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<!DOCTYPE html>
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<title>B Inference Examples | Statistical Inference via Data Science</title>
<meta name="description" content="An open-source and fully-reproducible electronic textbook for teaching statistical inference using tidyverse data science tools." />
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<meta name="twitter:description" content="An open-source and fully-reproducible electronic textbook for teaching statistical inference using tidyverse data science tools." />
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<meta name="author" content="Chester Ismay and Albert Y. Kim Foreword by Kelly S. McConville Adapted by William R. Morgan" />
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<li class="chapter" data-level="1" data-path="1-getting-started.html"><a href="1-getting-started.html"><i class="fa fa-check"></i><b>1</b> Getting Started with Data in R</a>
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<li class="chapter" data-level="1.1" data-path="1-getting-started.html"><a href="1-getting-started.html#r-rstudio"><i class="fa fa-check"></i><b>1.1</b> What are R and RStudio?</a>
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<li class="chapter" data-level="1.2" data-path="1-getting-started.html"><a href="1-getting-started.html#code"><i class="fa fa-check"></i><b>1.2</b> How do I code in R?</a>
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<li class="chapter" data-level="1.2.1" data-path="1-getting-started.html"><a href="1-getting-started.html#programming-concepts"><i class="fa fa-check"></i><b>1.2.1</b> Basic programming concepts and terminology</a></li>
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<li class="chapter" data-level="1.3" data-path="1-getting-started.html"><a href="1-getting-started.html#packages"><i class="fa fa-check"></i><b>1.3</b> What are R packages?</a>
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<li class="chapter" data-level="1.4" data-path="1-getting-started.html"><a href="1-getting-started.html#rfishbase"><i class="fa fa-check"></i><b>1.4</b> Explore your first datasets</a>
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<li class="chapter" data-level="1.4.3" data-path="1-getting-started.html"><a href="1-getting-started.html#exploredataframes"><i class="fa fa-check"></i><b>1.4.3</b> Exploring data frames</a></li>
<li class="chapter" data-level="1.4.4" data-path="1-getting-started.html"><a href="1-getting-started.html#identification-vs-measurement-variables"><i class="fa fa-check"></i><b>1.4.4</b> Identification and measurement variables</a></li>
<li class="chapter" data-level="1.4.5" data-path="1-getting-started.html"><a href="1-getting-started.html#help-files"><i class="fa fa-check"></i><b>1.4.5</b> Help files</a></li>
</ul></li>
<li class="chapter" data-level="1.5" data-path="1-getting-started.html"><a href="1-getting-started.html#conclusion"><i class="fa fa-check"></i><b>1.5</b> Conclusion</a>
<ul>
<li class="chapter" data-level="1.5.1" data-path="1-getting-started.html"><a href="1-getting-started.html#additional-resources"><i class="fa fa-check"></i><b>1.5.1</b> Additional resources</a></li>
<li class="chapter" data-level="1.5.2" data-path="1-getting-started.html"><a href="1-getting-started.html#whats-to-come"><i class="fa fa-check"></i><b>1.5.2</b> What’s to come?</a></li>
</ul></li>
</ul></li>
<li class="part"><span><b>I Data Science with tidyverse</b></span></li>
<li class="chapter" data-level="2" data-path="2-viz.html"><a href="2-viz.html"><i class="fa fa-check"></i><b>2</b> Data Visualization</a>
<ul>
<li class="chapter" data-level="" data-path="2-viz.html"><a href="2-viz.html#needed-packages"><i class="fa fa-check"></i>Needed packages</a></li>
<li class="chapter" data-level="2.1" data-path="2-viz.html"><a href="2-viz.html#grammarofgraphics"><i class="fa fa-check"></i><b>2.1</b> The grammar of graphics</a>
<ul>
<li class="chapter" data-level="2.1.1" data-path="2-viz.html"><a href="2-viz.html#components-of-the-grammar"><i class="fa fa-check"></i><b>2.1.1</b> Components of the grammar</a></li>
<li class="chapter" data-level="2.1.2" data-path="2-viz.html"><a href="2-viz.html#gapminder"><i class="fa fa-check"></i><b>2.1.2</b> Gapminder data</a></li>
<li class="chapter" data-level="2.1.3" data-path="2-viz.html"><a href="2-viz.html#other-components"><i class="fa fa-check"></i><b>2.1.3</b> Other components</a></li>
<li class="chapter" data-level="2.1.4" data-path="2-viz.html"><a href="2-viz.html#ggplot2-package"><i class="fa fa-check"></i><b>2.1.4</b> ggplot2 package</a></li>
</ul></li>
<li class="chapter" data-level="2.2" data-path="2-viz.html"><a href="2-viz.html#FiveNG"><i class="fa fa-check"></i><b>2.2</b> Five named graphs - the 5NG</a></li>
<li class="chapter" data-level="2.3" data-path="2-viz.html"><a href="2-viz.html#scatterplots"><i class="fa fa-check"></i><b>2.3</b> 5NG#1: Scatterplots</a>
<ul>
<li class="chapter" data-level="2.3.1" data-path="2-viz.html"><a href="2-viz.html#geompoint"><i class="fa fa-check"></i><b>2.3.1</b> Scatterplots via <code>geom_point</code></a></li>
<li class="chapter" data-level="2.3.2" data-path="2-viz.html"><a href="2-viz.html#overplotting"><i class="fa fa-check"></i><b>2.3.2</b> Overplotting</a></li>
<li class="chapter" data-level="2.3.3" data-path="2-viz.html"><a href="2-viz.html#summary"><i class="fa fa-check"></i><b>2.3.3</b> Summary</a></li>
</ul></li>
<li class="chapter" data-level="2.4" data-path="2-viz.html"><a href="2-viz.html#linegraphs"><i class="fa fa-check"></i><b>2.4</b> 5NG#2: Linegraphs</a>
<ul>
<li class="chapter" data-level="2.4.1" data-path="2-viz.html"><a href="2-viz.html#geomline"><i class="fa fa-check"></i><b>2.4.1</b> Linegraphs via <code>geom_line</code></a></li>
<li class="chapter" data-level="2.4.2" data-path="2-viz.html"><a href="2-viz.html#summary-1"><i class="fa fa-check"></i><b>2.4.2</b> Summary</a></li>
</ul></li>
<li class="chapter" data-level="2.5" data-path="2-viz.html"><a href="2-viz.html#facets"><i class="fa fa-check"></i><b>2.5</b> Facets</a></li>
<li class="chapter" data-level="2.6" data-path="2-viz.html"><a href="2-viz.html#histograms"><i class="fa fa-check"></i><b>2.6</b> 5NG#3: Histograms</a>
<ul>
<li class="chapter" data-level="2.6.1" data-path="2-viz.html"><a href="2-viz.html#geomhistogram"><i class="fa fa-check"></i><b>2.6.1</b> Histograms via <code>geom_histogram</code></a></li>
<li class="chapter" data-level="2.6.2" data-path="2-viz.html"><a href="2-viz.html#adjustbins"><i class="fa fa-check"></i><b>2.6.2</b> Adjusting the bins</a></li>
<li class="chapter" data-level="2.6.3" data-path="2-viz.html"><a href="2-viz.html#summary-2"><i class="fa fa-check"></i><b>2.6.3</b> Summary</a></li>
</ul></li>
<li class="chapter" data-level="2.7" data-path="2-viz.html"><a href="2-viz.html#boxplots"><i class="fa fa-check"></i><b>2.7</b> 5NG#4: Boxplots</a>
<ul>
