_PGML
+To use the normal approximation for a confidence interval for [``p``], we check that
+
+[``n\hat p \ge 10``] and [``n(1-\hat p)\ge 10``]
+
+Using your value of [``\hat p``], compute:
+
+[``n\hat p =``] [____]
+
+[``n(1-\hat p) =``] [____]
+
+Since both values are at least [``5``], the normal approximation is appropriate for this problem.
+END_PGML
+
+ANS($cmp_np, $cmp_nq);
+
+Section::End();
+
+# ============================================================
+# Step 5 — Find alpha and the cutoff
+# ============================================================
+Section::Begin("Section E — Find alpha and the cutoff");
+
+BEGIN_PGML
+For a [``97\%``] confidence interval:
+
+- [``1-\alpha = 0.97``]
+- each tail has area [``\alpha/2``]
+
+1) Enter the value of [``\alpha``]:
+
+[____]
+
+[@ image("cutoff.jpg", width=>520) @]*
+
+2) Use the normal table to find the positive cutoff [``z_{\alpha/2}``].
+
+Enter [``z_{\alpha/2}``]: [____]
+END_PGML
+
+ANS($cmp_alpha, $cmp_z);
+
+Section::End();
+
+# ============================================================
+# Step 6 — Compute the standard error and margin of error
+# ============================================================
+Section::Begin("Section F — Compute the standard error and margin of error");
+
+BEGIN_PGML
+Compute the standard error:
+
+[``\sqrt{\dfrac{\hat p(1-\hat p)}{n}}``]
+
+For this problem,
+
+[``\sqrt{\dfrac{(0.650)(0.350)}{120}} =``] [____]
+
+Now compute the margin of error:
+
+[``E = z_{\alpha/2}\sqrt{\dfrac{\hat p(1-\hat p)}{n}}``]
+
+For this problem,
+
+[``E = (2.170)\sqrt{\dfrac{(0.650)(0.350)}{120}} =``] [____]
+END_PGML
+
+ANS($cmp_se, $cmp_me);
+
+Section::End();
+
+# ============================================================
+# Step 7 — Construct the confidence interval
+# ============================================================
+Section::Begin("Section G — Construct the confidence interval");
+
+BEGIN_PGML
+Now construct the confidence interval:
+
+[``\hat p \pm E``]
+
+Using your values of [``\hat p``] and [``E``], enter the two endpoints.
+
+Lower endpoint: [____]
+
+Upper endpoint: [____]
+END_PGML
+
+ANS($cmp_lcl, $cmp_ucl);
+
+Section::End();
+
+# ============================================================
+# Step 8 — Interpretation
+# ============================================================
+Section::Begin("Section H — Interpret the confidence interval");
+
+BEGIN_PGML
+Choose the best interpretation of the confidence interval you constructed for [``p``].
+
+1) [``97\%``] of all individual library members must lie inside the interval.
+
+2) There is a [``97\%``] chance that the sample proportion [``\hat p``] lies inside the interval.
+
+3) The interval proves that the true population proportion is exactly equal to [``\hat p``].
+
+4) We are [``97\%``] confident that the true proportion of all library members who regularly use the self-checkout kiosk lies between the two endpoints of the interval.
+
+5) [``97\%``] of all possible samples will produce exactly the same value of [``\hat p``].
+
+Enter the number (1-5): [____]
+END_PGML
+
+ANS($cmp_interp);
+
+Section::End();
+
+# ============================================================
+# Feedback — hidden in Library Browser
+# ============================================================
+if ($ENABLE_GP_RATING && !$inLibraryBrowser) {
+ Section::Begin("Feedback");
+
+ BEGIN_PGML
+How useful was this Guided Problem for you?
+
+(1 = not useful, 2 = slightly useful, 3 = useful, 4 = very useful, 5 = extremely useful)
+
+Note: Consider using the Email Instructor button below to suggest improvements.
+
+Rating (1-5): [____]
+END_PGML
+
+ ANS($cmp_rating);
+
+ Section::End();
+}
+
+Scaffold::End();
+
+# ============================================================
+# Solution
+# ============================================================
+BEGIN_PGML_SOLUTION
+**1. Choose the correct formula**
+
+Because this is a confidence interval for a population proportion, the correct formula is
+
+[``\hat p \pm z_{\alpha/2}\sqrt{\dfrac{\hat p(1-\hat p)}{n}}``]
+
+**2. Compute the sample proportion**
+
+A success means the sampled member regularly uses the self-checkout kiosk.
+
+So
+
+[``\hat p = \dfrac{x}{n} = \dfrac{78}{120} = 0.650``]
+
+**3. Check the normal approximation conditions**
+
+We compute
+
+[``n\hat p = (120)(0.650) = 78``]
+
+and
+
+[``n(1-\hat p) = (120)(0.350) = 42``]
+
+Since both values are at least [``10``], the normal approximation is appropriate.
+
+**4. Find alpha and the cutoff**
+
+For a [``97\%``] confidence interval,
+
+[``1-\alpha = 0.97``]
+
+so
+
+[``\alpha = 0.03``]
+
+Then
+
+[``\alpha/2 = 0.015``]
+
+From the normal table,
+
+[``z_{\alpha/2} = z_{0.015} = 2.170``]
+
+**5. Compute the standard error**
+
+[``\sqrt{\dfrac{\hat p(1-\hat p)}{n}} = \sqrt{\dfrac{(0.650)(0.350)}{120}}``]
+
+So
+
+[``\sqrt{\dfrac{0.2275}{120}} = \sqrt{0.0018958333} \approx 0.044``]
+
+More precisely, the standard error is about [``0.043541``].
+
+**6. Compute the margin of error**
+
+[``E = z_{\alpha/2}\sqrt{\dfrac{\hat p(1-\hat p)}{n}}``]
+
+So
+
+[``E = (2.170)(0.043541) \approx 0.094``]
+
+More precisely, [``E \approx 0.094484``].
+
+**7. Construct the confidence interval**
+
+Use
+
+[``\hat p \pm E``]
+
+So
+
+[``0.650 \pm 0.094``]
+
+Lower endpoint:
+
+[``0.650 - 0.094484 \approx 0.556``]
+
+Upper endpoint:
+
+[``0.650 + 0.094484 \approx 0.744``]
+
+Therefore, the [``97\%``] confidence interval for [``p``] is
+
+[``(0.556,\ 0.744)``]
+
+**8. Interpret the confidence interval**
+
+The correct interpretation is:
+
+We are [``97\%``] confident that the true proportion of all library members who regularly use the self-checkout kiosk lies between [``0.556``] and [``0.744``].
+END_PGML_SOLUTION
+
+ENDDOCUMENT();
\ No newline at end of file
diff --git a/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/Estimation/ConfidenceInterval_SampleSize_GuidedProblem1.pg b/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/Estimation/ConfidenceInterval_SampleSize_GuidedProblem1.pg
new file mode 100644
index 0000000000..899e3fb703
--- /dev/null
+++ b/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/Estimation/ConfidenceInterval_SampleSize_GuidedProblem1.pg
@@ -0,0 +1,463 @@
+## DESCRIPTION
+## WeBWorK Guided Problems Project — Sample Size for Estimating a Mean:
+## Estimate for Sigma Given (Guided / Scaffolded)
+## ENDDESCRIPTION
+##
+## Institution(Dawson College)
+## Author(Mehdi Moodi)
+## DBsubject(Statistics)
+## DBchapter(Confidence Intervals)
+## DBsection(Sample Size for Estimating a Mean)
+## Level(2)
+## KEYWORDS(
+## 'sample size',
+## 'margin of error',
+## 'confidence interval',
+## 'population mean',
+## 'estimate for sigma',
+## 'z critical value'
+## )
+
+DOCUMENT();
+
+loadMacros(
+ "PGstandard.pl",
+ "PGanswermacros.pl",
+ "MathObjects.pl",
+ "PGML.pl",
+ "scaffold.pl",
+ "answerHints.pl",
+ "parserRadioButtons.pl",
+ "PGcourse.pl",
+);
+
+TEXT(beginproblem());
+
+# ============================================================
+# Settings
+# ============================================================
+
+$ENABLE_GP_RATING = 1;
+
+$inLibraryBrowser = 0;
+if ((defined($envir{isLibraryBrowser}) && $envir{isLibraryBrowser})
+ || (defined($envir{isLibrary}) && $envir{isLibrary})) {
+ $inLibraryBrowser = 1;
+}
+
+Context("Numeric");
+
+# ============================================================
+# Helper: show a hint only when the student's answer is wrong
+# ============================================================
+sub HintIfWrong {
+ my $msg = shift;
+
+ return sub {
+ my $ans = shift;
+
+ if (defined($ans->{score}) && $ans->{score} < 1) {
+ $ans->{ans_message} = "" unless defined($ans->{ans_message});
+ $ans->{ans_message} .= " $msg";
+ }
+
+ return $ans;
+ };
+}
+
+# ============================================================
+# Data
+# Context: coffee shop waiting time during lunch rush
+# ============================================================
+
+$conf = 0.95;
+$sigma_est = Real(4.8);
+$E = Real(1.5);
+$alpha = Real(0.05);
+$z_cut = Real(1.96);
+$n_raw = Real(39.337984);
+$n_final = 40;
+
+# ============================================================
+# Tolerances
+# ============================================================
+
+$tol_basic = 0.0005;
+$tol_z = 0.005;
+$tol_raw = 0.02;
+
+# ============================================================
+# Comparators
+# ============================================================
+
+# ------------------------------------------------------------
+# Formula-selection multiple choice
+# Correct answer: 1
+# ------------------------------------------------------------
+$cmp_formula = Real(1)->cmp(
+ checker => sub {
+ my ($correct, $student, $ansHash) = @_;
+
+ return 0 unless defined $student;
+
+ my $v = $student->value;
+ return 0 unless defined $v;
+ return 0 unless $v == int($v);
+ return 0 unless ($v >= 1 && $v <= 5);
+
+ return ($v == 1) ? 1 : 0;
+ }
+)->withPostFilter(
+ HintIfWrong(
+ "For planning sample size for a mean using an estimate of sigma, use n = (z_(alpha/2) sigma / E)^2."
+ )
+);
+
+# ------------------------------------------------------------
+# Identify ingredients
+# ------------------------------------------------------------
+$cmp_sigma = $sigma_est->cmp(
+ tolType => "absolute",
+ tolerance => $tol_basic
+)->withPostFilter(
+ HintIfWrong("Use the previous estimate of the population standard deviation given in the problem.")
+);
+
+$cmp_E = $E->cmp(
+ tolType => "absolute",
+ tolerance => $tol_basic
+)->withPostFilter(
+ HintIfWrong("Use the desired margin of error from the problem statement.")
+);
+
+$cmp_alpha = $alpha->cmp(
+ tolType => "absolute",
+ tolerance => $tol_basic
+)->withPostFilter(
+ HintIfWrong("For a 95% confidence level, 1 - alpha = 0.95, so alpha = 0.05.")
+);
+
+# ------------------------------------------------------------
+# z cutoff
+# ------------------------------------------------------------
+$cmp_z = $z_cut->cmp(
+ tolType => "absolute",
+ tolerance => $tol_z
+)->withPostFilter(
+ HintIfWrong(
+ "For a 95% confidence level, alpha = 0.05, so alpha/2 = 0.025 and z_(alpha/2) = 1.960."
+ )
+);
+
+# ------------------------------------------------------------
+# Raw sample size
+# ------------------------------------------------------------
+$cmp_n_raw = $n_raw->cmp(
+ tolType => "absolute",
+ tolerance => $tol_raw
+)->withPostFilter(
+ HintIfWrong("Substitute into n = (z_(alpha/2) sigma / E)^2 and compute before rounding.")
+);
+
+# ------------------------------------------------------------
+# Final sample size
+# ------------------------------------------------------------
+$cmp_n_final = Real($n_final)->cmp(
+ checker => sub {
+ my ($correct, $student, $ansHash) = @_;
+
+ return 0 unless defined $student;
+
+ my $v = $student->value;
+ return 0 unless defined $v;
+ return 0 unless $v == int($v);
+
+ return ($v == $n_final) ? 1 : 0;
+ }
+)->withPostFilter(
+ HintIfWrong("Always round the required sample size up to the next whole number.")
+);
+
+# ------------------------------------------------------------
+# Interpretation multiple choice
+# Correct answer: 2
+# ------------------------------------------------------------
+$cmp_interp = Real(2)->cmp(
+ checker => sub {
+ my ($correct, $student, $ansHash) = @_;
+
+ return 0 unless defined $student;
+
+ my $v = $student->value;
+ return 0 unless defined $v;
+ return 0 unless $v == int($v);
+ return 0 unless ($v >= 1 && $v <= 5);
+
+ return ($v == 2) ? 1 : 0;
+ }
+)->withPostFilter(
+ HintIfWrong(
+ "The final answer is the minimum whole-number sample size needed so the margin of error is at most the target value."
+ )
+);
+
+# ------------------------------------------------------------
+# Rating checker
+# Any integer 1 to 5 is accepted
+# ------------------------------------------------------------
+$cmp_rating = Real(3)->cmp(
+ checker => sub {
+ my ($correct, $student, $ansHash) = @_;
+
+ return 0 unless defined $student;
+
+ my $v = $student->value;
+ return 0 unless defined $v;
+ return 0 unless $v == int($v);
+ return 0 unless ($v >= 1 && $v <= 5);
+
+ return 1;
+ }
+);
+
+# ============================================================
+# Scaffold setup
+# ============================================================
+
+if ($inLibraryBrowser) {
+ Scaffold::Begin(
+ numbered => 1,
+ can_open => "always",
+ is_open => "always"
+ );
+}
+else {
+ Scaffold::Begin(
+ numbered => 1,
+ can_open => "when_previous_correct",
+ is_open => "correct_or_first_incorrect"
+ );
+}
+
+# ============================================================
+# Step 1 — Problem Statement
+# ============================================================
+Section::Begin("Problem Statement");
+
+BEGIN_PGML
+A coffee shop wants to estimate the true mean waiting time for customers during the lunch rush.
+
+A previous study suggests that the population standard deviation is about [``4.8``] minutes.
+
+The manager wants to estimate the true mean waiting time with a margin of error of at most [``1.5``] minutes at the [``95\%``] confidence level.
