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</head>
<body class="quarto-light">
<div class="reveal">
<div class="slides">
<section id="title-slide" class="quarto-title-block center">
<h1 class="title">Practicing Inference with Categorical Data</h1>
<p class="subtitle">One- and Two-Proportion Inference</p>
<div class="quarto-title-authors">
<div class="quarto-title-author">
<div class="quarto-title-author-name">
Dr. Gilbert
</div>
</div>
</div>
<p class="date">March 4, 2026</p>
</section>
<section class="slide level2">
<div class="cell">
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</section>
<section id="reminder-framework-for-constructing-confidence-intervals" class="slide level2">
<h2>Reminder: Framework for Constructing Confidence Intervals</h2>
<p><strong>General Strategy:</strong> Each time we seek to Construct a confidence interval, we will…</p>
<ol type="1">
<li class="fragment">Read the scenario very carefully</li>
<li class="fragment">Determine what it is we are trying to estimate – a mean? a proportion? a difference in means or proportions?</li>
<li class="fragment">Recall the general “formula” for the <em>confidence interval</em></li>
</ol>
<div class="fragment">
<p><span class="math display">\[\scriptsize{\left(\begin{array}{c}\texttt{point}\\ \texttt{estimate}\end{array}\right) \pm \left(\begin{array}{c}\texttt{critical}\\ \texttt{value}\end{array}\right)\cdot S_E}\]</span></p>
<ol start="4" type="1">
<li class="fragment">Re-read the scenario to identify the <em>point estimate</em>, which comes from the sample data</li>
<li class="fragment">Use <a href="https://agmath.github.io/SiteFiles/StdErrorDecisionTree.pdf" target="_blank">the Standard Error Decision Tree</a> to find how to compute the <em>standard error</em> (<span class="math inline">\(S_E\)</span>)</li>
<li class="fragment">Identify or compute the appropriate <em>critical value</em></li>
<li class="fragment">Find the <em>lower-</em> and <em>upper-bounds</em> for the confidence interval by evaluating the “formula”</li>
<li class="fragment">Interpret the confidence interval in the appropriate context</li>
</ol>
</div>
</section>
<section id="reminder-hypothesis-testing-framework" class="slide level2 smaller">
<h2>Reminder: Hypothesis Testing Framework</h2>
<p><strong>General Strategy:</strong> Each time we conduct a hypothesis test, we will…</p>
<ol type="1">
<li class="fragment"><p>Read the scenario carefully and determine whether you are testing a claim about a mean, a proportion, comparison of means, or comparison of proportions.</p></li>
<li class="fragment"><p>Identify the claim being tested and state the <em>null hypothesis</em> (<span class="math inline">\(H_0\)</span>) and <em>alternative hypothesis</em> (<span class="math inline">\(H_a\)</span>). The null hypothesis represents the “status quo” or “no difference” and always involves an equal sign, while the alternative hypothesis is the claim to be tested and involves one of: <span class="math inline">\(<,~>,~\neq\)</span></p></li>
<li class="fragment"><p>Draw a picture of your alternative hypothesis, shading in the tails corresponding to samples that would satisfy the alternative hypothesis.</p></li>
<li class="fragment"><p>Set the level of significance (<span class="math inline">\(\alpha\)</span>) for the test. The level of significance is the “cut-off” for a sample being unusual/unlikely/unexpected. The standard cutoff is <span class="math inline">\(\alpha = 0.05\)</span>, unless we are told otherwise.</p></li>
<li class="fragment"><p>Compute the <em>test statistic</em>, where <span class="math inline">\(\displaystyle{\texttt{test statistic} = \frac{\left(\texttt{point estimate}\right) - \left(\texttt{null value}\right)}{S_E}}\)</span></p>
<ul>
<li class="fragment">The <em>point estimate</em> comes from the sample data</li>
<li class="fragment">The <em>null value</em> comes from the null hypothesis</li>
<li class="fragment">The <em>standard error</em> (<span class="math inline">\(S_E\)</span>) formula is found on <a href="https://agmath.github.io/SiteFiles/StdErrorDecisionTree.pdf" target="_blank">the Standard Error Decision Tree</a></li>
</ul></li>
<li class="fragment"><p>The test statistic is just a boundary value, use it along with your picture from step 3 to calculate a <span class="math inline">\(p\)</span>-value (the probability of observing data at least as extreme as ours if the null hypothesis were true).</p></li>
<li class="fragment"><p>Compare your <span class="math inline">\(p\)</span>-value to the level of significance (<span class="math inline">\(\alpha\)</span>) demanded by your test and make a decision about whether or not to reject the null hypothesis.</p></li>
<li class="fragment"><p>Interpret the result of your hypothesis test in the context appropriate for your scenario.</p></li>
</ol>
</section>
<section id="inference-on-a-single-proportion-versus-comparisons-of-proportions" class="slide level2">
<h2>Inference on a Single Proportion versus Comparisons of Proportions</h2>
<div class="fragment">
<p>At our last class meeting, we saw one example of the construction of a confidence interval and another example conducting a hypothesis test</p>