<li class="chapter" data-level="2.7.1" data-path="2-viz.html"><a href="2-viz.html#geomboxplot"><i class="fa fa-check"></i><b>2.7.1</b> Boxplots via <code>geom_boxplot</code></a></li>
<li class="chapter" data-level="2.7.2" data-path="2-viz.html"><a href="2-viz.html#summary-3"><i class="fa fa-check"></i><b>2.7.2</b> Summary</a></li>
</ul></li>
<li class="chapter" data-level="2.8" data-path="2-viz.html"><a href="2-viz.html#geombar"><i class="fa fa-check"></i><b>2.8</b> 5NG#5: Barplots</a>
<ul>
<li class="chapter" data-level="2.8.1" data-path="2-viz.html"><a href="2-viz.html#barplots-via-geom_bar-or-geom_col"><i class="fa fa-check"></i><b>2.8.1</b> Barplots via <code>geom_bar</code> or <code>geom_col</code></a></li>
<li class="chapter" data-level="2.8.2" data-path="2-viz.html"><a href="2-viz.html#must-avoid-pie-charts"><i class="fa fa-check"></i><b>2.8.2</b> Must avoid pie charts!</a></li>
<li class="chapter" data-level="2.8.3" data-path="2-viz.html"><a href="2-viz.html#two-categ-barplot"><i class="fa fa-check"></i><b>2.8.3</b> Two categorical variables</a></li>
<li class="chapter" data-level="2.8.4" data-path="2-viz.html"><a href="2-viz.html#summary-4"><i class="fa fa-check"></i><b>2.8.4</b> Summary</a></li>
</ul></li>
<li class="chapter" data-level="2.9" data-path="2-viz.html"><a href="2-viz.html#data-vis-conclusion"><i class="fa fa-check"></i><b>2.9</b> Conclusion</a>
<ul>
<li class="chapter" data-level="2.9.1" data-path="2-viz.html"><a href="2-viz.html#summary-table"><i class="fa fa-check"></i><b>2.9.1</b> Summary table</a></li>
<li class="chapter" data-level="2.9.2" data-path="2-viz.html"><a href="2-viz.html#function-argument-specification"><i class="fa fa-check"></i><b>2.9.2</b> Function argument specification</a></li>
<li class="chapter" data-level="2.9.3" data-path="2-viz.html"><a href="2-viz.html#additional-resources-1"><i class="fa fa-check"></i><b>2.9.3</b> Additional resources</a></li>
<li class="chapter" data-level="2.9.4" data-path="2-viz.html"><a href="2-viz.html#whats-to-come-3"><i class="fa fa-check"></i><b>2.9.4</b> What’s to come</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="3" data-path="3-wrangling.html"><a href="3-wrangling.html"><i class="fa fa-check"></i><b>3</b> Data Wrangling</a>
<ul>
<li class="chapter" data-level="" data-path="3-wrangling.html"><a href="3-wrangling.html#wrangling-packages"><i class="fa fa-check"></i>Needed packages</a></li>
<li class="chapter" data-level="3.1" data-path="3-wrangling.html"><a href="3-wrangling.html#piping"><i class="fa fa-check"></i><b>3.1</b> The pipe operator: <code>%>%</code></a></li>
<li class="chapter" data-level="3.2" data-path="3-wrangling.html"><a href="3-wrangling.html#filter"><i class="fa fa-check"></i><b>3.2</b> <code>filter</code> rows</a></li>
<li class="chapter" data-level="3.3" data-path="3-wrangling.html"><a href="3-wrangling.html#slice-rows"><i class="fa fa-check"></i><b>3.3</b> <code>slice</code> rows</a></li>
<li class="chapter" data-level="3.4" data-path="3-wrangling.html"><a href="3-wrangling.html#select"><i class="fa fa-check"></i><b>3.4</b> <code>select</code> variables</a>
<ul>
<li class="chapter" data-level="3.4.1" data-path="3-wrangling.html"><a href="3-wrangling.html#rename"><i class="fa fa-check"></i><b>3.4.1</b> <code>rename</code> variables</a></li>
</ul></li>
<li class="chapter" data-level="3.5" data-path="3-wrangling.html"><a href="3-wrangling.html#summarize"><i class="fa fa-check"></i><b>3.5</b> <code>summarize</code> variables</a></li>
<li class="chapter" data-level="3.6" data-path="3-wrangling.html"><a href="3-wrangling.html#groupby"><i class="fa fa-check"></i><b>3.6</b> <code>group_by</code> rows</a>
<ul>
<li class="chapter" data-level="3.6.1" data-path="3-wrangling.html"><a href="3-wrangling.html#grouping-by-more-than-one-variable"><i class="fa fa-check"></i><b>3.6.1</b> Grouping by more than one variable</a></li>
</ul></li>
<li class="chapter" data-level="3.7" data-path="3-wrangling.html"><a href="3-wrangling.html#mutate"><i class="fa fa-check"></i><b>3.7</b> <code>mutate</code> existing variables</a></li>
<li class="chapter" data-level="3.8" data-path="3-wrangling.html"><a href="3-wrangling.html#arrange"><i class="fa fa-check"></i><b>3.8</b> <code>arrange</code> and sort rows</a></li>
<li class="chapter" data-level="3.9" data-path="3-wrangling.html"><a href="3-wrangling.html#joins"><i class="fa fa-check"></i><b>3.9</b> <code>join</code> data frames</a></li>
<li class="chapter" data-level="3.10" data-path="3-wrangling.html"><a href="3-wrangling.html#wrangling-conclusion"><i class="fa fa-check"></i><b>3.10</b> Conclusion</a>
<ul>
<li class="chapter" data-level="3.10.1" data-path="3-wrangling.html"><a href="3-wrangling.html#summary-table-1"><i class="fa fa-check"></i><b>3.10.1</b> Summary table</a></li>
<li class="chapter" data-level="3.10.2" data-path="3-wrangling.html"><a href="3-wrangling.html#additional-resources-2"><i class="fa fa-check"></i><b>3.10.2</b> Additional resources</a></li>
<li class="chapter" data-level="3.10.3" data-path="3-wrangling.html"><a href="3-wrangling.html#whats-to-come-1"><i class="fa fa-check"></i><b>3.10.3</b> What’s to come?</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="4" data-path="4-tidy.html"><a href="4-tidy.html"><i class="fa fa-check"></i><b>4</b> Data Importing and “Tidy” Data</a>
<ul>
<li class="chapter" data-level="" data-path="4-tidy.html"><a href="4-tidy.html#tidy-packages"><i class="fa fa-check"></i>Needed packages</a></li>
<li class="chapter" data-level="4.1" data-path="4-tidy.html"><a href="4-tidy.html#csv"><i class="fa fa-check"></i><b>4.1</b> Importing data</a>
<ul>
<li class="chapter" data-level="4.1.1" data-path="4-tidy.html"><a href="4-tidy.html#using-the-console"><i class="fa fa-check"></i><b>4.1.1</b> Using the console</a></li>
<li class="chapter" data-level="4.1.2" data-path="4-tidy.html"><a href="4-tidy.html#using-rstudios-interface"><i class="fa fa-check"></i><b>4.1.2</b> Using RStudio’s interface</a></li>
</ul></li>
<li class="chapter" data-level="4.2" data-path="4-tidy.html"><a href="4-tidy.html#tidy-data-ex"><i class="fa fa-check"></i><b>4.2</b> “Tidy” data</a>
<ul>
<li class="chapter" data-level="4.2.1" data-path="4-tidy.html"><a href="4-tidy.html#tidy-definition"><i class="fa fa-check"></i><b>4.2.1</b> Definition of “tidy” data</a></li>
<li class="chapter" data-level="4.2.2" data-path="4-tidy.html"><a href="4-tidy.html#converting-to-tidy-data"><i class="fa fa-check"></i><b>4.2.2</b> Converting to “tidy” data</a></li>
</ul></li>
<li class="chapter" data-level="4.3" data-path="4-tidy.html"><a href="4-tidy.html#case-study-tidy"><i class="fa fa-check"></i><b>4.3</b> Case study: Weight loss data</a></li>
<li class="chapter" data-level="4.4" data-path="4-tidy.html"><a href="4-tidy.html#tidyverse-package"><i class="fa fa-check"></i><b>4.4</b> <code>tidyverse</code> package</a></li>
<li class="chapter" data-level="4.5" data-path="4-tidy.html"><a href="4-tidy.html#tidy-data-conclusion"><i class="fa fa-check"></i><b>4.5</b> Conclusion</a>
<ul>