+
+Find the minimum sample size [``n``] required.
+
+Round the raw calculator value of [``n``] to 3 decimals, then round the final sample size up to the next whole number.
+END_PGML
+
+Section::End();
+
+# ============================================================
+# Step 2 — Choose the correct formula
+# ============================================================
+Section::Begin("Section B — Choose the correct formula");
+
+BEGIN_PGML
+Choose the correct sample-size formula for this problem.
+
+1) [``n=\left(\dfrac{z_{\alpha/2}\sigma}{E}\right)^2``]
+
+2) [``n=\left(\dfrac{t_{\alpha/2}s}{E}\right)^2``]
+
+3) [``n=\dfrac{z_{\alpha/2}\sigma}{E}``]
+
+4) [``n=\dfrac{z_{\alpha/2}^2\hat p(1-\hat p)}{E^2}``]
+
+5) [``n=\left(\dfrac{z_{\alpha/2}E}{\sigma}\right)^2``]
+
+Enter the number (1-5): [____]
+END_PGML
+
+ANS($cmp_formula);
+
+Section::End();
+
+# ============================================================
+# Step 3 — Identify the ingredients
+# ============================================================
+Section::Begin("Section C — Identify the ingredients");
+
+BEGIN_PGML
+For this problem, identify the values you will substitute into the formula.
+
+Estimated [``\sigma``]: [____]
+
+Desired margin of error [``E``]: [____]
+
+Value of [``\alpha``]: [____]
+END_PGML
+
+ANS($cmp_sigma, $cmp_E, $cmp_alpha);
+
+Section::End();
+
+# ============================================================
+# Step 4 — Find the cutoff
+# ============================================================
+Section::Begin("Section D — Find the cutoff");
+
+BEGIN_PGML
+For a [``95\%``] confidence level, use the normal table to find [``z_{\alpha/2}``].
+
+[@ image("cutoff.jpg", width=>520) @]*
+
+Enter [``z_{\alpha/2}``]: [____]
+END_PGML
+
+ANS($cmp_z);
+
+Section::End();
+
+# ============================================================
+# Step 5 — Compute the raw sample size
+# ============================================================
+Section::Begin("Section E — Compute the raw sample size");
+
+BEGIN_PGML
+Substitute into the sample-size formula:
+
+[``n=\left(\dfrac{z_{\alpha/2}\sigma}{E}\right)^2``]
+
+For this problem,
+
+[``n=\left(\dfrac{(1.960)(4.8)}{1.5}\right)^2 =``] [____]
+END_PGML
+
+ANS($cmp_n_raw);
+
+Section::End();
+
+# ============================================================
+# Step 6 — Round up to get the required sample size
+# ============================================================
+Section::Begin("Section F — Round up");
+
+BEGIN_PGML
+The raw calculator value of [``n``] is not a whole number.
+
+To guarantee that the margin of error is at most [``1.5``] minutes, we must round up.
+
+Enter the minimum whole-number sample size required: [____]
+END_PGML
+
+ANS($cmp_n_final);
+
+Section::End();
+
+# ============================================================
+# Step 7 — Interpretation
+# ============================================================
+Section::Begin("Section G — Interpret the answer");
+
+BEGIN_PGML
+Choose the best interpretation of your answer.
+
+1) The coffee shop should sample exactly [``39.338``] customers.
+
+2) The coffee shop must sample at least [``40``] customers so that a [``95\%``] confidence interval for the true mean waiting time has a margin of error of at most [``1.5``] minutes, using the estimate [``\sigma \approx 4.8``].
+
+3) The sample size [``40``] guarantees that the true mean waiting time will be found exactly.
+
+4) The answer means that [``95\%``] of all customers will wait less than [``1.5``] minutes.
+
+5) The answer means that the population mean waiting time is [``40``] minutes.
+
+Enter the number (1-5): [____]
+END_PGML
+
+ANS($cmp_interp);
+
+Section::End();
+
+# ============================================================
+# Feedback — hidden in Library Browser
+# ============================================================
+if ($ENABLE_GP_RATING && !$inLibraryBrowser) {
+ Section::Begin("Feedback");
+
+ BEGIN_PGML
+How useful was this Guided Problem for you?
+
+(1 = not useful, 2 = slightly useful, 3 = useful, 4 = very useful, 5 = extremely useful)
+
+Note: Consider using the Email Instructor button below to suggest improvements.
+
+Rating (1-5): [____]
+END_PGML
+
+ ANS($cmp_rating);
+
+ Section::End();
+}
+
+Scaffold::End();
+
+# ============================================================
+# Solution
+# ============================================================
+BEGIN_PGML_SOLUTION
+**1. Choose the correct formula**
+
+For planning a sample size for estimating a population mean, using an estimate of [``\sigma``], the formula is
+
+[``n=\left(\dfrac{z_{\alpha/2}\sigma}{E}\right)^2``]
+
+**2. Identify the ingredients**
+
+From the problem:
+
+- estimated [``\sigma = 4.8``]
+- desired margin of error [``E = 1.5``]
+- confidence level [``95\%``], so [``\alpha = 0.05``]
+
+**3. Find the cutoff**
+
+Since [``\alpha = 0.05``], we have
+
+[``\alpha/2 = 0.025``]
+
+From the normal table,
+
+[``z_{\alpha/2} = 1.960``]
+
+**4. Compute the raw sample size**
+
+Substitute into the formula:
+
+[``n=\left(\dfrac{z_{\alpha/2}\sigma}{E}\right)^2``]
+
+So
+
+[``n=\left(\dfrac{(1.960)(4.8)}{1.5}\right)^2``]
+
+[``n=\left(\dfrac{9.408}{1.5}\right)^2``]
+
+[``n=(6.272)^2``]
+
+[``n=39.338``]
+
+**5. Round up**
+
+Since sample size must be a whole number and must guarantee the desired margin of error, we round up:
+
+[``n=40``]
+
+**6. Interpretation**
+
+The correct interpretation is:
+
+The coffee shop must sample at least [``40``] customers so that a [``95\%``] confidence interval for the true mean waiting time has a margin of error of at most [``1.5``] minutes, using the estimate [``\sigma \approx 4.8``].
+END_PGML_SOLUTION
+
+ENDDOCUMENT();
\ No newline at end of file
diff --git a/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/Estimation/ConfidenceInterval_SampleSize_GuidedProblem2.pg b/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/Estimation/ConfidenceInterval_SampleSize_GuidedProblem2.pg
new file mode 100644
index 0000000000..0d1711e3a8
--- /dev/null
+++ b/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/Estimation/ConfidenceInterval_SampleSize_GuidedProblem2.pg
@@ -0,0 +1,465 @@
+## DESCRIPTION
+## WeBWorK Guided Problems Project — Sample Size for Estimating a Proportion:
+## Estimate for p-hat Given (Guided / Scaffolded)
+## ENDDESCRIPTION
+##
+## Institution(Dawson College)
+## Author(Mehdi Moodi)
+## DBsubject(Statistics)
+## DBchapter(Confidence Intervals)
+## DBsection(Sample Size for Estimating a Proportion)
+## Level(2)
+## KEYWORDS(
+## 'sample size',
+## 'margin of error',
+## 'confidence interval',
+## 'population proportion',
+## 'sample proportion',
+## 'z critical value'
+## )
+
+DOCUMENT();
+
+loadMacros(
+ "PGstandard.pl",
+ "PGanswermacros.pl",
+ "MathObjects.pl",
+ "PGML.pl",
+ "scaffold.pl",
+ "answerHints.pl",
+ "parserRadioButtons.pl",
+ "PGcourse.pl",
+);
+
+TEXT(beginproblem());
+
+# ============================================================
+# Settings
+# ============================================================
+
+$ENABLE_GP_RATING = 1;
+
+$inLibraryBrowser = 0;
+if ((defined($envir{isLibraryBrowser}) && $envir{isLibraryBrowser})
+ || (defined($envir{isLibrary}) && $envir{isLibrary})) {
+ $inLibraryBrowser = 1;
+}
+
+Context("Numeric");
+
+# ============================================================
+# Helper: show a hint only when the student's answer is wrong
+# ============================================================
+sub HintIfWrong {
+ my $msg = shift;
+
+ return sub {
+ my $ans = shift;
+
+ if (defined($ans->{score}) && $ans->{score} < 1) {
+ $ans->{ans_message} = "" unless defined($ans->{ans_message});
+ $ans->{ans_message} .= " $msg";
+ }
+
+ return $ans;
+ };
+}
+
+# ============================================================
+# Data
+# Context: transit app usage
+# ============================================================
+
+$conf = 0.98;
+$phat_est = Real(0.38);
+$qhat_est = Real(0.62);
+$E = Real(0.05);
+$alpha = Real(0.02);
+$z_cut = Real(2.325);
+$n_raw = Real(509.26612);
+$n_final = 510;
+
+# ============================================================
+# Tolerances
+# ============================================================
+
+$tol_basic = 0.0005;
+$tol_z = 0.005;
+$tol_raw = 0.05;
+
+# ============================================================
+# Comparators
+# ============================================================
+
+# ------------------------------------------------------------
+# Formula-selection multiple choice
+# Correct answer: 2
+# ------------------------------------------------------------
+$cmp_formula = Real(2)->cmp(
+ checker => sub {
+ my ($correct, $student, $ansHash) = @_;
+
+ return 0 unless defined $student;
+
+ my $v = $student->value;
+ return 0 unless defined $v;
+ return 0 unless $v == int($v);
+ return 0 unless ($v >= 1 && $v <= 5);
+
+ return ($v == 2) ? 1 : 0;
+ }
+)->withPostFilter(
+ HintIfWrong(
+ "For planning a sample size for a proportion using an estimate for p-hat, use n = z_(alpha/2)^2 p-hat(1-p-hat) / E^2."
+ )
+);
+
+# ------------------------------------------------------------
+# Identify ingredients
+# ------------------------------------------------------------
+$cmp_phat = $phat_est->cmp(
+ tolType => "absolute",
+ tolerance => $tol_basic
+)->withPostFilter(
+ HintIfWrong("Use the estimated sample proportion from the previous survey.")
+);
+
+$cmp_E = $E->cmp(
+ tolType => "absolute",
+ tolerance => $tol_basic
+)->withPostFilter(
+ HintIfWrong("Use the desired margin of error from the problem statement.")
+);
+
+$cmp_alpha = $alpha->cmp(
+ tolType => "absolute",
+ tolerance => $tol_basic
+)->withPostFilter(
+ HintIfWrong("For a 98% confidence level, 1 - alpha = 0.98, so alpha = 0.02.")
+);
+
+# ------------------------------------------------------------
+# z cutoff
+# ------------------------------------------------------------
+$cmp_z = $z_cut->cmp(
+ tolType => "absolute",
+ tolerance => $tol_z
+)->withPostFilter(
+ HintIfWrong(
+ "For a 98% confidence level, alpha = 0.02, so alpha/2 = 0.01 and z_(alpha/2) = 2.325."
+ )
+);
+
+# ------------------------------------------------------------
+# Raw sample size
+# ------------------------------------------------------------
+$cmp_n_raw = $n_raw->cmp(
+ tolType => "absolute",
+ tolerance => $tol_raw
+)->withPostFilter(
+ HintIfWrong("Substitute into n = z_(alpha/2)^2 p-hat(1-p-hat) / E^2 and compute before rounding.")
+);
+
+# ------------------------------------------------------------
+# Final sample size
+# ------------------------------------------------------------
+$cmp_n_final = Real($n_final)->cmp(
+ checker => sub {
+ my ($correct, $student, $ansHash) = @_;
+
+ return 0 unless defined $student;
+
+ my $v = $student->value;
+ return 0 unless defined $v;
+ return 0 unless $v == int($v);
+
+ return ($v == $n_final) ? 1 : 0;
+ }
+)->withPostFilter(
+ HintIfWrong("Always round the required sample size up to the next whole number.")
+);
+
+# ------------------------------------------------------------
+# Interpretation multiple choice
+# Correct answer: 3
+# ------------------------------------------------------------
+$cmp_interp = Real(3)->cmp(
+ checker => sub {
+ my ($correct, $student, $ansHash) = @_;
+
+ return 0 unless defined $student;
+
+ my $v = $student->value;
+ return 0 unless defined $v;
+ return 0 unless $v == int($v);
+ return 0 unless ($v >= 1 && $v <= 5);
+
+ return ($v == 3) ? 1 : 0;
+ }
+)->withPostFilter(
+ HintIfWrong(
+ "The final answer is the minimum whole-number sample size needed so the margin of error is at most the target value."
+ )
+);
+
+# ------------------------------------------------------------
+# Rating checker
+# Any integer 1 to 5 is accepted
+# ------------------------------------------------------------
+$cmp_rating = Real(3)->cmp(
+ checker => sub {
+ my ($correct, $student, $ansHash) = @_;
+
+ return 0 unless defined $student;
+
+ my $v = $student->value;
+ return 0 unless defined $v;
+ return 0 unless $v == int($v);
+ return 0 unless ($v >= 1 && $v <= 5);
+
+ return 1;
+ }
+);
+
+# ============================================================
+# Scaffold setup
+# ============================================================
+
+if ($inLibraryBrowser) {
+ Scaffold::Begin(
+ numbered => 1,
+ can_open => "always",
+ is_open => "always"
+ );
+}
+else {
+ Scaffold::Begin(
+ numbered => 1,
+ can_open => "when_previous_correct",
+ is_open => "correct_or_first_incorrect"
+ );
+}
+
+# ============================================================
+# Step 1 — Problem Statement
+# ============================================================
+Section::Begin("Problem Statement");
+
+BEGIN_PGML
+A city transit department wants to estimate the true proportion of commuters who use its mobile transit app at least once per week.
+
+A previous survey suggests that about [``38\%``] of commuters use the app weekly.
+
+The department wants to estimate the true population proportion with a margin of error of at most [``0.05``] at the [``98\%``] confidence level.
+
+Find the minimum sample size [``n``] required.
+
+Round the raw calculator value of [``n``] to 3 decimals, then round the final sample size up to the next whole number.