</div>
<div class="fragment">
<p>In both of those scenarios, the population parameter of interest was a single <em>population proportion</em></p>
</div>
<div class="fragment">
<p>We’ll see scenarios like this again today, but we’ll also consider scenarios in which we’d like to compare two population proportions.</p>
</div>
<div class="fragment">
<p>The general procedures for constructing confidence intervals or conducting hypothesis tests are the same in both scenarios, except that in the comparison scenarios…</p>
<ul>
<li class="fragment">the population parameter of interest is the <em>difference in proportions</em> between the two sub-populations</li>
<li class="fragment">the formula for computing the standard error (<span class="math inline">\(S_E\)</span>) is different</li>
</ul>
</div>
</section>
<section id="navigating-the-standard-error-tree" class="slide level2">
<h2>Navigating the Standard Error Tree</h2>
<div class="fragment">
<p>One of the main purposes of the <em>Standard Error Decision Tree</em> document is to help you identify the appropriate standard error (<span class="math inline">\(S_E\)</span>) to use in your calculation of a confidence interval or test statistic</p>
</div>
<div class="fragment">
<p>To navigate the tree, begin from the box at the top, which asks whether we are working with means (<span class="math inline">\(\mu\)</span>) or proportions (<span class="math inline">\(p\)</span>)</p>
</div>
<div class="fragment">
<p>Follow your answer down the tree and continue answering the questions until you land inside a box with a standard error formula (<span class="math inline">\(S_E = \cdots\)</span>)</p>
</div>
<div class="fragment">
<p>The side of the tree for working with proportions is less complicated than the side for means – we’ll stick to proportions for now</p>
</div>
<div class="fragment">
<p>At this point, we’ll either calculate <span class="math inline">\(\displaystyle{S_E = \sqrt{\frac{p\left(1 - p\right)}{n}}}\)</span> for a single proportion or <span class="math inline">\(\displaystyle{S_E = \sqrt{\frac{p_1\left(1 - p_1\right)}{n_1} + \frac{p_2\left(1 - p_2\right)}{n_2}}}\)</span> for a comparison between two population proportions</p>
</div>
</section>
<section id="a-completed-hypothesis-test" class="slide level2 smaller">
<h2>A Completed Hypothesis Test</h2>
<p><strong>Scenario:</strong> A group of students is curious about whether the success rates (finding a long-term relationship) differ between people using a swipe-based dating app versus a personality-based matchmaking app. They conduct a survey of users from both types of apps and compare the proportion of users who report being in a long-term relationship. On the swipe-based app, 46 out of 200 users surveyed reported being in a long-term relationship. On the personality-based site, 65 out of 180 users surveyed said the same. Is there significant evidence to suggest (at the 10% level of significance) that proportion of successful relationships are not the same for swipe-based apps and personality-based apps?</p>
<div class="columns">
<div class="column" style="width:50%;">
<ol type="1">
<li class="fragment">We’re testing a claim comparing two population<br> proportions</li>
<li class="fragment">The hypotheses are: <span class="math inline">\(\begin{array}{lcl} H_0 & : & p_{\text{swipe}} = p_{\text{personality}}\\ H_a & : & p_{\text{swipe}} \neq p_{\text{personality}}\end{array}\)</span></li>
</ol>
</div><div class="column" style="width:50%;">
</div>
</div>
</section>
<section id="a-completed-hypothesis-test-1" class="slide level2 smaller">
<h2>A Completed Hypothesis Test</h2>
<p><strong>Scenario:</strong> A group of students is curious about whether the success rates (finding a long-term relationship) differ between people using a swipe-based dating app versus a personality-based matchmaking app. They conduct a survey of users from both types of apps and compare the proportion of users who report being in a long-term relationship. On the swipe-based app, 46 out of 200 users surveyed reported being in a long-term relationship. On the personality-based site, 65 out of 180 users surveyed said the same. Is there significant evidence to suggest (at the 10% level of significance) that proportion of successful relationships are not the same for swipe-based apps and personality-based apps?</p>
<div class="columns">
<div class="column" style="width:50%;">
<div>
<ol type="1">
<li>We’re testing a claim comparing two population<br> proportions</li>
<li>The hypotheses are: <span class="math inline">\(\begin{array}{lcl} H_0 & : & p_{\text{swipe}} = p_{\text{personality}}\\ H_a & : & p_{\text{swipe}} \neq p_{\text{personality}}\end{array}\)</span>, but a<br> better way to write them is <span class="math inline">\(\begin{array}{lcl} H_0 & : & p_{\text{swipe}} - p_{\text{personality}} = 0\\ H_a & : & p_{\text{swipe}} - p_{\text{personality}} \neq 0\end{array}\)</span></li>
</ol>
</div>
<ol start="3" type="1">
<li class="fragment">See the picture of the alternative hypothesis below:</li>
</ol>