<li class="chapter" data-level="4.5.1" data-path="4-tidy.html"><a href="4-tidy.html#additional-resources-3"><i class="fa fa-check"></i><b>4.5.1</b> Additional resources</a></li>
<li class="chapter" data-level="4.5.2" data-path="4-tidy.html"><a href="4-tidy.html#whats-to-come-2"><i class="fa fa-check"></i><b>4.5.2</b> What’s to come?</a></li>
</ul></li>
</ul></li>
<li class="part"><span><b>II Data Modeling with moderndive</b></span></li>
<li class="chapter" data-level="5" data-path="5-regression.html"><a href="5-regression.html"><i class="fa fa-check"></i><b>5</b> Basic Regression</a>
<ul>
<li class="chapter" data-level="" data-path="5-regression.html"><a href="5-regression.html#reg-packages"><i class="fa fa-check"></i>Needed packages</a></li>
<li class="chapter" data-level="5.1" data-path="5-regression.html"><a href="5-regression.html#model1"><i class="fa fa-check"></i><b>5.1</b> One numerical explanatory variable</a>
<ul>
<li class="chapter" data-level="5.1.1" data-path="5-regression.html"><a href="5-regression.html#model1EDA"><i class="fa fa-check"></i><b>5.1.1</b> Exploratory data analysis</a></li>
<li class="chapter" data-level="5.1.2" data-path="5-regression.html"><a href="5-regression.html#model1table"><i class="fa fa-check"></i><b>5.1.2</b> Simple linear regression</a></li>
<li class="chapter" data-level="5.1.3" data-path="5-regression.html"><a href="5-regression.html#model1points"><i class="fa fa-check"></i><b>5.1.3</b> Observed/fitted values and residuals</a></li>
</ul></li>
<li class="chapter" data-level="5.2" data-path="5-regression.html"><a href="5-regression.html#model2"><i class="fa fa-check"></i><b>5.2</b> One categorical explanatory variable</a>
<ul>
<li class="chapter" data-level="5.2.1" data-path="5-regression.html"><a href="5-regression.html#model2EDA"><i class="fa fa-check"></i><b>5.2.1</b> Exploratory data analysis</a></li>
<li class="chapter" data-level="5.2.2" data-path="5-regression.html"><a href="5-regression.html#model2table"><i class="fa fa-check"></i><b>5.2.2</b> Linear regression</a></li>
<li class="chapter" data-level="5.2.3" data-path="5-regression.html"><a href="5-regression.html#model2points"><i class="fa fa-check"></i><b>5.2.3</b> Observed/fitted values and residuals</a></li>
</ul></li>
<li class="chapter" data-level="5.3" data-path="5-regression.html"><a href="5-regression.html#reg-related-topics"><i class="fa fa-check"></i><b>5.3</b> Related topics</a>
<ul>
<li class="chapter" data-level="5.3.1" data-path="5-regression.html"><a href="5-regression.html#correlation-is-not-causation"><i class="fa fa-check"></i><b>5.3.1</b> Correlation is not necessarily causation</a></li>
<li class="chapter" data-level="5.3.2" data-path="5-regression.html"><a href="5-regression.html#leastsquares"><i class="fa fa-check"></i><b>5.3.2</b> Best-fitting line</a></li>
<li class="chapter" data-level="5.3.3" data-path="5-regression.html"><a href="5-regression.html#underthehood"><i class="fa fa-check"></i><b>5.3.3</b> <code>get_regression_x()</code> functions</a></li>
</ul></li>
<li class="chapter" data-level="5.4" data-path="5-regression.html"><a href="5-regression.html#reg-conclusion"><i class="fa fa-check"></i><b>5.4</b> Conclusion</a>
<ul>
<li class="chapter" data-level="5.4.1" data-path="5-regression.html"><a href="5-regression.html#additional-resources-basic-regression"><i class="fa fa-check"></i><b>5.4.1</b> Additional resources</a></li>
<li class="chapter" data-level="5.4.2" data-path="5-regression.html"><a href="5-regression.html#whats-to-come-4"><i class="fa fa-check"></i><b>5.4.2</b> What’s to come?</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="6" data-path="6-multiple-regression.html"><a href="6-multiple-regression.html"><i class="fa fa-check"></i><b>6</b> Multiple Regression</a>
<ul>
<li class="chapter" data-level="" data-path="6-multiple-regression.html"><a href="6-multiple-regression.html#mult-reg-packages"><i class="fa fa-check"></i>Needed packages</a></li>
<li class="chapter" data-level="6.1" data-path="6-multiple-regression.html"><a href="6-multiple-regression.html#model4"><i class="fa fa-check"></i><b>6.1</b> One numerical and one categorical explanatory variable</a>
<ul>
<li class="chapter" data-level="6.1.1" data-path="6-multiple-regression.html"><a href="6-multiple-regression.html#model4EDA"><i class="fa fa-check"></i><b>6.1.1</b> Exploratory data analysis</a></li>
<li class="chapter" data-level="6.1.2" data-path="6-multiple-regression.html"><a href="6-multiple-regression.html#model4interactiontable"><i class="fa fa-check"></i><b>6.1.2</b> Interaction model</a></li>
<li class="chapter" data-level="6.1.3" data-path="6-multiple-regression.html"><a href="6-multiple-regression.html#model4table"><i class="fa fa-check"></i><b>6.1.3</b> Parallel slopes model</a></li>
<li class="chapter" data-level="6.1.4" data-path="6-multiple-regression.html"><a href="6-multiple-regression.html#model4points"><i class="fa fa-check"></i><b>6.1.4</b> Observed/fitted values and residuals</a></li>
</ul></li>
<li class="chapter" data-level="6.2" data-path="6-multiple-regression.html"><a href="6-multiple-regression.html#model3"><i class="fa fa-check"></i><b>6.2</b> Two categorical explanatory variables</a>
<ul>
<li class="chapter" data-level="6.2.1" data-path="6-multiple-regression.html"><a href="6-multiple-regression.html#model3EDA"><i class="fa fa-check"></i><b>6.2.1</b> Exploratory data analysis</a></li>
<li class="chapter" data-level="6.2.2" data-path="6-multiple-regression.html"><a href="6-multiple-regression.html#model3table"><i class="fa fa-check"></i><b>6.2.2</b> Regression lines</a></li>
<li class="chapter" data-level="6.2.3" data-path="6-multiple-regression.html"><a href="6-multiple-regression.html#model3points"><i class="fa fa-check"></i><b>6.2.3</b> Observed/fitted values and residuals</a></li>
</ul></li>
<li class="chapter" data-level="6.3" data-path="6-multiple-regression.html"><a href="6-multiple-regression.html#mult-reg-related-topics"><i class="fa fa-check"></i><b>6.3</b> Related topics</a>
<ul>
<li class="chapter" data-level="6.3.1" data-path="6-multiple-regression.html"><a href="6-multiple-regression.html#model-selection"><i class="fa fa-check"></i><b>6.3.1</b> Model selection using visualizations</a></li>
<li class="chapter" data-level="6.3.2" data-path="6-multiple-regression.html"><a href="6-multiple-regression.html#rsquared"><i class="fa fa-check"></i><b>6.3.2</b> Model selection using R-squared</a></li>
</ul></li>
<li class="chapter" data-level="6.4" data-path="6-multiple-regression.html"><a href="6-multiple-regression.html#mult-reg-conclusion"><i class="fa fa-check"></i><b>6.4</b> Conclusion</a>
<ul>
<li class="chapter" data-level="6.4.1" data-path="6-multiple-regression.html"><a href="6-multiple-regression.html#additional-resources-4"><i class="fa fa-check"></i><b>6.4.1</b> Additional resources</a></li>
<li class="chapter" data-level="6.4.2" data-path="6-multiple-regression.html"><a href="6-multiple-regression.html#whats-to-come-5"><i class="fa fa-check"></i><b>6.4.2</b> What’s to come?</a></li>
</ul></li>
</ul></li>