+END_PGML
+
+Section::End();
+
+# ============================================================
+# Step 2 — Choose the correct formula
+# ============================================================
+Section::Begin("Section B — Choose the correct formula");
+
+BEGIN_PGML
+Choose the correct sample-size formula for this problem.
+
+1) [``n=\left(\dfrac{z_{\alpha/2}\sigma}{E}\right)^2``]
+
+2) [``n=\dfrac{z_{\alpha/2}^{2}\hat p(1-\hat p)}{E^{2}}``]
+
+3) [``n=\dfrac{t_{\alpha/2}^{2}\hat p(1-\hat p)}{E^{2}}``]
+
+4) [``n=\dfrac{z_{\alpha/2}\hat p(1-\hat p)}{E}``]
+
+5) [``n=\left(\dfrac{z_{\alpha/2}E}{\hat p}\right)^2``]
+
+Enter the number (1-5): [____]
+END_PGML
+
+ANS($cmp_formula);
+
+Section::End();
+
+# ============================================================
+# Step 3 — Identify the ingredients
+# ============================================================
+Section::Begin("Section C — Identify the ingredients");
+
+BEGIN_PGML
+For this problem, identify the values you will substitute into the formula.
+
+Estimated [``\hat p``]: [____]
+
+Desired margin of error [``E``]: [____]
+
+Value of [``\alpha``]: [____]
+END_PGML
+
+ANS($cmp_phat, $cmp_E, $cmp_alpha);
+
+Section::End();
+
+# ============================================================
+# Step 4 — Find the cutoff
+# ============================================================
+Section::Begin("Section D — Find the cutoff");
+
+BEGIN_PGML
+For a [``98\%``] confidence level, use the normal table to find [``z_{\alpha/2}``].
+
+[@ image("cutoff.jpg", width=>520) @]*
+
+Enter [``z_{\alpha/2}``]: [____]
+END_PGML
+
+ANS($cmp_z);
+
+Section::End();
+
+# ============================================================
+# Step 5 — Compute the raw sample size
+# ============================================================
+Section::Begin("Section E — Compute the raw sample size");
+
+BEGIN_PGML
+Substitute into the sample-size formula:
+
+[``n=\dfrac{z_{\alpha/2}^{2}\hat p(1-\hat p)}{E^{2}}``]
+
+For this problem,
+
+[``n=\dfrac{(2.325)^2(0.38)(0.62)}{(0.05)^2} =``] [____]
+END_PGML
+
+ANS($cmp_n_raw);
+
+Section::End();
+
+# ============================================================
+# Step 6 — Round up to get the required sample size
+# ============================================================
+Section::Begin("Section F — Round up");
+
+BEGIN_PGML
+The raw calculator value of [``n``] is not a whole number.
+
+To guarantee that the margin of error is at most [``0.05``], we must round up.
+
+Enter the minimum whole-number sample size required: [____]
+END_PGML
+
+ANS($cmp_n_final);
+
+Section::End();
+
+# ============================================================
+# Step 7 — Interpretation
+# ============================================================
+Section::Begin("Section G — Interpret the answer");
+
+BEGIN_PGML
+Choose the best interpretation of your answer.
+
+1) The department should survey exactly [``509.266``] commuters.
+
+2) The answer means that [``98\%``] of commuters use the app.
+
+3) The department must survey at least [``510``] commuters so that a [``98\%``] confidence interval for the true proportion of weekly app users has a margin of error of at most [``0.05``], using the estimate [``\hat p \approx 0.38``].
+
+4) The sample size [``510``] guarantees that the true population proportion will be found exactly.
+
+5) The answer means that the true proportion is [``0.510``].
+
+Enter the number (1-5): [____]
+END_PGML
+
+ANS($cmp_interp);
+
+Section::End();
+
+# ============================================================
+# Feedback — hidden in Library Browser
+# ============================================================
+if ($ENABLE_GP_RATING && !$inLibraryBrowser) {
+ Section::Begin("Feedback");
+
+ BEGIN_PGML
+How useful was this Guided Problem for you?
+
+(1 = not useful, 2 = slightly useful, 3 = useful, 4 = very useful, 5 = extremely useful)
+
+Note: Consider using the Email Instructor button below to suggest improvements.
+
+Rating (1-5): [____]
+END_PGML
+
+ ANS($cmp_rating);
+
+ Section::End();
+}
+
+Scaffold::End();
+
+# ============================================================
+# Solution
+# ============================================================
+BEGIN_PGML_SOLUTION
+**1. Choose the correct formula**
+
+For planning a sample size for estimating a population proportion, when an estimate for [``\hat p``] is available, the formula is
+
+[``n=\dfrac{z_{\alpha/2}^{2}\hat p(1-\hat p)}{E^{2}}``]
+
+**2. Identify the ingredients**
+
+From the problem:
+
+- estimated [``\hat p = 0.38``]
+- so [``1-\hat p = 0.62``]
+- desired margin of error [``E = 0.05``]
+- confidence level [``98\%``], so [``\alpha = 0.02``]
+
+**3. Find the cutoff**
+
+Since [``\alpha = 0.02``], we have
+
+[``\alpha/2 = 0.01``]
+
+From the normal table,
+
+[``z_{\alpha/2} = 2.325``]
+
+**4. Compute the raw sample size**
+
+Substitute into the formula:
+
+[``n=\dfrac{z_{\alpha/2}^{2}\hat p(1-\hat p)}{E^{2}}``]
+
+So
+
+[``n=\dfrac{(2.325)^2(0.38)(0.62)}{(0.05)^2}``]
+
+[``n=\dfrac{5.405625(0.2356)}{0.0025}``]
+
+[``n=\dfrac{1.27316525}{0.0025}``]
+
+[``n=509.266``]
+
+**5. Round up**
+
+Since sample size must be a whole number and must guarantee the desired margin of error, we round up:
+
+[``n=510``]
+
+**6. Interpretation**
+
+The correct interpretation is:
+
+The department must survey at least [``510``] commuters so that a [``98\%``] confidence interval for the true proportion of weekly app users has a margin of error of at most [``0.05``], using the estimate [``\hat p \approx 0.38``].
+END_PGML_SOLUTION
+
+ENDDOCUMENT();
\ No newline at end of file
diff --git a/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/Estimation/ConfidenceInterval_SampleSize_GuidedProblem3.pg b/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/Estimation/ConfidenceInterval_SampleSize_GuidedProblem3.pg
new file mode 100644
index 0000000000..15bbbfdf5b
--- /dev/null
+++ b/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/Estimation/ConfidenceInterval_SampleSize_GuidedProblem3.pg
@@ -0,0 +1,488 @@
+## DESCRIPTION
+## WeBWorK Guided Problems Project — Sample Size for Estimating a Proportion:
+## No Estimate for p-hat Given (Guided / Scaffolded)
+## ENDDESCRIPTION
+##
+## Institution(Dawson College)
+## Author(Mehdi Moodi)
+## DBsubject(Statistics)
+## DBchapter(Confidence Intervals)
+## DBsection(Sample Size for Estimating a Proportion)
+## Level(2)
+## KEYWORDS(
+## 'sample size',
+## 'margin of error',
+## 'confidence interval',
+## 'population proportion',
+## 'no estimate for p-hat',
+## 'z critical value'
+## )
+
+DOCUMENT();
+
+loadMacros(
+ "PGstandard.pl",
+ "PGanswermacros.pl",
+ "MathObjects.pl",
+ "PGML.pl",
+ "scaffold.pl",
+ "answerHints.pl",
+ "parserRadioButtons.pl",
+ "PGcourse.pl",
+);
+
+TEXT(beginproblem());
+
+# ============================================================
+# Settings
+# ============================================================
+
+$ENABLE_GP_RATING = 1;
+
+$inLibraryBrowser = 0;
+if ((defined($envir{isLibraryBrowser}) && $envir{isLibraryBrowser})
+ || (defined($envir{isLibrary}) && $envir{isLibrary})) {
+ $inLibraryBrowser = 1;
+}
+
+Context("Numeric");
+
+# ============================================================
+# Helper: show a hint only when the student's answer is wrong
+# ============================================================
+sub HintIfWrong {
+ my $msg = shift;
+
+ return sub {
+ my $ans = shift;
+
+ if (defined($ans->{score}) && $ans->{score} < 1) {
+ $ans->{ans_message} = "" unless defined($ans->{ans_message});
+ $ans->{ans_message} .= " $msg";
+ }
+
+ return $ans;
+ };
+}
+
+# ============================================================
+# Data
+# Context: sunscreen use among young adults
+# ============================================================
+
+$conf = 0.90;
+$phat_est = Real(0.5);
+$qhat_est = Real(0.5);
+$E = Real(0.03);
+$alpha = Real(0.10);
+$z_cut = Real(1.645);
+$n_raw = Real(751.6736111111);
+$n_final = 752;
+
+# ============================================================
+# Tolerances
+# ============================================================
+
+$tol_basic = 0.0005;
+$tol_z = 0.005;
+$tol_raw = 0.05;
+
+# ============================================================
+# Comparators
+# ============================================================
+
+# ------------------------------------------------------------
+# Concept multiple choice: why use p-hat = 0.5?
+# Correct answer: 2
+# ------------------------------------------------------------
+$cmp_why_half = Real(2)->cmp(
+ checker => sub {
+ my ($correct, $student, $ansHash) = @_;
+
+ return 0 unless defined $student;
+
+ my $v = $student->value;
+ return 0 unless defined $v;
+ return 0 unless $v == int($v);
+ return 0 unless ($v >= 1 && $v <= 5);
+
+ return ($v == 2) ? 1 : 0;
+ }
+)->withPostFilter(
+ HintIfWrong(
+ "When no estimate for p-hat is available, using 0.5 makes p-hat(1-p-hat) as large as possible, which gives the largest and safest sample size."
+ )
+);
+
+# ------------------------------------------------------------
+# Identify ingredients
+# ------------------------------------------------------------
+$cmp_phat = $phat_est->cmp(
+ tolType => "absolute",
+ tolerance => $tol_basic
+)->withPostFilter(
+ HintIfWrong("When no estimate for p-hat is available, use p-hat = 0.5 for planning.")
+);
+
+$cmp_E = $E->cmp(
+ tolType => "absolute",
+ tolerance => $tol_basic
+)->withPostFilter(
+ HintIfWrong("Three percentage points means a margin of error of 0.03.")
+);
+
+$cmp_alpha = $alpha->cmp(
+ tolType => "absolute",
+ tolerance => $tol_basic
+)->withPostFilter(
+ HintIfWrong("For a 90% confidence level, 1 - alpha = 0.90, so alpha = 0.10.")
+);
+
+# ------------------------------------------------------------
+# z cutoff
+# ------------------------------------------------------------
+$cmp_z = $z_cut->cmp(
+ tolType => "absolute",
+ tolerance => $tol_z
+)->withPostFilter(
+ HintIfWrong(
+ "For a 90% confidence level, alpha = 0.10, so alpha/2 = 0.05 and z_(alpha/2) = 1.645."
+ )
+);
+
+# ------------------------------------------------------------
+# Raw sample size
+# ------------------------------------------------------------
+$cmp_n_raw = $n_raw->cmp(
+ tolType => "absolute",
+ tolerance => $tol_raw
+)->withPostFilter(
+ HintIfWrong("Substitute into n = z_(alpha/2)^2 p-hat(1-p-hat) / E^2 using p-hat = 0.5, then compute before rounding.")
+);
+
+# ------------------------------------------------------------
+# Final sample size
+# ------------------------------------------------------------
+$cmp_n_final = Real($n_final)->cmp(
+ checker => sub {
+ my ($correct, $student, $ansHash) = @_;
+
+ return 0 unless defined $student;
+
+ my $v = $student->value;
+ return 0 unless defined $v;
+ return 0 unless $v == int($v);
+
+ return ($v == $n_final) ? 1 : 0;
+ }
+)->withPostFilter(
+ HintIfWrong("Always round the required sample size up to the next whole number.")
+);
+
+# ------------------------------------------------------------
+# Interpretation multiple choice
+# Correct answer: 4
+# ------------------------------------------------------------
+$cmp_interp = Real(4)->cmp(
+ checker => sub {
+ my ($correct, $student, $ansHash) = @_;
+
+ return 0 unless defined $student;
+
+ my $v = $student->value;
+ return 0 unless defined $v;
+ return 0 unless $v == int($v);
+ return 0 unless ($v >= 1 && $v <= 5);
+
+ return ($v == 4) ? 1 : 0;
+ }
+)->withPostFilter(
+ HintIfWrong(
+ "The final answer is the minimum whole-number sample size needed so the margin of error is at most the target value."
+ )
+);
+
+# ------------------------------------------------------------
+# Rating checker
+# Any integer 1 to 5 is accepted
+# ------------------------------------------------------------
+$cmp_rating = Real(3)->cmp(
+ checker => sub {
+ my ($correct, $student, $ansHash) = @_;
+
+ return 0 unless defined $student;
+
+ my $v = $student->value;
+ return 0 unless defined $v;
+ return 0 unless $v == int($v);
+ return 0 unless ($v >= 1 && $v <= 5);
+
+ return 1;
+ }
+);
+
+# ============================================================
+# Scaffold setup
+# ============================================================
+
+if ($inLibraryBrowser) {
+ Scaffold::Begin(
+ numbered => 1,
+ can_open => "always",
+ is_open => "always"
+ );
+}
+else {
+ Scaffold::Begin(
+ numbered => 1,
+ can_open => "when_previous_correct",
+ is_open => "correct_or_first_incorrect"
+ );
+}
+
+# ============================================================
+# Step 1 — Problem Statement
+# ============================================================
+Section::Begin("Problem Statement");
+
+BEGIN_PGML
+A dermatologist wishes to estimate the proportion of young adults who apply sunscreen regularly before going out in the sun in the summer.
+
+Find the minimum sample size required to estimate the population proportion to within [``3``] percentage points at the [``90\%``] confidence level.
+
+Assume that no previous estimate of the population proportion is available.
+
+Round the raw calculator value of [``n``] to 3 decimals, then round the final sample size up to the next whole number.
+END_PGML
+
+Section::End();
+
+# ============================================================
+# Step 2 — Formula
+# ============================================================
+Section::Begin("Section B — Formula");
+
+BEGIN_PGML
+For planning sample size for a population proportion, we use
+
+[``n=\dfrac{z_{\alpha/2}^{2}\hat p(1-\hat p)}{E^{2}}``]
+
+When no estimate for [``\hat p``] is available, we use [``\hat p=0.5``].