</div><div class="column" style="width:50%;">
</div>
</div>
</section>
<section id="a-completed-hypothesis-test-2" class="slide level2 smaller">
<h2>A Completed Hypothesis Test</h2>
<p><strong>Scenario:</strong> A group of students is curious about whether the success rates (finding a long-term relationship) differ between people using a swipe-based dating app versus a personality-based matchmaking app. They conduct a survey of users from both types of apps and compare the proportion of users who report being in a long-term relationship. On the swipe-based app, 46 out of 200 users surveyed reported being in a long-term relationship. On the personality-based site, 65 out of 180 users surveyed said the same. Is there significant evidence to suggest (at the 10% level of significance) that proportion of successful relationships are not the same for swipe-based apps and personality-based apps?</p>
<div class="columns">
<div class="column" style="width:50%;">
<div>
<ol type="1">
<li><p>We’re testing a claim comparing two population<br> proportions</p></li>
<li><p>The hypotheses are: <span class="math inline">\(\begin{array}{lcl} H_0 & : & p_{\text{swipe}} = p_{\text{personality}}\\ H_a & : & p_{\text{swipe}} \neq p_{\text{personality}}\end{array}\)</span>, but a<br> better way to write them is <span class="math inline">\(\begin{array}{lcl} H_0 & : & p_{\text{swipe}} - p_{\text{personality}} = 0\\ H_a & : & p_{\text{swipe}} - p_{\text{personality}} \neq 0\end{array}\)</span></p></li>
<li><p>See the picture of the alternative hypothesis below:</p></li>
</ol>
<div class="columns">
<div class="column" style="width:20%;">
</div><div class="column" style="width:60%;">
<div class="cell">
<div class="cell-output-display">
<div>
<figure>
<p><img data-src="CategoricalInferencePractice_Slides_files/figure-revealjs/unnamed-chunk-2-1.png" width="960"></p>
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</div><div class="column" style="width:20%;">
</div>
</div>
</div>
<ol start="4" type="1">
<li class="fragment">Set <span class="math inline">\(\alpha = 0.10\)</span></li>
</ol>
</div><div class="column" style="width:50%;">
<ol start="5" type="1">
<li class="fragment">We’ll calculate the test statistic, where <span class="math inline">\(\texttt{test statistic} = \frac{\left(\texttt{point estimate}\right) - \left(\texttt{null value}\right)}{S_E}\)</span></li>
</ol>
</div>
</div>
</section>
<section id="a-completed-hypothesis-test-3" class="slide level2 smaller">
<h2>A Completed Hypothesis Test</h2>
<p><strong>Scenario:</strong> A group of students is curious about whether the success rates (finding a long-term relationship) differ between people using a swipe-based dating app versus a personality-based matchmaking app. They conduct a survey of users from both types of apps and compare the proportion of users who report being in a long-term relationship. On the swipe-based app, 46 out of 200 users surveyed reported being in a long-term relationship. On the personality-based site, 65 out of 180 users surveyed said the same. Is there significant evidence to suggest (at the 10% level of significance) that proportion of successful relationships are not the same for swipe-based apps and personality-based apps?</p>
<div class="columns">
<div class="column" style="width:50%;">
<div>
<ol type="1">
<li><p>We’re testing a claim comparing two population<br> proportions</p></li>
<li><p>The hypotheses are: <span class="math inline">\(\begin{array}{lcl} H_0 & : & p_{\text{swipe}} = p_{\text{personality}}\\ H_a & : & p_{\text{swipe}} \neq p_{\text{personality}}\end{array}\)</span>, but a<br> better way to write them is <span class="math inline">\(\begin{array}{lcl} H_0 & : & p_{\text{swipe}} - p_{\text{personality}} = 0\\ H_a & : & p_{\text{swipe}} - p_{\text{personality}} \neq 0\end{array}\)</span></p></li>
<li><p>See the picture of the alternative hypothesis below:</p></li>
</ol>
<div class="columns">
<div class="column" style="width:20%;">
</div><div class="column" style="width:60%;">
<div class="cell">
<div class="cell-output-display">
<div>
<figure>
<p><img data-src="CategoricalInferencePractice_Slides_files/figure-revealjs/unnamed-chunk-3-1.png" width="960"></p>
</figure>
</div>
</div>
</div>
</div><div class="column" style="width:20%;">
</div>
</div>
<ol start="4" type="1">
<li>Set <span class="math inline">\(\alpha = 0.10\)</span></li>
</ol>
</div>
</div><div class="column" style="width:50%;">
<div>
<ol start="5" type="1">
<li>We’ll calculate the test statistic, where <span class="math inline">\(\texttt{test statistic} = \frac{\left(\texttt{point estimate}\right) - \left(\texttt{null value}\right)}{S_E}\)</span></li>
</ol>
<p>Notice that <span class="math inline">\(\displaystyle{p_{\text{swipe}} = \frac{46}{200}}\)</span>, so <span class="math inline">\(p_{\text{swipe}} = 0.23\)</span> and<br> <span class="math inline">\(\displaystyle{p_{\text{personality}} = \frac{65}{180}}\)</span>, so <span class="math inline">\(p_{\text{personality}} \approx 0.3611\)</span></p>
</div>
</div>
</div>
</section>
<section id="a-completed-hypothesis-test-4" class="slide level2 smaller">