<li class="part"><span><b>III Statistical Inference with infer</b></span></li>
<li class="chapter" data-level="7" data-path="7-sampling.html"><a href="7-sampling.html"><i class="fa fa-check"></i><b>7</b> Sampling</a>
<ul>
<li class="chapter" data-level="" data-path="7-sampling.html"><a href="7-sampling.html#sampling-packages"><i class="fa fa-check"></i>Needed packages</a></li>
<li class="chapter" data-level="7.1" data-path="7-sampling.html"><a href="7-sampling.html#sampling-activity"><i class="fa fa-check"></i><b>7.1</b> Sampling bowl activity</a>
<ul>
<li class="chapter" data-level="7.1.1" data-path="7-sampling.html"><a href="7-sampling.html#what-proportion-of-this-bowls-balls-are-red"><i class="fa fa-check"></i><b>7.1.1</b> What proportion of this bowl’s balls are red?</a></li>
<li class="chapter" data-level="7.1.2" data-path="7-sampling.html"><a href="7-sampling.html#using-the-shovel-once"><i class="fa fa-check"></i><b>7.1.2</b> Using the shovel once</a></li>
<li class="chapter" data-level="7.1.3" data-path="7-sampling.html"><a href="7-sampling.html#student-shovels"><i class="fa fa-check"></i><b>7.1.3</b> Using the shovel 33 times</a></li>
<li class="chapter" data-level="7.1.4" data-path="7-sampling.html"><a href="7-sampling.html#sampling-what-did-we-just-do"><i class="fa fa-check"></i><b>7.1.4</b> What did we just do?</a></li>
</ul></li>
<li class="chapter" data-level="7.2" data-path="7-sampling.html"><a href="7-sampling.html#sampling-simulation"><i class="fa fa-check"></i><b>7.2</b> Virtual sampling</a>
<ul>
<li class="chapter" data-level="7.2.1" data-path="7-sampling.html"><a href="7-sampling.html#using-the-virtual-shovel-once"><i class="fa fa-check"></i><b>7.2.1</b> Using the virtual shovel once</a></li>
</ul></li>
<li class="chapter" data-level="7.3" data-path="7-sampling.html"><a href="7-sampling.html#sampling-framework"><i class="fa fa-check"></i><b>7.3</b> Sampling framework</a>
<ul>
<li class="chapter" data-level="7.3.1" data-path="7-sampling.html"><a href="7-sampling.html#terminology-and-notation"><i class="fa fa-check"></i><b>7.3.1</b> Terminology and notation</a></li>
<li class="chapter" data-level="7.3.2" data-path="7-sampling.html"><a href="7-sampling.html#sampling-definitions"><i class="fa fa-check"></i><b>7.3.2</b> Statistical definitions</a></li>
<li class="chapter" data-level="7.3.3" data-path="7-sampling.html"><a href="7-sampling.html#moral-of-the-story"><i class="fa fa-check"></i><b>7.3.3</b> The moral of the story</a></li>
</ul></li>
<li class="chapter" data-level="7.4" data-path="7-sampling.html"><a href="7-sampling.html#sampling-case-study"><i class="fa fa-check"></i><b>7.4</b> Case study: Polls</a></li>
<li class="chapter" data-level="7.5" data-path="7-sampling.html"><a href="7-sampling.html#sampling-conclusion-central-limit-theorem"><i class="fa fa-check"></i><b>7.5</b> Central Limit Theorem</a></li>
<li class="chapter" data-level="7.6" data-path="7-sampling.html"><a href="7-sampling.html#sampling-conclusion"><i class="fa fa-check"></i><b>7.6</b> Conclusion</a>
<ul>
<li class="chapter" data-level="7.6.1" data-path="7-sampling.html"><a href="7-sampling.html#sampling-conclusion-table"><i class="fa fa-check"></i><b>7.6.1</b> Sampling scenarios</a></li>
<li class="chapter" data-level="7.6.2" data-path="7-sampling.html"><a href="7-sampling.html#additional-resources-5"><i class="fa fa-check"></i><b>7.6.2</b> Additional resources</a></li>
<li class="chapter" data-level="7.6.3" data-path="7-sampling.html"><a href="7-sampling.html#whats-to-come-6"><i class="fa fa-check"></i><b>7.6.3</b> What’s to come?</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="8" data-path="8-confidence-intervals.html"><a href="8-confidence-intervals.html"><i class="fa fa-check"></i><b>8</b> Bootstrapping and Confidence Intervals</a>
<ul>
<li class="chapter" data-level="" data-path="8-confidence-intervals.html"><a href="8-confidence-intervals.html#CI-packages"><i class="fa fa-check"></i>Needed packages</a></li>
<li class="chapter" data-level="8.1" data-path="8-confidence-intervals.html"><a href="8-confidence-intervals.html#resampling-tactile"><i class="fa fa-check"></i><b>8.1</b> Pennies activity</a>
<ul>
<li class="chapter" data-level="8.1.1" data-path="8-confidence-intervals.html"><a href="8-confidence-intervals.html#what-is-the-average-year-on-us-pennies-in-2019"><i class="fa fa-check"></i><b>8.1.1</b> What is the average year on US pennies in 2019?</a></li>
<li class="chapter" data-level="8.1.2" data-path="8-confidence-intervals.html"><a href="8-confidence-intervals.html#resampling-once"><i class="fa fa-check"></i><b>8.1.2</b> Resampling once</a></li>
<li class="chapter" data-level="8.1.3" data-path="8-confidence-intervals.html"><a href="8-confidence-intervals.html#student-resamples"><i class="fa fa-check"></i><b>8.1.3</b> Resampling 35 times</a></li>
<li class="chapter" data-level="8.1.4" data-path="8-confidence-intervals.html"><a href="8-confidence-intervals.html#ci-what-did-we-just-do"><i class="fa fa-check"></i><b>8.1.4</b> What did we just do?</a></li>
</ul></li>
<li class="chapter" data-level="8.2" data-path="8-confidence-intervals.html"><a href="8-confidence-intervals.html#resampling-simulation"><i class="fa fa-check"></i><b>8.2</b> Computer simulation of resampling</a>
<ul>
<li class="chapter" data-level="8.2.1" data-path="8-confidence-intervals.html"><a href="8-confidence-intervals.html#virtually-resampling-once"><i class="fa fa-check"></i><b>8.2.1</b> Virtually resampling once</a></li>
<li class="chapter" data-level="8.2.2" data-path="8-confidence-intervals.html"><a href="8-confidence-intervals.html#bootstrap-35-replicates"><i class="fa fa-check"></i><b>8.2.2</b> Virtually resampling 35 times</a></li>
<li class="chapter" data-level="8.2.3" data-path="8-confidence-intervals.html"><a href="8-confidence-intervals.html#bootstrap-1000-replicates"><i class="fa fa-check"></i><b>8.2.3</b> Virtually resampling 1000 times</a></li>
</ul></li>
<li class="chapter" data-level="8.3" data-path="8-confidence-intervals.html"><a href="8-confidence-intervals.html#ci-build-up"><i class="fa fa-check"></i><b>8.3</b> Understanding confidence intervals</a>
<ul>
<li class="chapter" data-level="8.3.1" data-path="8-confidence-intervals.html"><a href="8-confidence-intervals.html#percentile-method"><i class="fa fa-check"></i><b>8.3.1</b> Percentile method</a></li>
<li class="chapter" data-level="8.3.2" data-path="8-confidence-intervals.html"><a href="8-confidence-intervals.html#se-method"><i class="fa fa-check"></i><b>8.3.2</b> Standard error method</a></li>
</ul></li>
<li class="chapter" data-level="8.4" data-path="8-confidence-intervals.html"><a href="8-confidence-intervals.html#bootstrap-process"><i class="fa fa-check"></i><b>8.4</b> Constructing confidence intervals</a>
<ul>
<li class="chapter" data-level="8.4.1" data-path="8-confidence-intervals.html"><a href="8-confidence-intervals.html#original-workflow"><i class="fa fa-check"></i><b>8.4.1</b> Original workflow</a></li>
