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---
layout: default
---
<div class="container">
<div class="row">
<div class="col">
<h1> Hypothesis!! </h1>
<!--########### Legends ##########-->
<h3><div class="wow zoomIn legend">
`H_a = tt"Alternative Hypothesis"`<br>
`H_o = tt"NULL Hypothesis"`<br>
`bar x_i = tt"Mean for incongruent scores"`<br>
`bar x_c = tt"Mean for congruent scores"`<br>
`bar x_d = tt"Point Estimate"`<br>
`t_(stat) = tt"t-statistics for the sample"`<br>
`delta = tt"The desired difference between both means"`
</div></h3>
<!--######### Hypothesis statements ############-->
<p>
<h3><span id="green" class="wow zoomIn delay"> NULL Hypothesis</span></h3>
<div class="tab wow bounceInRight">
`H_o: mu_i - mu_c = 0 -> tt"There is no"`
`tt"difference between"`
`tt"both the means"`<br><br>
`tt"here, "`<br>
`mu_c =`
`tt"Mean of the congruent score of population"`<br>
`mu_i =`
`tt"Mean of the incongruent score of population"`
</div>
</p>
<p>
<h3><span id="green" class="wow zoomIn delay"> ALTERNATIVE Hypothesis </span></h3>
<div class="tab wow bounceInRight">
`H_a: mu_c - mu_i lt 0 ->`
`tt"Congruent score"`
`tt"mean is lesser"`
`tt"than the incongruent"`
`tt"mean in 1"`
`tt"tailed t-test"`<br><br>
`tt"here, "`<br>
`mu_c =`
`tt"Mean of the congruent score of population"`<br>
`mu_i =`
`tt"Mean of the incongruent score of population"`
<br>
</div>
</p>
<!--########## dogrees of freedom ###########-->
<p>
<h3><span id="green" class="tab-up wow zoomIn delay">
Total degress of freedom
</span></h3>
<div class="tab wow bounceInRight">
`df = (n - 1) = tt"23"`
`if n = tt"Sample Size"`
</div>
</p>
<!--########## t statistics for alpha = 0.05 and df = 23 ###########-->
<p>
<h3><span id="green" class="tab-up wow zoomIn delay">
One tailed `t` value at `alpha` of 0.05
</span></h3>
<div class="tab wow bounceInRight">
`t_(alpha = 0.05) = +-tt"1.714"`
</div>
</p>
<!--- ######## Standard Deviation of the Difference -->
<p>
<h3><span id="green" class="tab-up wow zoomIn delay">
Standard Deviation of the difference
</span></h3>
<div class="tab wow bounceInRight">
`S = sqrt(sum_0^(n-1) (d_i - bar (md))^2 / (n -1)) = tt"4.8648"`<br>
`iff bar (md) = sum_0^(n-1) (x_i - y_i) / n`
`tt"= -7.9647"`
</div>
</p>
<!--########## Point of estimate ########### -->
<p>
<h3><span id="green" class="tab-up wow zoomIn delay">
Point of Estimate
</span></h3>
<div class="tab wow bounceInRight">
`bar x_d = bar x_c - bar x_i = bar (md) = tt"-7.9647"`
</div>
</p>
<!--########## Standard Error ###########-->
<p>
<h3><span id="green" class="tab-up wow zoomIn delay">
Standard Error
</span></h3>
<div class="tab wow bounceInRight">
`SE = S / sqrt(n) = tt"0.9930"`
</div>
</p>
<!--########### t statistics for the sample ##########-->
<p>
<h3><span id="green" class="tab-up wow zoomIn delay">
t-statistics for the sample
</span></h3>
<div class="tab wow bounceInRight">
`t_(stat) = ((bar x_c - bar x_i) - delta) / (SE) =tt"-8.0207"`
`iff delta = 0`
</div>
</p>
<!--########### Cohen's d for effect size ##########-->
<p>
<h3><span id="green" class="tab-up wow zoomIn delay">
Cohen's d for effect size
</span></h3>
<div class="tab wow bounceInRight">
`d = (bar x_c - bar x_i)/S = tt"-1.6372"`
</div>
</p>
<!--########### 95% confidence interval ##########-->
<p>
<h3><span id="green" class="tab-up wow zoomIn delay">
95% Cofidence Interval
</span></h3>
<div class="tab wow bounceInRight">
`CI = bar x_d +-` `t_(alpha = 0.05)` `xx SE`
` => CI in tt"[-9.666, -6.262]"`
</div>
</p>
<!--########### r^2 value ##########-->
<p>
<h3><span id="green" class="tab-up wow zoomIn delay">
Coefficient of Determination `r^2`
</span></h3>
<div class="tab wow bounceInRight">
`r^2 = t_(stat)^2 / (t_(stat)^2 + df) = tt"0.736"`
`~=tt"74%"`
</div>
</p>
<!-- Hypothesis Decision -->
<p>
<h3><span id="green" class="tab-up wow zoomIn delay">
Decision on the Hypothesis
</span></h3>
<div class="tab wow bounceInRight">
<p> As the `|t_(alpha=0.05)| < |t_(stat)|` we reject the `H_o` in favour of the `H_a`. </p>
</div>
</p>
</div>
</div>
<div class="row">
<div class="col">
<p>
<h3><span id="green" class="wow zoomIn"> Inference </span></h3>
<div class="wow zoomIn"> <p>
As we have rejected the `H_o` in favour of `H_a` and this is a controlled experiment, we can say the <em> incogrunet words does affect the reading time of the words as compared to the congruent words</em> beause of stroop effect and this statistically siginificant event. This is natural too because the t-score of `tt"-21"` is way more than the aplha value and hence the probability is way less for getting the difference of mean to be zero.
</p> </div><br>
<div class="wow zoomIn"> <p>
Consequently, as we have predicted in the scatter plot that most of the points are located along the regression axis and this would result in a very high value of `r^2`. We get the `r^2` to be almost `74%` which means that `74%` of the variation of the time in the incongruent score from the congruent score is actually because the words were changed to incongruent. The rest `26%` corresponds to some other factor namely lurking variables. This further proves the fact that if samples are taken without any bias or in other words randomly, the increase in time is mostly because of word type is changed from congruent to incongruent. Moreover the Cohen's d of negative 4 shows that there is a decerese in time and it is quite large. This is also relevant because the congrent time is supposed to be lesser than the incongruent time.</p>
</div><br>
<div class="wow zoomIn"><p> Another important factor is the confidence interval or CI. The confidence interval has always been a parameter for population metric capture. In other words, when we only have the samples but not the population parameter, we have to rely on the CI to make an assumption for the CI meteric if the experiment is performed on the population. Here the CI metric is the difference between `bar x_c - bar x_i` or the difference in the time between congruent and incongruent words. We saw that the value is `tt"[-9.666, -6.262]"`. <em>Hence it means that if we implement the same experiment on the population, then on average people would take almost `tt"6.2 sec to 9.6 sec"` less time in reading the congruent words as compared to incongruent words.</em></p>
</div>
</p>
</div>
</div>
</div>