<h2>A Completed Hypothesis Test</h2>
<p><strong>Scenario:</strong> A group of students is curious about whether the success rates (finding a long-term relationship) differ between people using a swipe-based dating app versus a personality-based matchmaking app. They conduct a survey of users from both types of apps and compare the proportion of users who report being in a long-term relationship. On the swipe-based app, 46 out of 200 users surveyed reported being in a long-term relationship. On the personality-based site, 65 out of 180 users surveyed said the same. Is there significant evidence to suggest (at the 10% level of significance) that proportion of successful relationships are not the same for swipe-based apps and personality-based apps?</p>
<div class="columns">
<div class="column" style="width:50%;">
<div>
<ol type="1">
<li><p>We’re testing a claim comparing two population<br> proportions</p></li>
<li><p>The hypotheses are: <span class="math inline">\(\begin{array}{lcl} H_0 & : & p_{\text{swipe}} = p_{\text{personality}}\\ H_a & : & p_{\text{swipe}} \neq p_{\text{personality}}\end{array}\)</span>, but a<br> better way to write them is <span class="math inline">\(\begin{array}{lcl} H_0 & : & p_{\text{swipe}} - p_{\text{personality}} = 0\\ H_a & : & p_{\text{swipe}} - p_{\text{personality}} \neq 0\end{array}\)</span></p></li>
<li><p>See the picture of the alternative hypothesis below:</p></li>
</ol>
<div class="columns">
<div class="column" style="width:20%;">
</div><div class="column" style="width:60%;">
<div class="cell">
<div class="cell-output-display">
<div>
<figure>
<p><img data-src="CategoricalInferencePractice_Slides_files/figure-revealjs/unnamed-chunk-4-1.png" width="960"></p>
</figure>
</div>
</div>
</div>
</div><div class="column" style="width:20%;">
</div>
</div>
<ol start="4" type="1">
<li>Set <span class="math inline">\(\alpha = 0.10\)</span></li>
</ol>
</div>
</div><div class="column" style="width:50%;">
<div>
<ol start="5" type="1">
<li>We’ll calculate the test statistic, where <span class="math inline">\(\texttt{test statistic} = \frac{\left(\texttt{point estimate}\right) - \left(\texttt{null value}\right)}{S_E}\)</span></li>
</ol>
<p>Notice that <span class="math inline">\(\displaystyle{p_{\text{swipe}} = \frac{46}{200}}\)</span>, so <span class="math inline">\(p_{\text{swipe}} = 0.23\)</span> and<br> <span class="math inline">\(\displaystyle{p_{\text{personality}} = \frac{65}{180}}\)</span>, so <span class="math inline">\(p_{\text{personality}} \approx 0.3611\)</span></p>
<p>Notice also that our <em>standard error</em> is calculated as</p>
<p><span class="math display">\[S_E = \sqrt{\frac{p_{\text{swipe}}\left(1 - p_{\text{swipe}}\right)}{n_{\text{swipe}}} + \frac{p_{\text{personality}}\left(1 - p_{\text{personality}}\right)}{n_{\text{personality}}}}\]</span></p>
</div>
</div>
</div>
</section>
<section id="a-completed-hypothesis-test-5" class="slide level2 smaller">
<h2>A Completed Hypothesis Test</h2>
<p><strong>Scenario:</strong> A group of students is curious about whether the success rates (finding a long-term relationship) differ between people using a swipe-based dating app versus a personality-based matchmaking app. They conduct a survey of users from both types of apps and compare the proportion of users who report being in a long-term relationship. On the swipe-based app, 46 out of 200 users surveyed reported being in a long-term relationship. On the personality-based site, 65 out of 180 users surveyed said the same. Is there significant evidence to suggest (at the 10% level of significance) that proportion of successful relationships are not the same for swipe-based apps and personality-based apps?</p>
<div class="columns">
<div class="column" style="width:50%;">
<div>
<ol type="1">
<li><p>We’re testing a claim comparing two population<br> proportions</p></li>
<li><p>The hypotheses are: <span class="math inline">\(\begin{array}{lcl} H_0 & : & p_{\text{swipe}} = p_{\text{personality}}\\ H_a & : & p_{\text{swipe}} \neq p_{\text{personality}}\end{array}\)</span>, but a<br> better way to write them is <span class="math inline">\(\begin{array}{lcl} H_0 & : & p_{\text{swipe}} - p_{\text{personality}} = 0\\ H_a & : & p_{\text{swipe}} - p_{\text{personality}} \neq 0\end{array}\)</span></p></li>
<li><p>See the picture of the alternative hypothesis below:</p></li>
</ol>
<div class="columns">
<div class="column" style="width:20%;">
</div><div class="column" style="width:60%;">
<div class="cell">
<div class="cell-output-display">
<div>
<figure>
<p><img data-src="CategoricalInferencePractice_Slides_files/figure-revealjs/unnamed-chunk-5-1.png" width="960"></p>
</figure>
</div>
</div>
</div>
</div><div class="column" style="width:20%;">
</div>
</div>
<ol start="4" type="1">
<li>Set <span class="math inline">\(\alpha = 0.10\)</span></li>
</ol>
</div>
</div><div class="column" style="width:50%;">
<div>
<ol start="5" type="1">
<li>We’ll calculate the test statistic, where <span class="math inline">\(\texttt{test statistic} = \frac{\left(\texttt{point estimate}\right) - \left(\texttt{null value}\right)}{S_E}\)</span></li>