<li class="chapter" data-level="8.4.2" data-path="8-confidence-intervals.html"><a href="8-confidence-intervals.html#infer-workflow"><i class="fa fa-check"></i><b>8.4.2</b> <code>infer</code> package workflow</a></li>
<li class="chapter" data-level="8.4.3" data-path="8-confidence-intervals.html"><a href="8-confidence-intervals.html#percentile-method-infer"><i class="fa fa-check"></i><b>8.4.3</b> Percentile method with <code>infer</code></a></li>
<li class="chapter" data-level="8.4.4" data-path="8-confidence-intervals.html"><a href="8-confidence-intervals.html#infer-se"><i class="fa fa-check"></i><b>8.4.4</b> Standard error method with <code>infer</code></a></li>
</ul></li>
<li class="chapter" data-level="8.5" data-path="8-confidence-intervals.html"><a href="8-confidence-intervals.html#one-prop-ci"><i class="fa fa-check"></i><b>8.5</b> Interpreting confidence intervals</a>
<ul>
<li class="chapter" data-level="8.5.1" data-path="8-confidence-intervals.html"><a href="8-confidence-intervals.html#ilyas-yohan"><i class="fa fa-check"></i><b>8.5.1</b> Did the net capture the fish?</a></li>
<li class="chapter" data-level="8.5.2" data-path="8-confidence-intervals.html"><a href="8-confidence-intervals.html#shorthand"><i class="fa fa-check"></i><b>8.5.2</b> Precise and shorthand interpretation</a></li>
<li class="chapter" data-level="8.5.3" data-path="8-confidence-intervals.html"><a href="8-confidence-intervals.html#ci-width"><i class="fa fa-check"></i><b>8.5.3</b> Width of confidence intervals</a></li>
</ul></li>
<li class="chapter" data-level="8.6" data-path="8-confidence-intervals.html"><a href="8-confidence-intervals.html#case-study-two-prop-ci"><i class="fa fa-check"></i><b>8.6</b> Case study: Is yawning contagious?</a>
<ul>
<li class="chapter" data-level="8.6.1" data-path="8-confidence-intervals.html"><a href="8-confidence-intervals.html#mythbusters-study-data"><i class="fa fa-check"></i><b>8.6.1</b> <em>Mythbusters</em> study data</a></li>
<li class="chapter" data-level="8.6.2" data-path="8-confidence-intervals.html"><a href="8-confidence-intervals.html#sampling-scenario"><i class="fa fa-check"></i><b>8.6.2</b> Sampling scenario</a></li>
<li class="chapter" data-level="8.6.3" data-path="8-confidence-intervals.html"><a href="8-confidence-intervals.html#ci-build"><i class="fa fa-check"></i><b>8.6.3</b> Constructing the confidence interval</a></li>
<li class="chapter" data-level="8.6.4" data-path="8-confidence-intervals.html"><a href="8-confidence-intervals.html#interpreting-the-confidence-interval"><i class="fa fa-check"></i><b>8.6.4</b> Interpreting the confidence interval</a></li>
</ul></li>
<li class="chapter" data-level="8.7" data-path="8-confidence-intervals.html"><a href="8-confidence-intervals.html#ci-conclusion"><i class="fa fa-check"></i><b>8.7</b> Conclusion</a>
<ul>
<li class="chapter" data-level="8.7.1" data-path="8-confidence-intervals.html"><a href="8-confidence-intervals.html#bootstrap-vs-sampling"><i class="fa fa-check"></i><b>8.7.1</b> Comparing bootstrap and sampling distributions</a></li>
<li class="chapter" data-level="8.7.2" data-path="8-confidence-intervals.html"><a href="8-confidence-intervals.html#theory-ci"><i class="fa fa-check"></i><b>8.7.2</b> Theory-based confidence intervals</a></li>
<li class="chapter" data-level="8.7.3" data-path="8-confidence-intervals.html"><a href="8-confidence-intervals.html#additional-resources-6"><i class="fa fa-check"></i><b>8.7.3</b> Additional resources</a></li>
<li class="chapter" data-level="8.7.4" data-path="8-confidence-intervals.html"><a href="8-confidence-intervals.html#whats-to-come-7"><i class="fa fa-check"></i><b>8.7.4</b> What’s to come?</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="9" data-path="9-hypothesis-testing.html"><a href="9-hypothesis-testing.html"><i class="fa fa-check"></i><b>9</b> Hypothesis Testing</a>
<ul>
<li class="chapter" data-level="" data-path="9-hypothesis-testing.html"><a href="9-hypothesis-testing.html#nhst-packages"><i class="fa fa-check"></i>Needed packages</a></li>
<li class="chapter" data-level="9.1" data-path="9-hypothesis-testing.html"><a href="9-hypothesis-testing.html#ht-activity"><i class="fa fa-check"></i><b>9.1</b> Promotions activity</a>
<ul>
<li class="chapter" data-level="9.1.1" data-path="9-hypothesis-testing.html"><a href="9-hypothesis-testing.html#does-gender-affect-promotions-at-a-bank"><i class="fa fa-check"></i><b>9.1.1</b> Does gender affect promotions at a bank?</a></li>
<li class="chapter" data-level="9.1.2" data-path="9-hypothesis-testing.html"><a href="9-hypothesis-testing.html#shuffling-once"><i class="fa fa-check"></i><b>9.1.2</b> Shuffling once</a></li>
<li class="chapter" data-level="9.1.3" data-path="9-hypothesis-testing.html"><a href="9-hypothesis-testing.html#shuffling-16-times"><i class="fa fa-check"></i><b>9.1.3</b> Shuffling 16 times</a></li>
<li class="chapter" data-level="9.1.4" data-path="9-hypothesis-testing.html"><a href="9-hypothesis-testing.html#ht-what-did-we-just-do"><i class="fa fa-check"></i><b>9.1.4</b> What did we just do?</a></li>
</ul></li>
<li class="chapter" data-level="9.2" data-path="9-hypothesis-testing.html"><a href="9-hypothesis-testing.html#understanding-ht"><i class="fa fa-check"></i><b>9.2</b> Understanding hypothesis tests</a></li>
<li class="chapter" data-level="9.3" data-path="9-hypothesis-testing.html"><a href="9-hypothesis-testing.html#ht-infer"><i class="fa fa-check"></i><b>9.3</b> Conducting hypothesis tests</a>
<ul>
<li class="chapter" data-level="9.3.1" data-path="9-hypothesis-testing.html"><a href="9-hypothesis-testing.html#infer-workflow-ht"><i class="fa fa-check"></i><b>9.3.1</b> <code>infer</code> package workflow</a></li>
<li class="chapter" data-level="9.3.2" data-path="9-hypothesis-testing.html"><a href="9-hypothesis-testing.html#comparing-infer-workflows"><i class="fa fa-check"></i><b>9.3.2</b> Comparison with confidence intervals</a></li>
<li class="chapter" data-level="9.3.3" data-path="9-hypothesis-testing.html"><a href="9-hypothesis-testing.html#only-one-test"><i class="fa fa-check"></i><b>9.3.3</b> “There is only one test”</a></li>
</ul></li>
<li class="chapter" data-level="9.4" data-path="9-hypothesis-testing.html"><a href="9-hypothesis-testing.html#ht-interpretation"><i class="fa fa-check"></i><b>9.4</b> Interpreting hypothesis tests</a>
<ul>
<li class="chapter" data-level="9.4.1" data-path="9-hypothesis-testing.html"><a href="9-hypothesis-testing.html#trial"><i class="fa fa-check"></i><b>9.4.1</b> Two possible outcomes</a></li>
<li class="chapter" data-level="9.4.2" data-path="9-hypothesis-testing.html"><a href="9-hypothesis-testing.html#types-of-errors"><i class="fa fa-check"></i><b>9.4.2</b> Types of errors</a></li>
<li class="chapter" data-level="9.4.3" data-path="9-hypothesis-testing.html"><a href="9-hypothesis-testing.html#choosing-alpha"><i class="fa fa-check"></i><b>9.4.3</b> How do we choose alpha?</a></li>
</ul></li>
<li class="chapter" data-level="9.5" data-path="9-hypothesis-testing.html"><a href="9-hypothesis-testing.html#ht-case-study"><i class="fa fa-check"></i><b>9.5</b> Case study: Are action or romance movies rated higher?</a>