+END_PGML
+
+Section::End();
+
+# ============================================================
+# Step 3 — Why do we use p-hat = 0.5?
+# ============================================================
+Section::Begin("Section C — Why use p-hat = 0.5?");
+
+BEGIN_PGML
+Why do we use [``\hat p = 0.5``] when no prior estimate is available?
+
+1) Because [``0.5``] is always the true population proportion.
+
+2) Because [``\hat p(1-\hat p)``] is largest when [``\hat p=0.5``], so this gives the largest and safest required sample size.
+
+3) Because [``0.5``] makes the margin of error equal to [``0``].
+
+4) Because the confidence level is [``90\%``].
+
+5) Because [``0.5``] makes the cutoff [``z_{\alpha/2}``] equal to [``1.645``].
+
+Enter the number (1-5): [____]
+END_PGML
+
+ANS($cmp_why_half);
+
+Section::End();
+
+# ============================================================
+# Step 4 — Identify the ingredients
+# ============================================================
+Section::Begin("Section D — Identify the ingredients");
+
+BEGIN_PGML
+For this problem, identify the values you will substitute into the formula.
+
+Value of [``\hat p``]: [____]
+
+Desired margin of error [``E``]: [____]
+
+Value of [``\alpha``]: [____]
+END_PGML
+
+ANS($cmp_phat, $cmp_E, $cmp_alpha);
+
+Section::End();
+
+# ============================================================
+# Step 5 — Find the cutoff
+# ============================================================
+Section::Begin("Section E — Find the cutoff");
+
+BEGIN_PGML
+For a [``90\%``] confidence level, use the normal table to find [``z_{\alpha/2}``].
+
+[@ image("cutoff.jpg", width=>520) @]*
+
+Enter [``z_{\alpha/2}``]: [____]
+END_PGML
+
+ANS($cmp_z);
+
+Section::End();
+
+# ============================================================
+# Step 6 — Compute the raw sample size
+# ============================================================
+Section::Begin("Section F — Compute the raw sample size");
+
+BEGIN_PGML
+Substitute into the sample-size formula:
+
+[``n=\dfrac{z_{\alpha/2}^{2}\hat p(1-\hat p)}{E^{2}}``]
+
+For this problem,
+
+[``n=\dfrac{(1.645)^2(0.5)(0.5)}{(0.03)^2} =``] [____]
+END_PGML
+
+ANS($cmp_n_raw);
+
+Section::End();
+
+# ============================================================
+# Step 7 — Round up to get the required sample size
+# ============================================================
+Section::Begin("Section G — Round up");
+
+BEGIN_PGML
+The raw calculator value of [``n``] is not a whole number.
+
+To guarantee that the margin of error is at most [``0.03``], we must round up.
+
+Enter the minimum whole-number sample size required: [____]
+END_PGML
+
+ANS($cmp_n_final);
+
+Section::End();
+
+# ============================================================
+# Step 8 — Interpretation
+# ============================================================
+Section::Begin("Section H — Interpret the answer");
+
+BEGIN_PGML
+Choose the best interpretation of your answer.
+
+1) The dermatologist should survey exactly [``751.674``] young adults.
+
+2) The answer means that [``90\%``] of young adults use sunscreen regularly.
+
+3) The sample size [``752``] guarantees that the true population proportion will be found exactly.
+
+4) The dermatologist must survey at least [``752``] young adults so that a [``90\%``] confidence interval for the true proportion has a margin of error of at most [``0.03``], using [``\hat p = 0.5``] for planning.
+
+5) The answer means that the true population proportion is [``0.752``].
+
+Enter the number (1-5): [____]
+END_PGML
+
+ANS($cmp_interp);
+
+Section::End();
+
+# ============================================================
+# Feedback — hidden in Library Browser
+# ============================================================
+if ($ENABLE_GP_RATING && !$inLibraryBrowser) {
+ Section::Begin("Feedback");
+
+ BEGIN_PGML
+How useful was this Guided Problem for you?
+
+(1 = not useful, 2 = slightly useful, 3 = useful, 4 = very useful, 5 = extremely useful)
+
+Note: Consider using the Email Instructor button below to suggest improvements.
+
+Rating (1-5): [____]
+END_PGML
+
+ ANS($cmp_rating);
+
+ Section::End();
+}
+
+Scaffold::End();
+
+# ============================================================
+# Solution
+# ============================================================
+BEGIN_PGML_SOLUTION
+**1. Use the formula**
+
+For planning a sample size for estimating a population proportion, we use
+
+[``n=\dfrac{z_{\alpha/2}^{2}\hat p(1-\hat p)}{E^{2}}``]
+
+Since no estimate for [``\hat p``] is available, we use
+
+[``\hat p=0.5``]
+
+**2. Why do we use [``0.5``]?**
+
+We use [``0.5``] because [``\hat p(1-\hat p)``] is largest when [``\hat p=0.5``].
+
+That gives the largest and most conservative required sample size.
+
+**3. Identify the ingredients**
+
+From the problem:
+
+- [``\hat p = 0.5``]
+- [``1-\hat p = 0.5``]
+- desired margin of error [``E = 0.03``]
+- confidence level [``90\%``], so [``\alpha = 0.10``]
+
+**4. Find the cutoff**
+
+Since [``\alpha = 0.10``], we have
+
+[``\alpha/2 = 0.05``]
+
+From the normal table,
+
+[``z_{\alpha/2} = 1.645``]
+
+**5. Compute the raw sample size**
+
+Substitute into the formula:
+
+[``n=\dfrac{z_{\alpha/2}^{2}\hat p(1-\hat p)}{E^{2}}``]
+
+So
+
+[``n=\dfrac{(1.645)^2(0.5)(0.5)}{(0.03)^2}``]
+
+[``n=\dfrac{2.706025(0.25)}{0.0009}``]
+
+[``n=\dfrac{0.67650625}{0.0009}``]
+
+[``n=751.674``]
+
+**6. Round up**
+
+Since sample size must be a whole number and must guarantee the desired margin of error, we round up:
+
+[``n=752``]
+
+**7. Interpretation**
+
+The correct interpretation is:
+
+The dermatologist must survey at least [``752``] young adults so that a [``90\%``] confidence interval for the true proportion has a margin of error of at most [``0.03``], using [``\hat p = 0.5``] for planning.
+END_PGML_SOLUTION
+
+ENDDOCUMENT();
\ No newline at end of file
diff --git a/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/Estimation/ConfidenceInterval_SmallSample_GuidedProblem1.pg b/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/Estimation/ConfidenceInterval_SmallSample_GuidedProblem1.pg
new file mode 100644
index 0000000000..742559a657
--- /dev/null
+++ b/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/Estimation/ConfidenceInterval_SmallSample_GuidedProblem1.pg
@@ -0,0 +1,529 @@
+## DESCRIPTION
+## WeBWorK Guided Problems Project — Small-Sample Confidence Intervals for the Mean (Guided / Scaffolded)
+## ENDDESCRIPTION
+##
+## Institution(Dawson College)
+## Author(Mehdi Moodi)
+## DBsubject(Statistics)
+## DBchapter(Confidence Intervals)
+## DBsection(Confidence Interval for the Mean)
+## Level(2)
+## KEYWORDS('confidence interval','small sample','mean','known sigma','unknown sigma','normal population','t distribution','degrees of freedom','confidence level','critical value')
+
+DOCUMENT();
+
+loadMacros(
+ "PGstandard.pl",
+ "PGanswermacros.pl",
+ "MathObjects.pl",
+ "PGML.pl",
+ "scaffold.pl",
+ "answerHints.pl",
+ "parserRadioButtons.pl",
+ "PGcourse.pl",
+);
+
+TEXT(beginproblem());
+
+# ----------------------------------------------------------------
+# PILOT: usefulness rating toggle (ACTIVE)
+# ----------------------------------------------------------------
+$ENABLE_GP_RATING = 1;
+
+# ----------------------------------------------------------------
+# Library Browser detection (LB stability)
+# ----------------------------------------------------------------
+$inLibraryBrowser = ($envir{isLibraryBrowser} || $envir{isLibrary} || 0);
+
+Context("Numeric");
+
+# =======================================================
+# Postfilter: add a hint ONLY when the answer is wrong
+# =======================================================
+sub HintIfWrong {
+ my $msg = shift;
+ return sub {
+ my $ans = shift;
+ if (defined($ans->{score}) && $ans->{score} < 1) {
+ $ans->{ans_message} = "" unless defined($ans->{ans_message});
+ $ans->{ans_message} .= " $msg";
+ }
+ return $ans;
+ };
+}
+
+# =======================================================
+# Data for sample standard deviation checkpoint
+# =======================================================
+$x1 = 6;
+$x2 = 9;
+$x3 = 11;
+$x4 = 13;
+$x5 = 14;
+$x6 = 18;
+$x7 = 20;
+
+$s_exact = 4.89897948556636;
+
+# =======================================================
+# Correct targets
+# =======================================================
+$alpha_98 = Real(0.02);
+
+$df_10 = Real(9);
+
+# z critical values
+$z_010 = Real(1.645);
+$z_005 = Real(1.960);
+$z_002 = Real(2.325);
+$z_001 = Real(2.575);
+
+# t critical values for df = 9
+$t_010 = Real(1.833);
+$t_005 = Real(2.262);
+$t_002 = Real(2.821);
+$t_001 = Real(3.250);
+
+# =======================================================
+# Tolerances
+# =======================================================
+$tol_alpha = 0.0005;
+$tol_s = 0.02;
+$tol_z = 0.01;
+$tol_t = 0.02;
+
+# =======================================================
+# Comparators
+# =======================================================
+
+# Section B MC: correct choice = 2 (xbar)
+$cmp_point_est = Real(2)->cmp(
+ checker => sub {
+ my ($correct, $student, $ansHash) = @_;
+ return 0 unless defined $student;
+ my $v = $student->value;
+ return 0 unless defined $v;
+ return 0 unless $v == int($v);
+ return 0 unless ($v >= 1 && $v <= 5);
+ return ($v == 2) ? 1 : 0;
+ }
+)->withPostFilter(
+ HintIfWrong("The appropriate point estimate for mu is the sample mean xbar.")
+);
+
+$cmp_alpha_98 = $alpha_98->cmp(
+ tolType => "absolute",
+ tolerance => $tol_alpha
+)->withPostFilter(
+ HintIfWrong("For a 98% confidence level, 1 - alpha = 0.98, so alpha = 0.02.")
+);
+
+# Section C MC: correct choice = 3
+$cmp_conf_meaning = Real(3)->cmp(
+ checker => sub {
+ my ($correct, $student, $ansHash) = @_;
+ return 0 unless defined $student;
+ my $v = $student->value;
+ return 0 unless defined $v;
+ return 0 unless $v == int($v);
+ return 0 unless ($v >= 1 && $v <= 5);
+ return ($v == 3) ? 1 : 0;
+ }
+)->withPostFilter(
+ HintIfWrong("The confidence level describes the long-run success rate of the method over many random samples.")
+);
+
+$cmp_s = Real($s_exact)->cmp(
+ tolType => "absolute",
+ tolerance => $tol_s
+)->withPostFilter(
+ HintIfWrong("Use the sample standard deviation on your calculator, not the population standard deviation.")
+);
+
+$cmp_df_10 = $df_10->cmp(
+ tolType => "absolute",
+ tolerance => 0
+)->withPostFilter(
+ HintIfWrong("For the t distribution, the degrees of freedom are n - 1.")
+);
+
+$cmp_z_010 = $z_010->cmp(
+ tolType => "absolute",
+ tolerance => $tol_z
+)->withPostFilter(
+ HintIfWrong("For alpha = 0.10, alpha/2 = 0.05, so the cutoff is z_(0.05) = 1.645.")
+);
+
+$cmp_z_005 = $z_005->cmp(
+ tolType => "absolute",
+ tolerance => $tol_z
+)->withPostFilter(
+ HintIfWrong("For alpha = 0.05, alpha/2 = 0.025, so the cutoff is z_(0.025) = 1.960.")
+);
+
+$cmp_z_002 = $z_002->cmp(
+ tolType => "absolute",
+ tolerance => $tol_z
+)->withPostFilter(
+ HintIfWrong("For alpha = 0.02, alpha/2 = 0.01, so the cutoff is z_(0.01) = 2.326.")
+);
+
+$cmp_z_001 = $z_001->cmp(
+ tolType => "absolute",
+ tolerance => $tol_z
+)->withPostFilter(
+ HintIfWrong("For alpha = 0.01, alpha/2 = 0.005, so the cutoff is z_(0.005) = 2.576.")
+);
+
+$cmp_t_010 = $t_010->cmp(
+ tolType => "absolute",
+ tolerance => $tol_t
+)->withPostFilter(
+ HintIfWrong("For df = 9 and alpha = 0.10, use t_(0.05,9) = 1.833.")
+);
+
+$cmp_t_005 = $t_005->cmp(
+ tolType => "absolute",
+ tolerance => $tol_t
+)->withPostFilter(
+ HintIfWrong("For df = 9 and alpha = 0.05, use t_(0.025,9) = 2.262.")
+);
+
+$cmp_t_002 = $t_002->cmp(
+ tolType => "absolute",
+ tolerance => $tol_t
+)->withPostFilter(
+ HintIfWrong("For df = 9 and alpha = 0.02, use t_(0.01,9) = 2.821.")
+);
+
+$cmp_t_001 = $t_001->cmp(
+ tolType => "absolute",
+ tolerance => $tol_t
+)->withPostFilter(
+ HintIfWrong("For df = 9 and alpha = 0.01, use t_(0.005,9) = 3.250.")