</ol>
<p>Notice that <span class="math inline">\(\displaystyle{p_{\text{swipe}} = \frac{46}{200}}\)</span>, so <span class="math inline">\(p_{\text{swipe}} = 0.23\)</span> and<br> <span class="math inline">\(\displaystyle{p_{\text{personality}} = \frac{65}{180}}\)</span>, so <span class="math inline">\(p_{\text{personality}} \approx 0.3611\)</span></p>
<p>Notice also that our <em>standard error</em> is calculated as</p>
<p><span class="math display">\[S_E = \sqrt{\frac{p_{\text{swipe}}\left(1 - p_{\text{swipe}}\right)}{n_{\text{swipe}}} + \frac{p_{\text{personality}}\left(1 - p_{\text{personality}}\right)}{n_{\text{personality}}}}\]</span></p>
<p>So,</p>
<p><span class="math display">\[\texttt{test statistic} = \frac{\left(0.23 - 0.3611\right) - 0}{\sqrt{\frac{0.23\left(1 - 0.23\right)}{200} + \frac{0.3611\left(1 - 0.3611\right)}{180}}} \approx -2.82\]</span></p>
</div>
</div>
</div>
</section>
<section id="a-completed-hypothesis-test-6" class="slide level2 smaller">
<h2>A Completed Hypothesis Test</h2>
<p><strong>Scenario:</strong> A group of students is curious about whether the success rates (finding a long-term relationship) differ between people using a swipe-based dating app versus a personality-based matchmaking app. They conduct a survey of users from both types of apps and compare the proportion of users who report being in a long-term relationship. On the swipe-based app, 46 out of 200 users surveyed reported being in a long-term relationship. On the personality-based site, 65 out of 180 users surveyed said the same. Is there significant evidence to suggest (at the 10% level of significance) that proportion of successful relationships are not the same for swipe-based apps and personality-based apps?</p>
<div class="columns">
<div class="column" style="width:50%;">
<div>
<ol type="1">
<li><p>We’re testing a claim comparing two population<br> proportions</p></li>
<li><p>The hypotheses are: <span class="math inline">\(\begin{array}{lcl} H_0 & : & p_{\text{swipe}} = p_{\text{personality}}\\ H_a & : & p_{\text{swipe}} \neq p_{\text{personality}}\end{array}\)</span>, but a<br> better way to write them is <span class="math inline">\(\begin{array}{lcl} H_0 & : & p_{\text{swipe}} - p_{\text{personality}} = 0\\ H_a & : & p_{\text{swipe}} - p_{\text{personality}} \neq 0\end{array}\)</span></p></li>
<li><p>See the picture of the alternative hypothesis below:</p></li>
</ol>
<div class="columns">
<div class="column" style="width:20%;">
</div><div class="column" style="width:60%;">
<div class="cell">
<div class="cell-output-display">
<div>
<figure>
<p><img data-src="CategoricalInferencePractice_Slides_files/figure-revealjs/unnamed-chunk-6-1.png" width="960"></p>
</figure>
</div>
</div>
</div>
</div><div class="column" style="width:20%;">
</div>
</div>
<ol start="4" type="1">
<li>Set <span class="math inline">\(\alpha = 0.10\)</span></li>
</ol>
</div>
</div><div class="column" style="width:50%;">
<div>
<ol start="5" type="1">
<li>We’ll calculate the test statistic, where <span class="math inline">\(\texttt{test statistic} = \frac{\left(\texttt{point estimate}\right) - \left(\texttt{null value}\right)}{S_E}\)</span></li>
</ol>
<p><span class="math display">\[\texttt{test statistic} \approx -2.82\]</span></p>
</div>
<ol start="6" type="1">
<li class="fragment">Now we’ll draw our test statistic on our standard normal distribution and calculate the <span class="math inline">\(p\)</span>-value</li>
</ol>
</div>
</div>
</section>
<section id="a-completed-hypothesis-test-7" class="slide level2 smaller">
<h2>A Completed Hypothesis Test</h2>
<p><strong>Scenario:</strong> A group of students is curious about whether the success rates (finding a long-term relationship) differ between people using a swipe-based dating app versus a personality-based matchmaking app. They conduct a survey of users from both types of apps and compare the proportion of users who report being in a long-term relationship. On the swipe-based app, 46 out of 200 users surveyed reported being in a long-term relationship. On the personality-based site, 65 out of 180 users surveyed said the same. Is there significant evidence to suggest (at the 10% level of significance) that proportion of successful relationships are not the same for swipe-based apps and personality-based apps?</p>
<div class="columns">
<div class="column" style="width:50%;">
<div>
<ol type="1">
<li><p>We’re testing a claim comparing two population<br> proportions</p></li>
<li><p>The hypotheses are: <span class="math inline">\(\begin{array}{lcl} H_0 & : & p_{\text{swipe}} = p_{\text{personality}}\\ H_a & : & p_{\text{swipe}} \neq p_{\text{personality}}\end{array}\)</span>, but a<br> better way to write them is <span class="math inline">\(\begin{array}{lcl} H_0 & : & p_{\text{swipe}} - p_{\text{personality}} = 0\\ H_a & : & p_{\text{swipe}} - p_{\text{personality}} \neq 0\end{array}\)</span></p></li>
<li><p>See the picture of the alternative hypothesis below:</p></li>
</ol>
<div class="columns">
<div class="column" style="width:20%;">
</div><div class="column" style="width:60%;">
<div class="cell">
<div class="cell-output-display">