<ul>
<li class="chapter" data-level="9.5.1" data-path="9-hypothesis-testing.html"><a href="9-hypothesis-testing.html#imdb-data"><i class="fa fa-check"></i><b>9.5.1</b> IMDb ratings data</a></li>
<li class="chapter" data-level="9.5.2" data-path="9-hypothesis-testing.html"><a href="9-hypothesis-testing.html#sampling-scenario-1"><i class="fa fa-check"></i><b>9.5.2</b> Sampling scenario</a></li>
<li class="chapter" data-level="9.5.3" data-path="9-hypothesis-testing.html"><a href="9-hypothesis-testing.html#conducting-the-hypothesis-test"><i class="fa fa-check"></i><b>9.5.3</b> Conducting the hypothesis test</a></li>
</ul></li>
<li class="chapter" data-level="9.6" data-path="9-hypothesis-testing.html"><a href="9-hypothesis-testing.html#nhst-conclusion"><i class="fa fa-check"></i><b>9.6</b> Conclusion</a>
<ul>
<li class="chapter" data-level="9.6.1" data-path="9-hypothesis-testing.html"><a href="9-hypothesis-testing.html#theory-hypo"><i class="fa fa-check"></i><b>9.6.1</b> Theory-based hypothesis tests</a></li>
<li class="chapter" data-level="9.6.2" data-path="9-hypothesis-testing.html"><a href="9-hypothesis-testing.html#when-inference-is-not-needed"><i class="fa fa-check"></i><b>9.6.2</b> When inference is not needed</a></li>
<li class="chapter" data-level="9.6.3" data-path="9-hypothesis-testing.html"><a href="9-hypothesis-testing.html#problems-with-p-values"><i class="fa fa-check"></i><b>9.6.3</b> Problems with p-values</a></li>
<li class="chapter" data-level="9.6.4" data-path="9-hypothesis-testing.html"><a href="9-hypothesis-testing.html#additional-resources-7"><i class="fa fa-check"></i><b>9.6.4</b> Additional resources</a></li>
<li class="chapter" data-level="9.6.5" data-path="9-hypothesis-testing.html"><a href="9-hypothesis-testing.html#whats-to-come-8"><i class="fa fa-check"></i><b>9.6.5</b> What’s to come</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="10" data-path="10-inference-for-regression.html"><a href="10-inference-for-regression.html"><i class="fa fa-check"></i><b>10</b> Inference for Regression</a>
<ul>
<li class="chapter" data-level="" data-path="10-inference-for-regression.html"><a href="10-inference-for-regression.html#inf-packages"><i class="fa fa-check"></i>Needed packages</a></li>
<li class="chapter" data-level="10.1" data-path="10-inference-for-regression.html"><a href="10-inference-for-regression.html#regression-refresher"><i class="fa fa-check"></i><b>10.1</b> Regression refresher</a>
<ul>
<li class="chapter" data-level="10.1.1" data-path="10-inference-for-regression.html"><a href="10-inference-for-regression.html#teaching-evaluations-analysis"><i class="fa fa-check"></i><b>10.1.1</b> Teaching evaluations analysis</a></li>
<li class="chapter" data-level="10.1.2" data-path="10-inference-for-regression.html"><a href="10-inference-for-regression.html#sampling-scenario-2"><i class="fa fa-check"></i><b>10.1.2</b> Sampling scenario</a></li>
</ul></li>
<li class="chapter" data-level="10.2" data-path="10-inference-for-regression.html"><a href="10-inference-for-regression.html#regression-interp"><i class="fa fa-check"></i><b>10.2</b> Interpreting regression tables</a>
<ul>
<li class="chapter" data-level="10.2.1" data-path="10-inference-for-regression.html"><a href="10-inference-for-regression.html#regression-se"><i class="fa fa-check"></i><b>10.2.1</b> Standard error</a></li>
<li class="chapter" data-level="10.2.2" data-path="10-inference-for-regression.html"><a href="10-inference-for-regression.html#regression-test-statistic"><i class="fa fa-check"></i><b>10.2.2</b> Test statistic</a></li>
<li class="chapter" data-level="10.2.3" data-path="10-inference-for-regression.html"><a href="10-inference-for-regression.html#p-value"><i class="fa fa-check"></i><b>10.2.3</b> p-value</a></li>
<li class="chapter" data-level="10.2.4" data-path="10-inference-for-regression.html"><a href="10-inference-for-regression.html#confidence-interval"><i class="fa fa-check"></i><b>10.2.4</b> Confidence interval</a></li>
<li class="chapter" data-level="10.2.5" data-path="10-inference-for-regression.html"><a href="10-inference-for-regression.html#regression-table-computation"><i class="fa fa-check"></i><b>10.2.5</b> How does R compute the table?</a></li>
</ul></li>
<li class="chapter" data-level="10.3" data-path="10-inference-for-regression.html"><a href="10-inference-for-regression.html#regression-conditions"><i class="fa fa-check"></i><b>10.3</b> Conditions for inference for regression</a>
<ul>
<li class="chapter" data-level="10.3.1" data-path="10-inference-for-regression.html"><a href="10-inference-for-regression.html#residuals-refresher"><i class="fa fa-check"></i><b>10.3.1</b> Residuals refresher</a></li>
<li class="chapter" data-level="10.3.2" data-path="10-inference-for-regression.html"><a href="10-inference-for-regression.html#linearity-of-relationship"><i class="fa fa-check"></i><b>10.3.2</b> Linearity of relationship</a></li>
<li class="chapter" data-level="10.3.3" data-path="10-inference-for-regression.html"><a href="10-inference-for-regression.html#independence-of-residuals"><i class="fa fa-check"></i><b>10.3.3</b> Independence of residuals</a></li>
<li class="chapter" data-level="10.3.4" data-path="10-inference-for-regression.html"><a href="10-inference-for-regression.html#normality-of-residuals"><i class="fa fa-check"></i><b>10.3.4</b> Normality of residuals</a></li>
<li class="chapter" data-level="10.3.5" data-path="10-inference-for-regression.html"><a href="10-inference-for-regression.html#equality-of-variance"><i class="fa fa-check"></i><b>10.3.5</b> Equality of variance</a></li>
<li class="chapter" data-level="10.3.6" data-path="10-inference-for-regression.html"><a href="10-inference-for-regression.html#what-is-the-conclusion"><i class="fa fa-check"></i><b>10.3.6</b> What’s the conclusion?</a></li>
</ul></li>
<li class="chapter" data-level="10.4" data-path="10-inference-for-regression.html"><a href="10-inference-for-regression.html#infer-regression"><i class="fa fa-check"></i><b>10.4</b> Simulation-based inference for regression</a>
<ul>
<li class="chapter" data-level="10.4.1" data-path="10-inference-for-regression.html"><a href="10-inference-for-regression.html#confidence-interval-for-slope"><i class="fa fa-check"></i><b>10.4.1</b> Confidence interval for slope</a></li>
<li class="chapter" data-level="10.4.2" data-path="10-inference-for-regression.html"><a href="10-inference-for-regression.html#hypothesis-test-for-slope"><i class="fa fa-check"></i><b>10.4.2</b> Hypothesis test for slope</a></li>
</ul></li>
<li class="chapter" data-level="10.5" data-path="10-inference-for-regression.html"><a href="10-inference-for-regression.html#inference-conclusion"><i class="fa fa-check"></i><b>10.5</b> Conclusion</a>
<ul>
<li class="chapter" data-level="10.5.1" data-path="10-inference-for-regression.html"><a href="10-inference-for-regression.html#theory-regression"><i class="fa fa-check"></i><b>10.5.1</b> Theory-based inference for regression</a></li>