+);
+
+# Rating checker
+$cmp_rating = Real(3)->cmp(
+ checker => sub {
+ my ($correct, $student, $ansHash) = @_;
+ return 0 unless defined $student;
+ my $v = $student->value;
+ return 0 unless defined $v;
+ return 0 unless $v == int($v);
+ return 0 unless ($v >= 1 && $v <= 5);
+ return 1;
+ }
+);
+
+# =======================================================
+# Scaffold setup
+# =======================================================
+if ($inLibraryBrowser) {
+ Scaffold::Begin(
+ numbered => 1,
+ can_open => "always",
+ is_open => "always"
+ );
+} else {
+ Scaffold::Begin(
+ numbered => 1,
+ can_open => "when_previous_correct",
+ is_open => "correct_or_first_incorrect"
+ );
+}
+
+# -------------------------------------------------------
+# Step 1 — Problem Statement
+# -------------------------------------------------------
+Section::Begin("Problem Statement");
+
+BEGIN_PGML
+In this Guided Problem, we introduce the idea of a small-sample confidence interval for the population mean [``\mu``].
+
+Assume a random sample of size [``n < 30``] is taken from a population.
+
+For the small-sample methods introduced here to be valid, the population must be normally distributed.
+
+In this GP, you will learn:
+
+- what a [``100(1-\alpha)\%``] confidence interval means,
+- what the confidence level represents,
+- the two small-sample formulas for a confidence interval for [``\mu``],
+- why normality of the population matters for small samples,
+- what degrees of freedom mean,
+- and how to find the correct cutoff using either the normal table or the t table.
+
+Rounding: Round cutoff values to 3 decimals.
+END_PGML
+
+Section::End();
+
+# -------------------------------------------------------
+# Step 2 — Section B
+# -------------------------------------------------------
+Section::Begin("Section B — Appropriate point estimate for mu");
+
+BEGIN_PGML
+A confidence interval for [``\mu``] has the form
+
+[``\text{estimate} \pm \text{margin of error}``]
+
+What is the appropriate point estimate for the population mean [``\mu``]?
+
+1) [``\mu``]
+2) [``\bar x``]
+3) [``\sigma``]
+4) [``s``]
+5) [``\alpha``]
+
+Enter the number (1-5): [____]
+END_PGML
+
+ANS($cmp_point_est);
+
+Section::End();
+
+# -------------------------------------------------------
+# Step 3 — Section C
+# -------------------------------------------------------
+Section::Begin("Section C — Level of confidence");
+
+BEGIN_PGML
+When we build a confidence interval, we must consider the required level of confidence.
+
+Common values are usually [``90\%``] ([``\alpha = 0.10``]), [``95\%``] ([``\alpha = 0.05``]), and [``99\%``] ([``\alpha = 0.01``]).
+
+The confidence level describes how the method behaves over many random samples. For example, a [``95\%``] level of confidence means [``95\%``] of intervals built using this method, by repeatedly sampling the population, will capture the actual [``\mu``].
+
+1) If the confidence level is [``98\%``], enter the value of [``\alpha``]:
+
+[____]
+
+2) Choose the best interpretation of a [``100(1-\alpha)\%``] confidence interval:
+
+1) [``100(1-\alpha)\%``] of all sample means are equal to [``\mu``].
+2) There is a [``100(1-\alpha)\%``] chance that [``\mu``] changes from sample to sample.
+3) If we repeatedly took many random samples and built intervals the same way each time, about [``100(1-\alpha)\%``] of those intervals would contain [``\mu``].
+4) A [``100(1-\alpha)\%``] confidence interval always contains [``\mu``].
+5) The confidence level tells us that this particular sample must be correct.
+
+Enter the number (1-5): [____]
+END_PGML
+
+ANS($cmp_alpha_98, $cmp_conf_meaning);
+
+Section::End();
+
+# -------------------------------------------------------
+# Step 4 — Section D
+# -------------------------------------------------------
+Section::Begin("Section D — Small-sample formulas and sample standard deviation");
+
+BEGIN_PGML
+For small samples, the Central Limit Theorem does not apply. If the population is normally distributed, [``\bar X``] follows the normal distribution for known [``\sigma``] and the t-student distribution for unknown [``\sigma``].
+
+There are two common small-sample formulas for a confidence interval for [``\mu``]:
+
+**Case 1: sigma known**
+If the population is normal and [``\sigma``] is known, use
+
+[``\bar x \pm z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}``]
+
+So in this case, the cutoff comes from the **normal table**.
+
+**Case 2: sigma unknown**
+If the population is normal and [``\sigma``] is unknown, use
+
+[``\bar x \pm t_{\alpha/2}\dfrac{s}{\sqrt{n}}``]
+
+So in this case, the cutoff comes from the **t table**.
+
+For the t distribution, we also need the degrees of freedom:
+
+[``df = n - 1``]
+
+Use your calculator to find the **sample standard deviation** of the following data:
+
+[``6,\ 9,\ 11,\ 13,\ 14,\ 18,\ 20``]
+
+Enter [``s``]: [____]
+END_PGML
+
+ANS($cmp_s);
+
+Section::End();
+
+# -------------------------------------------------------
+# Step 5 — Sections E
+# -------------------------------------------------------
+Section::Begin("Sections E — Degrees of freedom and cutoff values");
+
+BEGIN_PGML
+For a [``100(1-\alpha)\%``] confidence interval:
+
+- the middle area is [``1-\alpha``]
+- the total area in the two tails is [``\alpha``]
+- each tail has area [``\alpha/2``]
+
+For **small samples**, remember:
+
+- if [``\sigma``] is known, use the **normal table** and the cutoff [``z_{\alpha/2}``]
+- if [``\sigma``] is unknown, use the **t table** and the cutoff [``t_{\alpha/2}``]
+- for the t distribution, [``df=n-1``]
+
+In this section, assume sample size [``n=10``].
+
+[@ image("cutoff_t.jpg", width=>520) @]*
+
+1) If [``n=10``], enter the degrees of freedom:
+
+[____]
+
+2) Assume [``\sigma``] is known. Use the **normal table** to find the positive cutoff [``z_{\alpha/2}``] for each value of [``\alpha``]:
+
+For [``\alpha = 0.10``], enter [``z_{\alpha/2}``]: [____]
+
+For [``\alpha = 0.05``], enter [``z_{\alpha/2}``]: [____]
+
+For [``\alpha = 0.02``], enter [``z_{\alpha/2}``]: [____]
+
+For [``\alpha = 0.01``], enter [``z_{\alpha/2}``]: [____]
+
+3) Assume [``\sigma``] is unknown. Use the **t table** with your degrees of freedom to find the positive cutoff [``t_{\alpha/2}``] for each value of [``\alpha``]:
+
+For [``\alpha = 0.10``], enter [``t_{\alpha/2}``]: [____]
+
+For [``\alpha = 0.05``], enter [``t_{\alpha/2}``]: [____]
+
+For [``\alpha = 0.02``], enter [``t_{\alpha/2}``]: [____]
+
+For [``\alpha = 0.01``], enter [``t_{\alpha/2}``]: [____]
+
+Note that smaller [``\alpha``] means higher confidence level, which is desirable. However, smaller [``\alpha``] also means a bigger cutoff value that will result in a bigger margin of error for the confidence interval. This is a trade-off. Higher confidence level comes at the expense of a wider, and thus less precise, confidence interval.
+END_PGML
+
+ANS(
+ $cmp_df_10,
+ $cmp_z_010, $cmp_z_005, $cmp_z_002, $cmp_z_001,
+ $cmp_t_010, $cmp_t_005, $cmp_t_002, $cmp_t_001
+);
+
+Section::End();
+
+# -------------------------------------------------------
+# Feedback (rating) — hidden in Library Browser
+# -------------------------------------------------------
+if ($ENABLE_GP_RATING && !$inLibraryBrowser) {
+ Section::Begin("Feedback");
+
+ BEGIN_PGML
+How useful was this Guided Problem for you?
+
+(1 = not useful, 2 = slightly useful, 3 = useful, 4 = very useful, 5 = extremely useful)
+
+Note: Consider using the Email Instructor button below to suggest improvements.
+
+Rating (1-5): [____]
+END_PGML
+
+ ANS($cmp_rating);
+
+ Section::End();
+}
+
+Scaffold::End();
+
+BEGIN_PGML_SOLUTION
+**1. Appropriate point estimate**
+
+A confidence interval for [``\mu``] is centered at the sample mean.
+
+So the appropriate point estimate for the population mean [``\mu``] is
+
+[``\bar x``]
+
+**2. Confidence level and alpha**
+
+If the confidence level is [``98\%``], then
+
+[``1-\alpha = 0.98``]
+
+so
+
+[``\alpha = 0.02``]
+
+The confidence level describes the long-run success rate of the method. For example, a [``95\%``] confidence level means that if we repeatedly sample the population and build intervals the same way each time, about [``95\%``] of those intervals will capture the true [``\mu``].
+
+**3. Small-sample formulas**
+
+For small samples, the population should be normally distributed.
+
+If [``\sigma``] is known, use
+
+[``\bar x \pm z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}``]
+
+So the cutoff comes from the normal table.
+
+If [``\sigma``] is unknown, use
+
+[``\bar x \pm t_{\alpha/2}\dfrac{s}{\sqrt{n}}``]
+
+So the cutoff comes from the t table.
+
+This second formula uses the t distribution because [``\sigma``] is unknown and we replace it with [``s``].
+
+**4. Degrees of freedom**
+
+For the t distribution,
+
+[``df = n - 1``]
+
+So if [``n=10``], then
+
+[``df = 10 - 1 = 9``]
+
+**5. Compute the sample standard deviation**
+
+For the data
+
+[``6,\ 9,\ 11,\ 13,\ 14,\ 18,\ 20``]
+
+the sample mean is
+
+[``\bar x = 13``]
+
+So the sample variance is
+
+[``s^2 = \dfrac{144}{7-1} = 24``]
+
+and the sample standard deviation is
+
+[``s = \sqrt{24} \approx 4.899``]
+
+**6. Cutoff values when sigma is known**
+
+If [``\sigma``] is known, we use the normal table.
+
+The common positive cutoffs are:
+
+- for [``\alpha = 0.10``]: [``z_{0.05} = 1.645``]
+- for [``\alpha = 0.05``]: [``z_{0.025} = 1.960``]
+- for [``\alpha = 0.02``]: [``z_{0.01} = 2.325``]
+- for [``\alpha = 0.01``]: [``z_{0.005} = 2.575``]
+
+**7. Cutoff values when sigma is unknown**
+
+If [``\sigma``] is unknown and [``n=10``], then [``df=9``] and we use the t table.
+
+The common positive cutoffs are:
+
+- for [``\alpha = 0.10``]: [``t_{0.05,9} = 1.833``]
+- for [``\alpha = 0.05``]: [``t_{0.025,9} = 2.262``]
+- for [``\alpha = 0.02``]: [``t_{0.01,9} = 2.821``]
+- for [``\alpha = 0.01``]: [``t_{0.005,9} = 3.250``]
+
+These t cutoffs are larger than the corresponding z cutoffs because small samples with unknown [``\sigma``] involve more uncertainty.
+END_PGML_SOLUTION
+
+ENDDOCUMENT();
\ No newline at end of file
diff --git a/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/Estimation/ConfidenceInterval_SmallSample_GuidedProblem2.pg b/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/Estimation/ConfidenceInterval_SmallSample_GuidedProblem2.pg
new file mode 100644
index 0000000000..8465ea99bb
--- /dev/null
+++ b/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/Estimation/ConfidenceInterval_SmallSample_GuidedProblem2.pg
@@ -0,0 +1,426 @@
+## DESCRIPTION
+## WeBWorK Guided Problems Project — Confidence Interval for the Mean: Small Sample, Sigma Known (Guided / Scaffolded)
+## ENDDESCRIPTION
+##
+## Institution(Dawson College)
+## Author(Mehdi Moodi)
+## DBsubject(Statistics)
+## DBchapter(Confidence Intervals)
+## DBsection(Confidence Interval for the Mean)
+## Level(2)
+## KEYWORDS('confidence interval','population mean','small sample','sigma known','normal population','z critical value','standard error','margin of error')
+
+DOCUMENT();
+
+loadMacros(
+ "PGstandard.pl",
+ "PGanswermacros.pl",
+ "MathObjects.pl",
+ "PGML.pl",
+ "scaffold.pl",
+ "answerHints.pl",
+ "parserRadioButtons.pl",
+ "PGcourse.pl",
+);
+
+TEXT(beginproblem());
+
+# ----------------------------------------------------------------
+# PILOT: usefulness rating toggle
+# ----------------------------------------------------------------
+$ENABLE_GP_RATING = 1;
+
+# ----------------------------------------------------------------
+# Library Browser detection (LB stability)
+# ----------------------------------------------------------------
+$inLibraryBrowser = ($envir{isLibraryBrowser} || $envir{isLibrary} || 0);
+
+Context("Numeric");
+
+# =======================================================
+# Postfilter: add a hint ONLY when the answer is wrong
+# =======================================================
+sub HintIfWrong {
+ my $msg = shift;
+ return sub {
+ my $ans = shift;
+ if (defined($ans->{score}) && $ans->{score} < 1) {
+ $ans->{ans_message} = "" unless defined($ans->{ans_message});
+ $ans->{ans_message} .= " $msg";
+ }
+ return $ans;
+ };
+}
+
+# =======================================================
+# Data
+# =======================================================
+$xbar = 38;
+$sigma = 6;
+$n = 16;
+$conf = 0.90;
+
+$alpha = Real(0.10);
+$z_cutoff = Real(1.645);
+$se = Real(1.5);
+$me = Real(2.4675);
+$lcl = Real(35.5325);
+$ucl = Real(40.4675);
+
+# =======================================================
+# Tolerances
+# =======================================================
+$tol_alpha = 0.0005;
+$tol_z = 0.005;
+$tol_se = 0.005;
+$tol_me = 0.01;
+$tol_end = 0.02;
+
+# =======================================================
+# Comparators
+# =======================================================
+
+# Formula-selection MC: correct choice = 4
+$cmp_formula = Real(4)->cmp(
+ checker => sub {
+ my ($correct, $student, $ansHash) = @_;
+ return 0 unless defined $student;
+ my $v = $student->value;
+ return 0 unless defined $v;
+ return 0 unless $v == int($v);
+ return 0 unless ($v >= 1 && $v <= 5);
+ return ($v == 4) ? 1 : 0;
+ }
+)->withPostFilter(
+ HintIfWrong("Because the population is normal and sigma is known, use xbar +/- z_(alpha/2)(sigma/sqrt(n)).")