<div>
<figure>
<p><img data-src="CategoricalInferencePractice_Slides_files/figure-revealjs/unnamed-chunk-7-1.png" width="960"></p>
</figure>
</div>
</div>
</div>
</div><div class="column" style="width:20%;">
</div>
</div>
<ol start="4" type="1">
<li>Set <span class="math inline">\(\alpha = 0.10\)</span></li>
</ol>
</div>
</div><div class="column" style="width:50%;">
<div>
<ol start="5" type="1">
<li>We’ll calculate the test statistic, where <span class="math inline">\(\texttt{test statistic} = \frac{\left(\texttt{point estimate}\right) - \left(\texttt{null value}\right)}{S_E}\)</span></li>
</ol>
<p><span class="math display">\[\texttt{test statistic} \approx -2.82\]</span></p>
<ol start="6" type="1">
<li>Now we’ll draw our test statistic on our standard normal distribution and calculate the <span class="math inline">\(p\)</span>-value</li>
</ol>
<div class="columns">
<div class="column" style="width:20%;">
</div><div class="column" style="width:60%;">
<div class="cell">
<div class="cell-output-display">
<div>
<figure>
<p><img data-src="CategoricalInferencePractice_Slides_files/figure-revealjs/unnamed-chunk-8-1.png" width="960"></p>
</figure>
</div>
</div>
</div>
</div><div class="column" style="width:20%;">
</div>
</div>
</div>
<ol start="7" type="1">
<li class="fragment"><span class="math inline">\(p\)</span>-value = …</li>
</ol>
</div>
</div>
</section>
<section id="a-completed-hypothesis-test-8" class="slide level2 smaller">
<h2>A Completed Hypothesis Test</h2>
<p><strong>Scenario:</strong> A group of students is curious about whether the success rates (finding a long-term relationship) differ between people using a swipe-based dating app versus a personality-based matchmaking app. They conduct a survey of users from both types of apps and compare the proportion of users who report being in a long-term relationship. On the swipe-based app, 46 out of 200 users surveyed reported being in a long-term relationship. On the personality-based site, 65 out of 180 users surveyed said the same. Is there significant evidence to suggest (at the 10% level of significance) that proportion of successful relationships are not the same for swipe-based apps and personality-based apps?</p>
<div class="columns">
<div class="column" style="width:50%;">
<div>
<ol type="1">
<li><p>We’re testing a claim comparing two population<br> proportions</p></li>
<li><p>The hypotheses are: <span class="math inline">\(\begin{array}{lcl} H_0 & : & p_{\text{swipe}} = p_{\text{personality}}\\ H_a & : & p_{\text{swipe}} \neq p_{\text{personality}}\end{array}\)</span>, but a<br> better way to write them is <span class="math inline">\(\begin{array}{lcl} H_0 & : & p_{\text{swipe}} - p_{\text{personality}} = 0\\ H_a & : & p_{\text{swipe}} - p_{\text{personality}} \neq 0\end{array}\)</span></p></li>
<li><p>See the picture of the alternative hypothesis below:</p></li>
</ol>
<div class="columns">
<div class="column" style="width:20%;">
</div><div class="column" style="width:60%;">
<div class="cell">
<div class="cell-output-display">
<div>
<figure>
<p><img data-src="CategoricalInferencePractice_Slides_files/figure-revealjs/unnamed-chunk-9-1.png" width="960"></p>
</figure>
</div>
</div>
</div>
</div><div class="column" style="width:20%;">
</div>
</div>
<ol start="4" type="1">
<li>Set <span class="math inline">\(\alpha = 0.10\)</span></li>
</ol>
</div>
</div><div class="column" style="width:50%;">
<div>
<ol start="5" type="1">
<li>We’ll calculate the test statistic, where <span class="math inline">\(\texttt{test statistic} = \frac{\left(\texttt{point estimate}\right) - \left(\texttt{null value}\right)}{S_E}\)</span></li>
</ol>
<p><span class="math display">\[\texttt{test statistic} \approx -2.82\]</span></p>
<ol start="6" type="1">
<li>Now we’ll draw our test statistic on our standard normal distribution and calculate the <span class="math inline">\(p\)</span>-value</li>
</ol>
<div class="columns">
<div class="column" style="width:20%;">
</div><div class="column" style="width:60%;">
<div class="cell">
<div class="cell-output-display">
<div>
<figure>
<p><img data-src="CategoricalInferencePractice_Slides_files/figure-revealjs/unnamed-chunk-10-1.png" width="960"></p>
</figure>
</div>
</div>
</div>
</div><div class="column" style="width:20%;">
</div>
</div>
<ol start="7" type="1">
<li><span class="math inline">\(p\)</span>-value = <code>2*pnorm(-2.82, 0, 1)</code> <span class="math inline">\(\approx\)</span> 0.0048</li>
</ol>
</div>
</div>
</div>
</section>
<section id="a-completed-hypothesis-test-9" class="slide level2 smaller">
<h2>A Completed Hypothesis Test</h2>
<p><strong>Scenario:</strong> A group of students is curious about whether the success rates (finding a long-term relationship) differ between people using a swipe-based dating app versus a personality-based matchmaking app. They conduct a survey of users from both types of apps and compare the proportion of users who report being in a long-term relationship. On the swipe-based app, 46 out of 200 users surveyed reported being in a long-term relationship. On the personality-based site, 65 out of 180 users surveyed said the same. Is there significant evidence to suggest (at the 10% level of significance) that proportion of successful relationships are not the same for swipe-based apps and personality-based apps?</p>