<li class="chapter" data-level="10.5.2" data-path="10-inference-for-regression.html"><a href="10-inference-for-regression.html#summary-of-statistical-inference"><i class="fa fa-check"></i><b>10.5.2</b> Summary of statistical inference</a></li>
<li class="chapter" data-level="10.5.3" data-path="10-inference-for-regression.html"><a href="10-inference-for-regression.html#additional-resources-8"><i class="fa fa-check"></i><b>10.5.3</b> Additional resources</a></li>
<li class="chapter" data-level="10.5.4" data-path="10-inference-for-regression.html"><a href="10-inference-for-regression.html#whats-to-come-9"><i class="fa fa-check"></i><b>10.5.4</b> What’s to come</a></li>
</ul></li>
</ul></li>
<li class="part"><span><b>IV Conclusion</b></span></li>
<li class="chapter" data-level="11" data-path="11-thinking-with-data.html"><a href="11-thinking-with-data.html"><i class="fa fa-check"></i><b>11</b> Tell Your Story with Data</a>
<ul>
<li class="chapter" data-level="11.1" data-path="11-thinking-with-data.html"><a href="11-thinking-with-data.html#review"><i class="fa fa-check"></i><b>11.1</b> Review</a>
<ul>
<li class="chapter" data-level="" data-path="11-thinking-with-data.html"><a href="11-thinking-with-data.html#story-packages"><i class="fa fa-check"></i>Needed packages</a></li>
</ul></li>
<li class="chapter" data-level="11.2" data-path="11-thinking-with-data.html"><a href="11-thinking-with-data.html#seattle-house-prices"><i class="fa fa-check"></i><b>11.2</b> Case study: Seattle house prices</a>
<ul>
<li class="chapter" data-level="11.2.1" data-path="11-thinking-with-data.html"><a href="11-thinking-with-data.html#house-prices-EDA-I"><i class="fa fa-check"></i><b>11.2.1</b> Exploratory data analysis: Part I</a></li>
<li class="chapter" data-level="11.2.2" data-path="11-thinking-with-data.html"><a href="11-thinking-with-data.html#house-prices-EDA-II"><i class="fa fa-check"></i><b>11.2.2</b> Exploratory data analysis: Part II</a></li>
<li class="chapter" data-level="11.2.3" data-path="11-thinking-with-data.html"><a href="11-thinking-with-data.html#house-prices-regression"><i class="fa fa-check"></i><b>11.2.3</b> Regression modeling</a></li>
<li class="chapter" data-level="11.2.4" data-path="11-thinking-with-data.html"><a href="11-thinking-with-data.html#house-prices-making-predictions"><i class="fa fa-check"></i><b>11.2.4</b> Making predictions</a></li>
</ul></li>
<li class="chapter" data-level="11.3" data-path="11-thinking-with-data.html"><a href="11-thinking-with-data.html#data-journalism"><i class="fa fa-check"></i><b>11.3</b> Case study: Effective data storytelling</a>
<ul>
<li class="chapter" data-level="11.3.1" data-path="11-thinking-with-data.html"><a href="11-thinking-with-data.html#bechdel-test-for-hollywood-gender-representation"><i class="fa fa-check"></i><b>11.3.1</b> Bechdel test for Hollywood gender representation</a></li>
<li class="chapter" data-level="11.3.2" data-path="11-thinking-with-data.html"><a href="11-thinking-with-data.html#us-births-in-1999"><i class="fa fa-check"></i><b>11.3.2</b> US Births in 1999</a></li>
<li class="chapter" data-level="11.3.3" data-path="11-thinking-with-data.html"><a href="11-thinking-with-data.html#scripts-of-r-code"><i class="fa fa-check"></i><b>11.3.3</b> Scripts of R code</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="11-thinking-with-data.html"><a href="11-thinking-with-data.html#concluding-remarks"><i class="fa fa-check"></i>Concluding remarks</a></li>
</ul></li>
<li class="appendix"><span><b>Appendix</b></span></li>
<li class="chapter" data-level="A" data-path="A-appendixA.html"><a href="A-appendixA.html"><i class="fa fa-check"></i><b>A</b> Statistical Background</a>
<ul>
<li class="chapter" data-level="A.1" data-path="A-appendixA.html"><a href="A-appendixA.html#appendix-stat-terms"><i class="fa fa-check"></i><b>A.1</b> Basic statistical terms</a>
<ul>
<li class="chapter" data-level="A.1.1" data-path="A-appendixA.html"><a href="A-appendixA.html#mean"><i class="fa fa-check"></i><b>A.1.1</b> Mean</a></li>
<li class="chapter" data-level="A.1.2" data-path="A-appendixA.html"><a href="A-appendixA.html#median"><i class="fa fa-check"></i><b>A.1.2</b> Median</a></li>
<li class="chapter" data-level="A.1.3" data-path="A-appendixA.html"><a href="A-appendixA.html#appendix-sd-variance"><i class="fa fa-check"></i><b>A.1.3</b> Standard deviation and variance</a></li>
<li class="chapter" data-level="A.1.4" data-path="A-appendixA.html"><a href="A-appendixA.html#five-number-summary"><i class="fa fa-check"></i><b>A.1.4</b> Five-number summary</a></li>
<li class="chapter" data-level="A.1.5" data-path="A-appendixA.html"><a href="A-appendixA.html#distribution"><i class="fa fa-check"></i><b>A.1.5</b> Distribution</a></li>
<li class="chapter" data-level="A.1.6" data-path="A-appendixA.html"><a href="A-appendixA.html#outliers"><i class="fa fa-check"></i><b>A.1.6</b> Outliers</a></li>
</ul></li>
<li class="chapter" data-level="A.2" data-path="A-appendixA.html"><a href="A-appendixA.html#appendix-normal-curve"><i class="fa fa-check"></i><b>A.2</b> Normal distribution</a></li>
<li class="chapter" data-level="A.3" data-path="A-appendixA.html"><a href="A-appendixA.html#appendix-log10-transformations"><i class="fa fa-check"></i><b>A.3</b> log10 transformations</a></li>
</ul></li>
<li class="chapter" data-level="B" data-path="B-appendixB.html"><a href="B-appendixB.html"><i class="fa fa-check"></i><b>B</b> Inference Examples</a>
<ul>
<li class="chapter" data-level="" data-path="B-appendixB.html"><a href="B-appendixB.html#needed-packages-1"><i class="fa fa-check"></i>Needed packages</a></li>
<li class="chapter" data-level="B.1" data-path="B-appendixB.html"><a href="B-appendixB.html#inference-mind-map"><i class="fa fa-check"></i><b>B.1</b> Inference mind map</a></li>
<li class="chapter" data-level="B.2" data-path="B-appendixB.html"><a href="B-appendixB.html#one-mean"><i class="fa fa-check"></i><b>B.2</b> One mean</a>
<ul>
<li class="chapter" data-level="B.2.1" data-path="B-appendixB.html"><a href="B-appendixB.html#problem-statement"><i class="fa fa-check"></i><b>B.2.1</b> Problem statement</a></li>
<li class="chapter" data-level="B.2.2" data-path="B-appendixB.html"><a href="B-appendixB.html#competing-hypotheses"><i class="fa fa-check"></i><b>B.2.2</b> Competing hypotheses</a></li>
<li class="chapter" data-level="B.2.3" data-path="B-appendixB.html"><a href="B-appendixB.html#exploring-the-sample-data"><i class="fa fa-check"></i><b>B.2.3</b> Exploring the sample data</a></li>
<li class="chapter" data-level="B.2.4" data-path="B-appendixB.html"><a href="B-appendixB.html#non-traditional-methods"><i class="fa fa-check"></i><b>B.2.4</b> Non-traditional methods</a></li>
<li class="chapter" data-level="B.2.5" data-path="B-appendixB.html"><a href="B-appendixB.html#traditional-methods"><i class="fa fa-check"></i><b>B.2.5</b> Traditional methods</a></li>
<li class="chapter" data-level="B.2.6" data-path="B-appendixB.html"><a href="B-appendixB.html#comparing-results"><i class="fa fa-check"></i><b>B.2.6</b> Comparing results</a></li>