+);
+
+$cmp_alpha = $alpha->cmp(
+ tolType => "absolute",
+ tolerance => $tol_alpha
+)->withPostFilter(
+ HintIfWrong("For a 90% confidence level, 1 - alpha = 0.90, so alpha = 0.10.")
+);
+
+$cmp_z = $z_cutoff->cmp(
+ tolType => "absolute",
+ tolerance => $tol_z
+)->withPostFilter(
+ HintIfWrong("For a 90% confidence interval, alpha = 0.10, so alpha/2 = 0.05 and the table cutoff is 1.645.")
+);
+
+$cmp_se = $se->cmp(
+ tolType => "absolute",
+ tolerance => $tol_se
+)->withPostFilter(
+ HintIfWrong("The standard error here is sigma/sqrt(n).")
+);
+
+$cmp_me = $me->cmp(
+ tolType => "absolute",
+ tolerance => $tol_me
+)->withPostFilter(
+ HintIfWrong("The margin of error is z_(alpha/2)(sigma/sqrt(n)).")
+);
+
+$cmp_lcl = $lcl->cmp(
+ tolType => "absolute",
+ tolerance => $tol_end
+)->withPostFilter(
+ HintIfWrong("The lower endpoint is xbar minus the margin of error.")
+);
+
+$cmp_ucl = $ucl->cmp(
+ tolType => "absolute",
+ tolerance => $tol_end
+)->withPostFilter(
+ HintIfWrong("The upper endpoint is xbar plus the margin of error.")
+);
+
+# Interpretation MC: correct choice = 2
+$cmp_interp = Real(2)->cmp(
+ checker => sub {
+ my ($correct, $student, $ansHash) = @_;
+ return 0 unless defined $student;
+ my $v = $student->value;
+ return 0 unless defined $v;
+ return 0 unless $v == int($v);
+ return 0 unless ($v >= 1 && $v <= 5);
+ return ($v == 2) ? 1 : 0;
+ }
+)->withPostFilter(
+ HintIfWrong("A confidence interval for mu is interpreted as a plausible range for the true population mean.")
+);
+
+# Rating checker
+$cmp_rating = Real(3)->cmp(
+ checker => sub {
+ my ($correct, $student, $ansHash) = @_;
+ return 0 unless defined $student;
+ my $v = $student->value;
+ return 0 unless defined $v;
+ return 0 unless $v == int($v);
+ return 0 unless ($v >= 1 && $v <= 5);
+ return 1;
+ }
+);
+
+# =======================================================
+# Scaffold setup
+# =======================================================
+if ($inLibraryBrowser) {
+ Scaffold::Begin(
+ numbered => 1,
+ can_open => "always",
+ is_open => "always"
+ );
+} else {
+ Scaffold::Begin(
+ numbered => 1,
+ can_open => "when_previous_correct",
+ is_open => "correct_or_first_incorrect"
+ );
+}
+
+# -------------------------------------------------------
+# Step 1 — Problem Statement
+# -------------------------------------------------------
+Section::Begin("Problem Statement");
+
+BEGIN_PGML
+A university parking office wants to estimate the true mean morning commute time for students who drive to campus.
+
+A random sample of [``n=16``] students is selected. The sample mean commute time is [``\bar x = 38``] minutes.
+The population standard deviation is known to be [``\sigma = 6``] minutes.
+
+Assume the population of commute times is normally distributed.
+
+Construct a [``90\%``] confidence interval for the true population mean [``\mu``].
+
+Round the cutoff, standard error, margin of error, and interval endpoints to 3 decimals.
+END_PGML
+
+Section::End();
+
+# -------------------------------------------------------
+# Step 2 — Choose the correct formula
+# -------------------------------------------------------
+Section::Begin("Section B — Choose the correct formula");
+
+BEGIN_PGML
+Choose the correct confidence-interval formula for this problem, considering that the sample is small, the population is normal, and the population standard deviation [``\sigma``] is known.
+
+1) [``\bar x \pm t_{\alpha/2}\dfrac{s}{\sqrt{n}}``]
+
+2) [``\hat p \pm z_{\alpha/2}\sqrt{\dfrac{\hat p(1-\hat p)}{n}}``]
+
+3) [``\bar x \pm z_{\alpha/2}\dfrac{s}{\sqrt{n}}``]
+
+4) [``\bar x \pm z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}``]
+
+5) [``\bar x \pm t_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}``]
+
+Enter the number (1-5): [____]
+END_PGML
+
+ANS($cmp_formula);
+
+Section::End();
+
+# -------------------------------------------------------
+# Step 3 — Find alpha and the cutoff
+# -------------------------------------------------------
+Section::Begin("Section C — Find alpha and the cutoff");
+
+BEGIN_PGML
+For a [``90\%``] confidence interval:
+
+- [``1-\alpha = 0.90``]
+- each tail has area [``\alpha/2``]
+
+1) Enter the value of [``\alpha``]:
+
+[____]
+
+[@ image("cutoff.jpg", width=>520) @]*
+
+2) Use the normal table to find the positive cutoff [``z_{\alpha/2}``].
+
+Enter [``z_{\alpha/2}``]: [____]
+END_PGML
+
+ANS($cmp_alpha, $cmp_z);
+
+Section::End();
+
+# -------------------------------------------------------
+# Step 4 — Compute the standard error and margin of error
+# -------------------------------------------------------
+Section::Begin("Section D — Compute the standard error and margin of error");
+
+BEGIN_PGML
+Compute the standard error:
+
+[``\dfrac{\sigma}{\sqrt{n}}``]
+
+For this problem,
+
+[``\dfrac{6}{\sqrt{16}} =``] [____]
+
+Now compute the margin of error:
+
+[``E = z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}``]
+
+For this problem,
+
+[``E = (1.645)\left(\dfrac{6}{\sqrt{16}}\right) =``] [____]
+END_PGML
+
+ANS($cmp_se, $cmp_me);
+
+Section::End();
+
+# -------------------------------------------------------
+# Step 5 — Construct the confidence interval
+# -------------------------------------------------------
+Section::Begin("Section E — Construct the confidence interval");
+
+BEGIN_PGML
+Now construct the confidence interval:
+
+[``\bar x \pm E``]
+
+Using [``\bar x = 38``] and your margin of error, enter the two endpoints.
+
+Lower endpoint: [____]
+
+Upper endpoint: [____]
+END_PGML
+
+ANS($cmp_lcl, $cmp_ucl);
+
+Section::End();
+
+# -------------------------------------------------------
+# Step 6 — Interpretation
+# -------------------------------------------------------
+Section::Begin("Section F — Interpret the confidence interval");
+
+BEGIN_PGML
+The [``90\%``] confidence interval is
+
+[``(35.533,\ 40.468)``]
+
+Choose the best interpretation.
+
+1) [``90\%``] of all commuting students take between [``35.533``] and [``40.468``] minutes to get to campus.
+
+2) We are [``90\%``] confident that the true mean commute time for all students who drive to campus is between [``35.533``] and [``40.468``] minutes.
+
+3) There is a [``90\%``] chance that the sample mean [``\bar x``] is between [``35.533``] and [``40.468``].
+
+4) The interval proves that the true population mean is exactly [``38``] minutes.
+
+5) [``90\%``] of all possible sample means are exactly equal to [``\mu``].
+
+Enter the number (1-5): [____]
+END_PGML
+
+ANS($cmp_interp);
+
+Section::End();
+
+# -------------------------------------------------------
+# Feedback (rating) — hidden in Library Browser
+# -------------------------------------------------------
+if ($ENABLE_GP_RATING && !$inLibraryBrowser) {
+ Section::Begin("Feedback");
+
+ BEGIN_PGML
+How useful was this Guided Problem for you?
+
+(1 = not useful, 2 = slightly useful, 3 = useful, 4 = very useful, 5 = extremely useful)
+
+Note: Consider using the Email Instructor button below to suggest improvements.
+
+Rating (1-5): [____]
+END_PGML
+
+ ANS($cmp_rating);
+
+ Section::End();
+}
+
+Scaffold::End();
+
+BEGIN_PGML_SOLUTION
+**1. Choose the correct formula**
+
+Because the population is normal and the population standard deviation [``\sigma``] is known, the correct formula is
+
+[``\bar x \pm z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}``]
+
+**2. Find alpha and the cutoff**
+
+For a [``90\%``] confidence interval,
+
+[``1-\alpha = 0.90``]
+
+so
+
+[``\alpha = 0.10``]
+
+Then
+
+[``\alpha/2 = 0.05``]
+
+From the normal table,
+
+[``z_{\alpha/2} = z_{0.05} = 1.645``]
+
+**3. Compute the standard error**
+
+[``\dfrac{\sigma}{\sqrt{n}} = \dfrac{6}{\sqrt{16}} = \dfrac{6}{4} = 1.500``]
+
+So the standard error is [``1.500``].
+
+**4. Compute the margin of error**
+
+[``E = z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}``]
+
+So
+
+[``E = (1.645)(1.500) = 2.468``]
+
+**5. Construct the confidence interval**
+
+Use
+
+[``\bar x \pm E``]
+
+So
+
+[``38 \pm 2.468``]
+
+Lower endpoint:
+
+[``38 - 2.468 = 35.533``]
+
+Upper endpoint:
+
+[``38 + 2.468 = 40.468``]
+
+Therefore, the [``90\%``] confidence interval for [``\mu``] is
+
+[``(35.533,\ 40.468)``]
+
+**6. Interpret the confidence interval**
+
+The correct interpretation is:
+
+We are [``90\%``] confident that the true mean commute time for all students who drive to campus is between [``35.533``] and [``40.468``] minutes.
+
+This interval gives a plausible range for the population mean [``\mu``]. It does **not** describe where [``90\%``] of individual commute times fall, and it does **not** mean that the sample mean has a [``90\%``] chance of being in that interval.
+END_PGML_SOLUTION
+
+ENDDOCUMENT();
diff --git a/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/Estimation/ConfidenceInterval_SmallSample_GuidedProblem3.pg b/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/Estimation/ConfidenceInterval_SmallSample_GuidedProblem3.pg
new file mode 100644
index 0000000000..754ad499bf
--- /dev/null
+++ b/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/Estimation/ConfidenceInterval_SmallSample_GuidedProblem3.pg
@@ -0,0 +1,432 @@
+## DESCRIPTION
+## WeBWorK Guided Problems Project — Confidence Interval for the Mean: Small Sample, Sigma Unknown (Guided / Scaffolded)
+## ENDDESCRIPTION
+##
+## Institution(Dawson College)
+## Author(Mehdi Moodi)
+## DBsubject(Statistics)
+## DBchapter(Confidence Intervals)
+## DBsection(Confidence Interval for the Mean)
+## Level(2)
+## KEYWORDS('confidence interval','population mean','small sample','sigma unknown','normal population','t critical value','standard error','margin of error')
+
+DOCUMENT();
+
+loadMacros(
+ "PGstandard.pl",
+ "PGanswermacros.pl",
+ "MathObjects.pl",
+ "PGML.pl",
+ "scaffold.pl",
+ "answerHints.pl",
+ "parserRadioButtons.pl",
+ "PGcourse.pl",
+);
+
+TEXT(beginproblem());
+
+# ----------------------------------------------------------------
+# PILOT: usefulness rating toggle
+# ----------------------------------------------------------------
+$ENABLE_GP_RATING = 1;
+
+# ----------------------------------------------------------------
+# Library Browser detection (LB stability)
+# ----------------------------------------------------------------
+$inLibraryBrowser = ($envir{isLibraryBrowser} || $envir{isLibrary} || 0);
+
+Context("Numeric");
+
+# =======================================================
+# Postfilter: add a hint ONLY when the answer is wrong
+# =======================================================
+sub HintIfWrong {
+ my $msg = shift;
+ return sub {
+ my $ans = shift;
+ if (defined($ans->{score}) && $ans->{score} < 1) {
+ $ans->{ans_message} = "" unless defined($ans->{ans_message});
+ $ans->{ans_message} .= " $msg";
+ }
+ return $ans;
+ };
+}
+
+# =======================================================
+# Data
+# =======================================================
+$xbar = 68;
+$s = 5.4;
+$n = 9;
+$df = 8;
+$conf = 0.95;
+
+$alpha = Real(0.05);
+$t_cutoff = Real(2.306);
+$se = Real(1.8);
+$me = Real(4.1508);
+$lcl = Real(63.8492);
+$ucl = Real(72.1508);
+
+# =======================================================
+# Tolerances
+# =======================================================
+$tol_alpha = 0.0005;
+$tol_t = 0.005;
+$tol_se = 0.005;
+$tol_me = 0.01;
+$tol_end = 0.02;
+
+# =======================================================
+# Comparators
+# =======================================================
+
+# Formula-selection MC: correct choice = 1
+$cmp_formula = Real(1)->cmp(
+ checker => sub {
+ my ($correct, $student, $ansHash) = @_;
+ return 0 unless defined $student;
+ my $v = $student->value;
+ return 0 unless defined $v;
+ return 0 unless $v == int($v);
+ return 0 unless ($v >= 1 && $v <= 5);
+ return ($v == 1) ? 1 : 0;
+ }
+)->withPostFilter(
+ HintIfWrong("Because the sample is small, the population is normal, and sigma is unknown, use xbar +/- t_(alpha/2)(s/sqrt(n)).")
+);
+
+$cmp_alpha = $alpha->cmp(
+ tolType => "absolute",
+ tolerance => $tol_alpha
+)->withPostFilter(
+ HintIfWrong("For a 95% confidence level, 1 - alpha = 0.95, so alpha = 0.05.")
+);
+
+$cmp_t = $t_cutoff->cmp(
+ tolType => "absolute",
+ tolerance => $tol_t
+)->withPostFilter(
+ HintIfWrong("For a 95% confidence interval, alpha = 0.05, so alpha/2 = 0.025. With n = 9, the degrees of freedom are 8, so use t_(0.025,8) = 2.306.")
+);
+
+$cmp_se = $se->cmp(
+ tolType => "absolute",
+ tolerance => $tol_se
+)->withPostFilter(
+ HintIfWrong("The standard error here is s/sqrt(n).")
+);
+
+$cmp_me = $me->cmp(
+ tolType => "absolute",
+ tolerance => $tol_me
+)->withPostFilter(
+ HintIfWrong("The margin of error is t_(alpha/2)(s/sqrt(n)).")