<div class="columns">
<div class="column" style="width:50%;">
<div>
<ol type="1">
<li><p>We’re testing a claim comparing two population<br> proportions</p></li>
<li><p>The hypotheses are: <span class="math inline">\(\begin{array}{lcl} H_0 & : & p_{\text{swipe}} = p_{\text{personality}}\\ H_a & : & p_{\text{swipe}} \neq p_{\text{personality}}\end{array}\)</span>, but a<br> better way to write them is <span class="math inline">\(\begin{array}{lcl} H_0 & : & p_{\text{swipe}} - p_{\text{personality}} = 0\\ H_a & : & p_{\text{swipe}} - p_{\text{personality}} \neq 0\end{array}\)</span></p></li>
<li><p>See the picture of the alternative hypothesis below:</p></li>
</ol>
<div class="columns">
<div class="column" style="width:20%;">
</div><div class="column" style="width:60%;">
<div class="cell">
<div class="cell-output-display">
<div>
<figure>
<p><img data-src="CategoricalInferencePractice_Slides_files/figure-revealjs/unnamed-chunk-11-1.png" width="960"></p>
</figure>
</div>
</div>
</div>
</div><div class="column" style="width:20%;">
</div>
</div>
<ol start="4" type="1">
<li>Set <span class="math inline">\(\alpha = 0.10\)</span></li>
</ol>
</div>
</div><div class="column" style="width:50%;">
<div>
<ol start="5" type="1">
<li>We’ll calculate the test statistic, where <span class="math inline">\(\texttt{test statistic} = \frac{\left(\texttt{point estimate}\right) - \left(\texttt{null value}\right)}{S_E}\)</span></li>
</ol>
<p><span class="math display">\[\texttt{test statistic} \approx -2.82\]</span></p>
<ol start="6" type="1">
<li>Now we’ll draw our test statistic on our standard normal distribution and calculate the <span class="math inline">\(p\)</span>-value</li>
</ol>
<div class="columns">
<div class="column" style="width:20%;">
</div><div class="column" style="width:60%;">
<div class="cell">
<div class="cell-output-display">
<div>
<figure>
<p><img data-src="CategoricalInferencePractice_Slides_files/figure-revealjs/unnamed-chunk-12-1.png" width="960"></p>
</figure>
</div>
</div>
</div>
</div><div class="column" style="width:20%;">
</div>
</div>
<ol start="7" type="1">
<li><p><span class="math inline">\(p\)</span>-value = <code>2*pnorm(-2.82, 0, 1)</code> <span class="math inline">\(\approx\)</span> 0.0048</p></li>
<li><p><span class="math inline">\(p\)</span>-value <span class="math inline">\(< \alpha\)</span>, we reject <span class="math inline">\(H_0\)</span> and accept <span class="math inline">\(H_a\)</span></p></li>
</ol>
</div>
</div>
</div>
</section>
<section id="a-completed-hypothesis-test-10" class="slide level2 smaller">
<h2>A Completed Hypothesis Test</h2>
<p><strong>Scenario:</strong> A group of students is curious about whether the success rates (finding a long-term relationship) differ between people using a swipe-based dating app versus a personality-based matchmaking app. They conduct a survey of users from both types of apps and compare the proportion of users who report being in a long-term relationship. On the swipe-based app, 46 out of 200 users surveyed reported being in a long-term relationship. On the personality-based site, 65 out of 180 users surveyed said the same. Is there significant evidence to suggest (at the 10% level of significance) that proportion of successful relationships are not the same for swipe-based apps and personality-based apps?</p>
<div class="columns">
<div class="column" style="width:50%;">
<div>
<ol type="1">
<li><p>We’re testing a claim comparing two population<br> proportions</p></li>
<li><p>The hypotheses are: <span class="math inline">\(\begin{array}{lcl} H_0 & : & p_{\text{swipe}} = p_{\text{personality}}\\ H_a & : & p_{\text{swipe}} \neq p_{\text{personality}}\end{array}\)</span>, but a<br> better way to write them is <span class="math inline">\(\begin{array}{lcl} H_0 & : & p_{\text{swipe}} - p_{\text{personality}} = 0\\ H_a & : & p_{\text{swipe}} - p_{\text{personality}} \neq 0\end{array}\)</span></p></li>
<li><p>See the picture of the alternative hypothesis below:</p></li>
</ol>
<div class="columns">
<div class="column" style="width:20%;">
</div><div class="column" style="width:60%;">
<div class="cell">
<div class="cell-output-display">
<div>
<figure>
<p><img data-src="CategoricalInferencePractice_Slides_files/figure-revealjs/unnamed-chunk-13-1.png" width="960"></p>
</figure>
</div>
</div>
</div>
</div><div class="column" style="width:20%;">
</div>
</div>
<ol start="4" type="1">
<li>Set <span class="math inline">\(\alpha = 0.10\)</span></li>
</ol>
</div>
</div><div class="column" style="width:50%;">
<div>
<ol start="5" type="1">
<li>We’ll calculate the test statistic, where <span class="math inline">\(\texttt{test statistic} = \frac{\left(\texttt{point estimate}\right) - \left(\texttt{null value}\right)}{S_E}\)</span></li>
</ol>
<p><span class="math display">\[\texttt{test statistic} \approx -2.82\]</span></p>
<ol start="6" type="1">