+);
+
+$cmp_lcl = $lcl->cmp(
+ tolType => "absolute",
+ tolerance => $tol_end
+)->withPostFilter(
+ HintIfWrong("The lower endpoint is xbar minus the margin of error.")
+);
+
+$cmp_ucl = $ucl->cmp(
+ tolType => "absolute",
+ tolerance => $tol_end
+)->withPostFilter(
+ HintIfWrong("The upper endpoint is xbar plus the margin of error.")
+);
+
+# Interpretation MC: correct choice = 2
+$cmp_interp = Real(2)->cmp(
+ checker => sub {
+ my ($correct, $student, $ansHash) = @_;
+ return 0 unless defined $student;
+ my $v = $student->value;
+ return 0 unless defined $v;
+ return 0 unless $v == int($v);
+ return 0 unless ($v >= 1 && $v <= 5);
+ return ($v == 2) ? 1 : 0;
+ }
+)->withPostFilter(
+ HintIfWrong("A confidence interval for mu is interpreted as a plausible range for the true population mean.")
+);
+
+# Rating checker
+$cmp_rating = Real(3)->cmp(
+ checker => sub {
+ my ($correct, $student, $ansHash) = @_;
+ return 0 unless defined $student;
+ my $v = $student->value;
+ return 0 unless defined $v;
+ return 0 unless $v == int($v);
+ return 0 unless ($v >= 1 && $v <= 5);
+ return 1;
+ }
+);
+
+# =======================================================
+# Scaffold setup
+# =======================================================
+if ($inLibraryBrowser) {
+ Scaffold::Begin(
+ numbered => 1,
+ can_open => "always",
+ is_open => "always"
+ );
+} else {
+ Scaffold::Begin(
+ numbered => 1,
+ can_open => "when_previous_correct",
+ is_open => "correct_or_first_incorrect"
+ );
+}
+
+# -------------------------------------------------------
+# Step 1 — Problem Statement
+# -------------------------------------------------------
+Section::Begin("Problem Statement");
+
+BEGIN_PGML
+A museum wants to estimate the true mean amount of time visitors spend inside a special exhibition.
+
+A random sample of [``n=9``] visitors is selected. The sample mean time is [``\bar x = 68``] minutes.
+The sample standard deviation is [``s = 5.4``] minutes.
+
+Assume the population of visit times is normally distributed.
+
+Construct a [``95\%``] confidence interval for the true population mean [``\mu``].
+
+Round the cutoff, standard error, margin of error, and interval endpoints to 3 decimals.
+END_PGML
+
+Section::End();
+
+# -------------------------------------------------------
+# Step 2 — Choose the correct formula
+# -------------------------------------------------------
+Section::Begin("Section B — Choose the correct formula");
+
+BEGIN_PGML
+Choose the correct confidence-interval formula for this problem, considering that the sample is small, the population is normal, and the population standard deviation [``\sigma``] is unknown.
+
+1) [``\bar x \pm t_{\alpha/2}\dfrac{s}{\sqrt{n}}``]
+
+2) [``\hat p \pm z_{\alpha/2}\sqrt{\dfrac{\hat p(1-\hat p)}{n}}``]
+
+3) [``\bar x \pm z_{\alpha/2}\dfrac{s}{\sqrt{n}}``]
+
+4) [``\bar x \pm z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}``]
+
+5) [``\bar x \pm t_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}``]
+
+Enter the number (1-5): [____]
+END_PGML
+
+ANS($cmp_formula);
+
+Section::End();
+
+# -------------------------------------------------------
+# Step 3 — Find alpha and the cutoff
+# -------------------------------------------------------
+Section::Begin("Section C — Find alpha and the cutoff");
+
+BEGIN_PGML
+For a [``95\%``] confidence interval:
+
+- [``1-\alpha = 0.95``]
+- each tail has area [``\alpha/2``]
+- since [``n=9``], the degrees of freedom are [``df = 8``]
+
+1) Enter the value of [``\alpha``]:
+
+[____]
+
+[@ image("cutoff_t.jpg", width=>520) @]*
+
+2) Use the t table to find the positive cutoff [``t_{\alpha/2}``].
+
+Enter [``t_{\alpha/2}``]: [____]
+END_PGML
+
+ANS($cmp_alpha, $cmp_t);
+
+Section::End();
+
+# -------------------------------------------------------
+# Step 4 — Compute the standard error and margin of error
+# -------------------------------------------------------
+Section::Begin("Section D — Compute the standard error and margin of error");
+
+BEGIN_PGML
+Compute the standard error:
+
+[``\dfrac{s}{\sqrt{n}}``]
+
+For this problem,
+
+[``\dfrac{5.4}{\sqrt{9}} =``] [____]
+
+Now compute the margin of error:
+
+[``E = t_{\alpha/2}\dfrac{s}{\sqrt{n}}``]
+
+For this problem,
+
+[``E = (2.306)\left(\dfrac{5.4}{\sqrt{9}}\right) =``] [____]
+END_PGML
+
+ANS($cmp_se, $cmp_me);
+
+Section::End();
+
+# -------------------------------------------------------
+# Step 5 — Construct the confidence interval
+# -------------------------------------------------------
+Section::Begin("Section E — Construct the confidence interval");
+
+BEGIN_PGML
+Now construct the confidence interval:
+
+[``\bar x \pm E``]
+
+Using [``\bar x = 68``] and your margin of error, enter the two endpoints.
+
+Lower endpoint: [____]
+
+Upper endpoint: [____]
+END_PGML
+
+ANS($cmp_lcl, $cmp_ucl);
+
+Section::End();
+
+# -------------------------------------------------------
+# Step 6 — Interpretation
+# -------------------------------------------------------
+Section::Begin("Section F — Interpret the confidence interval");
+
+BEGIN_PGML
+The [``95\%``] confidence interval is
+
+[``(63.849,\ 72.151)``]
+
+Choose the best interpretation.
+
+1) [``95\%``] of all museum visitors spend between [``63.849``] and [``72.151``] minutes in the special exhibition.
+
+2) We are [``95\%``] confident that the true mean time all visitors spend in the special exhibition is between [``63.849``] and [``72.151``] minutes.
+
+3) There is a [``95\%``] chance that the sample mean [``\bar x``] is between [``63.849``] and [``72.151``].
+
+4) The interval proves that the true population mean is exactly [``68``] minutes.
+
+5) [``95\%``] of all possible sample means are exactly equal to [``\mu``].
+
+Enter the number (1-5): [____]
+END_PGML
+
+ANS($cmp_interp);
+
+Section::End();
+
+# -------------------------------------------------------
+# Feedback (rating) — hidden in Library Browser
+# -------------------------------------------------------
+if ($ENABLE_GP_RATING && !$inLibraryBrowser) {
+ Section::Begin("Feedback");
+
+ BEGIN_PGML
+How useful was this Guided Problem for you?
+
+(1 = not useful, 2 = slightly useful, 3 = useful, 4 = very useful, 5 = extremely useful)
+
+Note: Consider using the Email Instructor button below to suggest improvements.
+
+Rating (1-5): [____]
+END_PGML
+
+ ANS($cmp_rating);
+
+ Section::End();
+}
+
+Scaffold::End();
+
+BEGIN_PGML_SOLUTION
+**1. Choose the correct formula**
+
+Because the sample is small, the population is normal, and the population standard deviation [``\sigma``] is unknown, the correct formula is
+
+[``\bar x \pm t_{\alpha/2}\dfrac{s}{\sqrt{n}}``]
+
+**2. Find alpha and the cutoff**
+
+For a [``95\%``] confidence interval,
+
+[``1-\alpha = 0.95``]
+
+so
+
+[``\alpha = 0.05``]
+
+Then
+
+[``\alpha/2 = 0.025``]
+
+Since [``n=9``], the degrees of freedom are
+
+[``df = n-1 = 8``]
+
+From the t table,
+
+[``t_{\alpha/2} = t_{0.025,8} = 2.306``]
+
+**3. Compute the standard error**
+
+[``\dfrac{s}{\sqrt{n}} = \dfrac{5.4}{\sqrt{9}} = \dfrac{5.4}{3} = 1.800``]
+
+So the standard error is [``1.800``].
+
+**4. Compute the margin of error**
+
+[``E = t_{\alpha/2}\dfrac{s}{\sqrt{n}}``]
+
+So
+
+[``E = (2.306)(1.800) = 4.151``]
+
+**5. Construct the confidence interval**
+
+Use
+
+[``\bar x \pm E``]
+
+So
+
+[``68 \pm 4.151``]
+
+Lower endpoint:
+
+[``68 - 4.151 = 63.849``]
+
+Upper endpoint:
+
+[``68 + 4.151 = 72.151``]
+
+Therefore, the [``95\%``] confidence interval for [``\mu``] is
+
+[``(63.849,\ 72.151)``]
+
+**6. Interpret the confidence interval**
+
+The correct interpretation is:
+
+We are [``95\%``] confident that the true mean time all visitors spend in the special exhibition is between [``63.849``] and [``72.151``] minutes.
+
+This interval gives a plausible range for the population mean [``\mu``]. It does **not** describe where [``95\%``] of individual visit times fall, and it does **not** mean that the sample mean has a [``95\%``] chance of being in that interval.
+END_PGML_SOLUTION
+
+ENDDOCUMENT();
diff --git a/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/Estimation/ConfidenceInterval_SmallSample_GuidedProblem4.pg b/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/Estimation/ConfidenceInterval_SmallSample_GuidedProblem4.pg
new file mode 100644
index 0000000000..2f401117da
--- /dev/null
+++ b/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/Estimation/ConfidenceInterval_SmallSample_GuidedProblem4.pg
@@ -0,0 +1,548 @@
+## DESCRIPTION
+## WeBWorK Guided Problems Project — Confidence Interval for the Mean:
+## Small Sample, Sigma Unknown, Raw Data Given (Guided / Scaffolded)
+## ENDDESCRIPTION
+##
+## Institution(Dawson College)
+## Author(Mehdi Moodi)
+## DBsubject(Statistics)
+## DBchapter(Confidence Intervals)
+## DBsection(Confidence Interval for the Mean)
+## Level(2)
+## KEYWORDS(
+## 'confidence interval',
+## 'population mean',
+## 'small sample',
+## 'sigma unknown',
+## 'sample standard deviation',
+## 't critical value',
+## 'standard error',
+## 'margin of error'
+## )
+
+DOCUMENT();
+
+loadMacros(
+ "PGstandard.pl",
+ "PGanswermacros.pl",
+ "MathObjects.pl",
+ "PGML.pl",
+ "scaffold.pl",
+ "answerHints.pl",
+ "parserRadioButtons.pl",
+ "PGcourse.pl",
+);
+
+TEXT(beginproblem());
+
+# ============================================================
+# Settings
+# ============================================================
+
+$ENABLE_GP_RATING = 1;
+
+$inLibraryBrowser = 0;
+if ((defined($envir{isLibraryBrowser}) && $envir{isLibraryBrowser})
+ || (defined($envir{isLibrary}) && $envir{isLibrary})) {
+ $inLibraryBrowser = 1;
+}
+
+Context("Numeric");
+
+# ============================================================
+# Helper: show a hint only when the student's answer is wrong
+# ============================================================
+sub HintIfWrong {
+ my $msg = shift;
+
+ return sub {
+ my $ans = shift;
+
+ if (defined($ans->{score}) && $ans->{score} < 1) {
+ $ans->{ans_message} = "" unless defined($ans->{ans_message});
+ $ans->{ans_message} .= " $msg";
+ }
+
+ return $ans;
+ };
+}
+
+# ============================================================
+# Data
+# Context: large poster printing times (in minutes)
+# ============================================================
+
+$n = 10;
+$conf = 0.99;
+
+# Raw data:
+# 18.2, 20.1, 19.5, 21.0, 20.4, 18.9, 19.8, 20.7, 21.3, 19.1
+
+$xbar = Real(19.9);
+$s = Real(0.9888264649);
+$alpha = Real(0.01);
+$df = 9;
+$t_cut = Real(3.250);
+$se = Real(0.3126890567);
+$me = Real(1.0162394342);
+$lcl = Real(18.8837605658);
+$ucl = Real(20.9162394342);
+
+# ============================================================
+# Tolerances
+# ============================================================
+
+$tol_xbar = 0.005;
+$tol_s = 0.005;
+$tol_alpha = 0.0005;
+$tol_t = 0.005;
+$tol_se = 0.005;
+$tol_me = 0.01;
+$tol_end = 0.02;
+
+# ============================================================
+# Comparators
+# ============================================================
+
+# ------------------------------------------------------------
+# Formula-selection multiple choice
+# Correct answer: 3
+# ------------------------------------------------------------
+$cmp_formula = Real(3)->cmp(
+ checker => sub {
+ my ($correct, $student, $ansHash) = @_;
+
+ return 0 unless defined $student;
+
+ my $v = $student->value;
+ return 0 unless defined $v;
+ return 0 unless $v == int($v);
+ return 0 unless ($v >= 1 && $v <= 5);
+
+ return ($v == 3) ? 1 : 0;
+ }
+)->withPostFilter(
+ HintIfWrong(
+ "Because the sample is small, the population is normal, and sigma is unknown, use xbar +/- t_(alpha/2)(s/sqrt(n))."
+ )
+);
+
+# ------------------------------------------------------------
+# Sample mean
+# ------------------------------------------------------------
+$cmp_xbar = $xbar->cmp(
+ tolType => "absolute",
+ tolerance => $tol_xbar
+)->withPostFilter(
+ HintIfWrong("Use your calculator's 1-variable statistics and enter the sample mean xbar.")
+);
+
+# ------------------------------------------------------------
+# Sample standard deviation
+# ------------------------------------------------------------
+$cmp_s = $s->cmp(
+ tolType => "absolute",
+ tolerance => $tol_s
+)->withPostFilter(
+ HintIfWrong("Be sure to use the sample standard deviation s, not the population standard deviation.")
+);
+
+# ------------------------------------------------------------
+# Alpha
+# ------------------------------------------------------------
+$cmp_alpha = $alpha->cmp(
+ tolType => "absolute",
+ tolerance => $tol_alpha
+)->withPostFilter(
+ HintIfWrong("For a 99% confidence level, 1 - alpha = 0.99, so alpha = 0.01.")
+);
+
+# ------------------------------------------------------------
+# t cutoff
+# ------------------------------------------------------------
+$cmp_t = $t_cut->cmp(
+ tolType => "absolute",
+ tolerance => $tol_t
+)->withPostFilter(
+ HintIfWrong(
+ "For a 99% confidence interval with n = 10, use df = 9 and alpha/2 = 0.005. Then read t_(0.005,9) from the t table."
+ )
+);
+
+# ------------------------------------------------------------
+# Standard error
+# ------------------------------------------------------------
+$cmp_se = $se->cmp(
+ tolType => "absolute",
+ tolerance => $tol_se
+)->withPostFilter(
+ HintIfWrong("The standard error here is s/sqrt(n).")
+);
+
+# ------------------------------------------------------------
+# Margin of error
+# ------------------------------------------------------------
+$cmp_me = $me->cmp(
+ tolType => "absolute",
+ tolerance => $tol_me
+)->withPostFilter(
+ HintIfWrong("The margin of error is t_(alpha/2) times s/sqrt(n).")
+);
+
+# ------------------------------------------------------------
+# Lower endpoint
+# ------------------------------------------------------------
+$cmp_lcl = $lcl->cmp(
+ tolType => "absolute",
+ tolerance => $tol_end
+)->withPostFilter(
+ HintIfWrong("The lower endpoint is xbar minus the margin of error.")
+);
+
+# ------------------------------------------------------------
+# Upper endpoint
+# ------------------------------------------------------------
+$cmp_ucl = $ucl->cmp(
+ tolType => "absolute",
+ tolerance => $tol_end
+)->withPostFilter(
+ HintIfWrong("The upper endpoint is xbar plus the margin of error.")
+);
+
+# ------------------------------------------------------------
+# Interpretation multiple choice
+# Correct answer: 4
+# ------------------------------------------------------------
+$cmp_interp = Real(4)->cmp(
+ checker => sub {
+ my ($correct, $student, $ansHash) = @_;
+
+ return 0 unless defined $student;
+
+ my $v = $student->value;
+ return 0 unless defined $v;
+ return 0 unless $v == int($v);
+ return 0 unless ($v >= 1 && $v <= 5);
+
+ return ($v == 4) ? 1 : 0;
+ }
+)->withPostFilter(
+ HintIfWrong(
+ "A confidence interval for mu is interpreted as a plausible range for the true population mean."
+ )
+);
+
+# ------------------------------------------------------------
+# Rating checker
+# Any integer 1 to 5 is accepted
+# ------------------------------------------------------------
+$cmp_rating = Real(3)->cmp(
+ checker => sub {
+ my ($correct, $student, $ansHash) = @_;
+
+ return 0 unless defined $student;
+
+ my $v = $student->value;
+ return 0 unless defined $v;
+ return 0 unless $v == int($v);
+ return 0 unless ($v >= 1 && $v <= 5);
+
+ return 1;
+ }
+);
+
+# ============================================================
+# Scaffold setup
+# ============================================================
+
+if ($inLibraryBrowser) {
+ Scaffold::Begin(
+ numbered => 1,
+ can_open => "always",
+ is_open => "always"
+ );
+}
+else {
+ Scaffold::Begin(
+ numbered => 1,
+ can_open => "when_previous_correct",
+ is_open => "correct_or_first_incorrect"
+ );
+}
+
+# ============================================================
+# Step 1 — Problem Statement
+# ============================================================
+Section::Begin("Problem Statement");
+
+BEGIN_PGML
+A university print shop wants to estimate the true mean time required to print a large conference poster.
+
+A random sample of [``n=10``] poster-printing jobs is selected. The printing times, in minutes, are:
+
+[``18.2,\ 20.1,\ 19.5,\ 21.0,\ 20.4,\ 18.9,\ 19.8,\ 20.7,\ 21.3,\ 19.1``]
+
+Assume the population of printing times is normally distributed.
+
+Construct a [``99\%``] confidence interval for the true population mean [``\mu``].
+
+Round [``\bar x``], [``s``], the cutoff, the standard error, the margin of error, and the interval endpoints to 3 decimals.
+END_PGML
+
+Section::End();
+
+# ============================================================
+# Step 2 — Choose the correct formula
+# ============================================================
+Section::Begin("Section B — Choose the correct formula");
+
+BEGIN_PGML
+Choose the correct confidence-interval formula for this problem.
+
+1) [``\bar x \pm z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}``]
+
+2) [``\bar x \pm t_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}``]
+
+3) [``\bar x \pm t_{\alpha/2}\dfrac{s}{\sqrt{n}}``]
+
+4) [``\hat p \pm z_{\alpha/2}\sqrt{\dfrac{\hat p(1-\hat p)}{n}}``]
+
+5) [``\bar x \pm z_{\alpha/2}\dfrac{s}{\sqrt{n}}``]
+
+Enter the number (1-5): [____]
+END_PGML
+
+ANS($cmp_formula);
+
+Section::End();
+
+# ============================================================
+# Step 3 — Use a calculator to find xbar and s
+# ============================================================
+Section::Begin("Section C — Use a calculator to find xbar and s");
+
+BEGIN_PGML
+Use your calculator's 1-variable statistics on the 10 printing times.
+
+Enter the sample mean [``\bar x``]: [____]
+
+Enter the sample standard deviation [``s``]: [____]
+END_PGML
+
+ANS($cmp_xbar, $cmp_s);
+
+Section::End();
+
+# ============================================================
+# Step 4 — Find alpha and the t cutoff
+# ============================================================
+Section::Begin("Section D — Find alpha and the cutoff");
+
+BEGIN_PGML
+For a [``99\%``] confidence interval:
+
+- [``1-\alpha = 0.99``]
+- each tail has area [``\alpha/2``]
+- the degrees of freedom are [``n-1 = 9``]
+
+1) Enter the value of [``\alpha``]:
+
+[____]
+
+[@ image("cutoff_t.jpg", width=>520) @]*
+
+2) Use the [``t``] table to find the positive cutoff [``t_{\alpha/2}``] for [``df=9``].
+
+Enter [``t_{\alpha/2}``]: [____]
+END_PGML
+
+ANS($cmp_alpha, $cmp_t);
+
+Section::End();
+
+# ============================================================
+# Step 5 — Compute the standard error and margin of error
+# ============================================================
+Section::Begin("Section E — Compute the standard error and margin of error");
+
+BEGIN_PGML
+Compute the standard error:
+
+[``\dfrac{s}{\sqrt{n}}``]
+
+For this problem,
+
+[``\dfrac{s}{\sqrt{10}} =``] [____]
+
+Now compute the margin of error:
+
+[``E = t_{\alpha/2}\dfrac{s}{\sqrt{n}}``]
+
+For this problem,
+
+[``E = t_{\alpha/2}\left(\dfrac{s}{\sqrt{10}}\right) =``] [____]
+END_PGML
+
+ANS($cmp_se, $cmp_me);
+
+Section::End();
+
+# ============================================================
+# Step 6 — Construct the confidence interval
+# ============================================================
+Section::Begin("Section F — Construct the confidence interval");
+
+BEGIN_PGML
+Now construct the confidence interval:
+
+[``\bar x \pm E``]
+
+Using your values of [``\bar x``] and [``E``], enter the two endpoints.
+
+Lower endpoint: [____]
+
+Upper endpoint: [____]
+END_PGML
+
+ANS($cmp_lcl, $cmp_ucl);
+
+Section::End();
+
+# ============================================================
+# Step 7 — Interpretation
+# ============================================================
+Section::Begin("Section G — Interpret the confidence interval");
+
+BEGIN_PGML
+Choose the best interpretation of the confidence interval you constructed for [``\mu``].
+
+1) [``99\%``] of all individual poster-printing times must lie inside the interval.
+
+2) There is a [``99\%``] chance that the sample mean [``\bar x``] lies inside the interval.
+
+3) The interval proves that the true mean printing time is exactly equal to [``\bar x``].
+
+4) We are [``99\%``] confident that the true mean time required to print a large conference poster lies between the two endpoints of the interval.
+
+5) [``99\%``] of all possible sample means are equal to the true mean.
+
+Enter the number (1-5): [____]
+END_PGML
+
+ANS($cmp_interp);
+
+Section::End();
+
+# ============================================================
+# Feedback — hidden in Library Browser
+# ============================================================
+if ($ENABLE_GP_RATING && !$inLibraryBrowser) {
+ Section::Begin("Feedback");
+
+ BEGIN_PGML
+How useful was this Guided Problem for you?
+
+(1 = not useful, 2 = slightly useful, 3 = useful, 4 = very useful, 5 = extremely useful)
+
+Note: Consider using the Email Instructor button below to suggest improvements.
+
+Rating (1-5): [____]
+END_PGML
+
+ ANS($cmp_rating);
+
+ Section::End();
+}
+
+Scaffold::End();
+
+# ============================================================
+# Solution
+# ============================================================
+BEGIN_PGML_SOLUTION
+**1. Choose the correct formula**
+
+Because the sample is small, the population is normal, and the population standard deviation is unknown, the correct formula is
+
+[``\bar x \pm t_{\alpha/2}\dfrac{s}{\sqrt{n}}``]
+
+**2. Compute the sample mean and sample standard deviation**
+
+Using the 10 data values in 1-variable statistics, we get
+
+[``\bar x = 19.900``]
+
+and
+
+[``s = 0.989``]
+
+(more precisely, [``s \approx 0.988826``]).
+
+**3. Find alpha and the t cutoff**
+
+For a [``99\%``] confidence interval,
+
+[``1-\alpha = 0.99``]
+
+so
+
+[``\alpha = 0.01``]
+
+Then
+
+[``\alpha/2 = 0.005``]
+
+and the degrees of freedom are
+
+[``df = n-1 = 10-1 = 9``]
+
+From the [``t``] table,
+
+[``t_{\alpha/2} = t_{0.005,9} = 3.250``]
+
+**4. Compute the standard error**
+
+[``\dfrac{s}{\sqrt{n}} = \dfrac{0.988826}{\sqrt{10}} \approx 0.313``]
+
+So the standard error is [``0.313``].
+
+**5. Compute the margin of error**
+
+[``E = t_{\alpha/2}\dfrac{s}{\sqrt{n}}``]
+
+So
+
+[``E = (3.250)(0.312689) \approx 1.016``]
+
+Therefore, the margin of error is [``1.016``].
+
+**6. Construct the confidence interval**
+
+Use
+
+[``\bar x \pm E``]
+
+So
+
+[``19.900 \pm 1.016``]
+
+Lower endpoint:
+
+[``19.900 - 1.016 = 18.884``]
+
+Upper endpoint:
+
+[``19.900 + 1.016 = 20.916``]
+
+Therefore, the [``99\%``] confidence interval for [``\mu``] is
+
+[``(18.884,\ 20.916)``]
+
+**7. Interpret the confidence interval**
+
+The correct interpretation is:
+
+We are [``99\%``] confident that the true mean time required to print a large conference poster lies between [``18.884``] minutes and [``20.916``] minutes.
+END_PGML_SOLUTION
+
+ENDDOCUMENT();
\ No newline at end of file
diff --git a/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/Estimation/cutoff.jpg b/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/Estimation/cutoff.jpg
new file mode 100644
index 0000000000..6a650be0ea
Binary files /dev/null and b/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/Estimation/cutoff.jpg differ
diff --git a/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/Estimation/cutoff_t.jpg b/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/Estimation/cutoff_t.jpg
new file mode 100644
index 0000000000..a856b2b0c3
Binary files /dev/null and b/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/Estimation/cutoff_t.jpg differ
diff --git a/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/NormalRandomVariables/NormalRandomVariables_NonStandard_GuidedProblem1.pg b/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/NormalRandomVariables/NormalRandomVariables_NonStandard_GuidedProblem1.pg
new file mode 100644
index 0000000000..ddbabc4c65
--- /dev/null
+++ b/Contrib/DawsonCollege/Probability_Statistics/GuidedProblems/NormalRandomVariables/NormalRandomVariables_NonStandard_GuidedProblem1.pg
@@ -0,0 +1,333 @@
+## DESCRIPTION
+## WeBWorK Guided Problems Project — Standardizing: Using the z-table for Non-Standard Normal Variables (Guided / Scaffolded)
+## ENDDESCRIPTION
+##
+## Institution(Dawson College)
+## Author(Mehdi Moodi)
+## DBsubject(Statistics)
+## DBchapter(Probability)
+## DBsection(Normal Distribution)
+## Level(2)
+## KEYWORDS('normal distribution','standardization','z score','z table','non-standard normal','cutoff')
+
+DOCUMENT();
+
+loadMacros(
+ "PGstandard.pl",
+ "PGanswermacros.pl",
+ "MathObjects.pl",
+ "PGML.pl",
+ "scaffold.pl",
+ "answerHints.pl",
+ "parserRadioButtons.pl",
+ "PGcourse.pl",
+);
+
+TEXT(beginproblem());
+
+# ----------------------------------------------------------------
+# PILOT: usefulness rating toggle (ACTIVE)
+# ----------------------------------------------------------------
+$ENABLE_GP_RATING = 1;
+
+# ----------------------------------------------------------------
+# Library Browser detection (LB stability)
+# ----------------------------------------------------------------
+$inLibraryBrowser = ($envir{isLibraryBrowser} || $envir{isLibrary} || 0);
+
+# =======================================================
+# Data
+# =======================================================
+Context("Numeric");
+
+# Images (put in same folder as this .pg file)
+$IMG_STD = "non_st_to_st.jpg";
+$IMG_THEORY = "nonstandard_theory.jpg"; # <-- rename this to match your actual filename if different
+
+# Main variable for Practices 1-6
+$mu = Real(70);
+$sigma = Real(8);
+
+# Warm-up z-scores
+$z_78 = Real("1.00"); # (78-70)/8
+$z_62 = Real("-1.00"); # (62-70)/8
+
+# Probabilities (computed internally; not shown to students)
+$p_X_lt_78 = Real("0.8413"); # P(X<78)
+$p_X_gt_62 = Real("0.8413"); # P(X>62)
+$p_66_82 = Real("0.6247"); # P(66