<li>Now we’ll draw our test statistic on our standard normal distribution and calculate the <span class="math inline">\(p\)</span>-value</li>
</ol>
<div class="columns">
<div class="column" style="width:20%;">
</div><div class="column" style="width:60%;">
<div class="cell">
<div class="cell-output-display">
<div>
<figure>
<p><img data-src="CategoricalInferencePractice_Slides_files/figure-revealjs/unnamed-chunk-14-1.png" width="960"></p>
</figure>
</div>
</div>
</div>
</div><div class="column" style="width:20%;">
</div>
</div>
<ol start="7" type="1">
<li><p><span class="math inline">\(p\)</span>-value = <code>2*pnorm(-2.82, 0, 1)</code> <span class="math inline">\(\approx\)</span> 0.0048</p></li>
<li><p>Our observed data is incompatible with the proportion of successful relationships being equal. The relationship success rates differ across the two apps.</p></li>
</ol>
</div>
</div>
</div>
</section>
<section id="a-completed-confidence-interval" class="slide level2 smaller">
<h2>A Completed Confidence Interval</h2>
<p><strong>Scenario:</strong> A college wants to understand if there is a difference in the proportion of students who attend office hours in-person versus virtually. Faculty members noticed varying attendance patterns and want to explore whether virtual office hours increase engagement. Out of 120 students surveyed, 38 said that they attend in-person office hours regularly. A separate random sample of 140 students revealed that 62 attended virtual office hours regularly. Construct a 90% confidence interval for the difference in proportions of students attending virtual and face-to-face office hours regularly.</p>
<div class="columns">
<div class="column" style="width:50%;">
<ol type="1">
<li class="fragment">Read the problem carefully <span class="math inline">\(\checkmark\)</span></li>
<li class="fragment">Estimate the <em>difference between the proportions</em> of students regularly attending virtual office hours versus in-person office hours</li>
<li class="fragment">The “formula” for the confidence<br> interval is</li>
</ol>
</div><div class="column" style="width:50%;">
</div>
</div>
</section>
<section id="a-completed-confidence-interval-1" class="slide level2 smaller">
<h2>A Completed Confidence Interval</h2>
<p><strong>Scenario:</strong> A college wants to understand if there is a difference in the proportion of students who attend office hours in-person versus virtually. Faculty members noticed varying attendance patterns and want to explore whether virtual office hours increase engagement. Out of 120 students surveyed, 38 said that they attend in-person office hours regularly. A separate random sample of 140 students revealed that 62 attended virtual office hours regularly. Construct a 90% confidence interval for the difference in proportions of students attending virtual and face-to-face office hours regularly.</p>
<div class="columns">
<div class="column" style="width:50%;">
<div>
<ol type="1">
<li>Read the problem carefully <span class="math inline">\(\checkmark\)</span></li>
<li>Estimate the <em>difference between the proportions</em> of students regularly attending virtual office hours versus in-person office hours</li>
<li>The “formula” for the confidence<br> interval is</li>
</ol>
</div>
<p><span class="math display">\[\scriptsize{\left(\begin{array}{c}\texttt{point}\\ \texttt{estimate}\end{array}\right) \pm \left(\begin{array}{c}\texttt{critical}\\ \texttt{value}\end{array}\right)\cdot \left(\begin{array}{c}\texttt{standard}\\ \texttt{error}\end{array}\right)}\]</span></p>
<ol start="4" type="1">
<li class="fragment">The <em>point estimate</em> is the difference in proportions…</li>
</ol>
</div><div class="column" style="width:50%;">
</div>
</div>
</section>
<section id="a-completed-confidence-interval-2" class="slide level2 smaller">
<h2>A Completed Confidence Interval</h2>
<p><strong>Scenario:</strong> A college wants to understand if there is a difference in the proportion of students who attend office hours in-person versus virtually. Faculty members noticed varying attendance patterns and want to explore whether virtual office hours increase engagement. Out of 120 students surveyed, 38 said that they attend in-person office hours regularly. A separate random sample of 140 students revealed that 62 attended virtual office hours regularly. Construct a 90% confidence interval for the difference in proportions of students attending virtual and face-to-face office hours regularly.</p>
<div class="columns">
<div class="column" style="width:50%;">
<div>
<ol type="1">
<li>Read the problem carefully <span class="math inline">\(\checkmark\)</span></li>
<li>Estimate the <em>difference between the proportions</em> of students regularly attending virtual office hours versus in-person office hours</li>
<li>The “formula” for the confidence<br> interval is</li>
</ol>
<p><span class="math display">\[\scriptsize{\left(\begin{array}{c}\texttt{point}\\ \texttt{estimate}\end{array}\right) \pm \left(\begin{array}{c}\texttt{critical}\\ \texttt{value}\end{array}\right)\cdot \left(\begin{array}{c}\texttt{standard}\\ \texttt{error}\end{array}\right)}\]</span></p>
<ol start="4" type="1">