</ul></li>
<li class="chapter" data-level="B.3" data-path="B-appendixB.html"><a href="B-appendixB.html#one-proportion"><i class="fa fa-check"></i><b>B.3</b> One proportion</a>
<ul>
<li class="chapter" data-level="B.3.1" data-path="B-appendixB.html"><a href="B-appendixB.html#problem-statement-1"><i class="fa fa-check"></i><b>B.3.1</b> Problem statement</a></li>
<li class="chapter" data-level="B.3.2" data-path="B-appendixB.html"><a href="B-appendixB.html#competing-hypotheses-1"><i class="fa fa-check"></i><b>B.3.2</b> Competing hypotheses</a></li>
<li class="chapter" data-level="B.3.3" data-path="B-appendixB.html"><a href="B-appendixB.html#exploring-the-sample-data-1"><i class="fa fa-check"></i><b>B.3.3</b> Exploring the sample data</a></li>
<li class="chapter" data-level="B.3.4" data-path="B-appendixB.html"><a href="B-appendixB.html#non-traditional-methods-1"><i class="fa fa-check"></i><b>B.3.4</b> Non-traditional methods</a></li>
<li class="chapter" data-level="B.3.5" data-path="B-appendixB.html"><a href="B-appendixB.html#traditional-methods-1"><i class="fa fa-check"></i><b>B.3.5</b> Traditional methods</a></li>
<li class="chapter" data-level="B.3.6" data-path="B-appendixB.html"><a href="B-appendixB.html#comparing-results-1"><i class="fa fa-check"></i><b>B.3.6</b> Comparing results</a></li>
</ul></li>
<li class="chapter" data-level="B.4" data-path="B-appendixB.html"><a href="B-appendixB.html#two-proportions"><i class="fa fa-check"></i><b>B.4</b> Two proportions</a>
<ul>
<li class="chapter" data-level="B.4.1" data-path="B-appendixB.html"><a href="B-appendixB.html#problem-statement-2"><i class="fa fa-check"></i><b>B.4.1</b> Problem statement</a></li>
<li class="chapter" data-level="B.4.2" data-path="B-appendixB.html"><a href="B-appendixB.html#competing-hypotheses-2"><i class="fa fa-check"></i><b>B.4.2</b> Competing hypotheses</a></li>
<li class="chapter" data-level="B.4.3" data-path="B-appendixB.html"><a href="B-appendixB.html#exploring-the-sample-data-2"><i class="fa fa-check"></i><b>B.4.3</b> Exploring the sample data</a></li>
<li class="chapter" data-level="B.4.4" data-path="B-appendixB.html"><a href="B-appendixB.html#non-traditional-methods-2"><i class="fa fa-check"></i><b>B.4.4</b> Non-traditional methods</a></li>
<li class="chapter" data-level="B.4.5" data-path="B-appendixB.html"><a href="B-appendixB.html#traditional-methods-2"><i class="fa fa-check"></i><b>B.4.5</b> Traditional methods</a></li>
<li class="chapter" data-level="B.4.6" data-path="B-appendixB.html"><a href="B-appendixB.html#test-statistic-2"><i class="fa fa-check"></i><b>B.4.6</b> Test statistic</a></li>
<li class="chapter" data-level="B.4.7" data-path="B-appendixB.html"><a href="B-appendixB.html#state-conclusion-2"><i class="fa fa-check"></i><b>B.4.7</b> State conclusion</a></li>
<li class="chapter" data-level="B.4.8" data-path="B-appendixB.html"><a href="B-appendixB.html#comparing-results-2"><i class="fa fa-check"></i><b>B.4.8</b> Comparing results</a></li>
</ul></li>
<li class="chapter" data-level="B.5" data-path="B-appendixB.html"><a href="B-appendixB.html#two-means-independent-samples"><i class="fa fa-check"></i><b>B.5</b> Two means (independent samples)</a>
<ul>
<li class="chapter" data-level="B.5.1" data-path="B-appendixB.html"><a href="B-appendixB.html#problem-statement-3"><i class="fa fa-check"></i><b>B.5.1</b> Problem statement</a></li>
<li class="chapter" data-level="B.5.2" data-path="B-appendixB.html"><a href="B-appendixB.html#competing-hypotheses-3"><i class="fa fa-check"></i><b>B.5.2</b> Competing hypotheses</a></li>
<li class="chapter" data-level="B.5.3" data-path="B-appendixB.html"><a href="B-appendixB.html#exploring-the-sample-data-3"><i class="fa fa-check"></i><b>B.5.3</b> Exploring the sample data</a></li>
<li class="chapter" data-level="B.5.4" data-path="B-appendixB.html"><a href="B-appendixB.html#non-traditional-methods-3"><i class="fa fa-check"></i><b>B.5.4</b> Non-traditional methods</a></li>
<li class="chapter" data-level="B.5.5" data-path="B-appendixB.html"><a href="B-appendixB.html#traditional-methods-3"><i class="fa fa-check"></i><b>B.5.5</b> Traditional methods</a></li>
<li class="chapter" data-level="B.5.6" data-path="B-appendixB.html"><a href="B-appendixB.html#test-statistic-3"><i class="fa fa-check"></i><b>B.5.6</b> Test statistic</a></li>
<li class="chapter" data-level="B.5.7" data-path="B-appendixB.html"><a href="B-appendixB.html#compute-p-value-1"><i class="fa fa-check"></i><b>B.5.7</b> Compute <span class="math inline">\(p\)</span>-value</a></li>
<li class="chapter" data-level="B.5.8" data-path="B-appendixB.html"><a href="B-appendixB.html#state-conclusion-3"><i class="fa fa-check"></i><b>B.5.8</b> State conclusion</a></li>
<li class="chapter" data-level="B.5.9" data-path="B-appendixB.html"><a href="B-appendixB.html#comparing-results-3"><i class="fa fa-check"></i><b>B.5.9</b> Comparing results</a></li>
</ul></li>
<li class="chapter" data-level="B.6" data-path="B-appendixB.html"><a href="B-appendixB.html#two-means-paired-samples"><i class="fa fa-check"></i><b>B.6</b> Two means (paired samples)</a>
<ul>
<li class="chapter" data-level="" data-path="B-appendixB.html"><a href="B-appendixB.html#problem-statement-4"><i class="fa fa-check"></i>Problem statement</a></li>
<li class="chapter" data-level="B.6.1" data-path="B-appendixB.html"><a href="B-appendixB.html#competing-hypotheses-4"><i class="fa fa-check"></i><b>B.6.1</b> Competing hypotheses</a></li>
<li class="chapter" data-level="B.6.2" data-path="B-appendixB.html"><a href="B-appendixB.html#exploring-the-sample-data-4"><i class="fa fa-check"></i><b>B.6.2</b> Exploring the sample data</a></li>
<li class="chapter" data-level="B.6.3" data-path="B-appendixB.html"><a href="B-appendixB.html#non-traditional-methods-4"><i class="fa fa-check"></i><b>B.6.3</b> Non-traditional methods</a></li>
<li class="chapter" data-level="B.6.4" data-path="B-appendixB.html"><a href="B-appendixB.html#traditional-methods-4"><i class="fa fa-check"></i><b>B.6.4</b> Traditional methods</a></li>
<li class="chapter" data-level="B.6.5" data-path="B-appendixB.html"><a href="B-appendixB.html#comparing-results-4"><i class="fa fa-check"></i><b>B.6.5</b> Comparing results</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="C" data-path="C-appendixC.html"><a href="C-appendixC.html"><i class="fa fa-check"></i><b>C</b> Tips and Tricks</a>
<ul>
<li class="chapter" data-level="" data-path="C-appendixC.html"><a href="C-appendixC.html#needed-packages-2"><i class="fa fa-check"></i>Needed packages</a></li>
<li class="chapter" data-level="C.1" data-path="C-appendixC.html"><a href="C-appendixC.html#data-wrangling"><i class="fa fa-check"></i><b>C.1</b> Data wrangling</a>
<ul>
<li class="chapter" data-level="C.1.1" data-path="C-appendixC.html"><a href="C-appendixC.html#appendix-missing-values"><i class="fa fa-check"></i><b>C.1.1</b> Dealing with missing values</a></li>
<li class="chapter" data-level="C.1.2" data-path="C-appendixC.html"><a href="C-appendixC.html#appendix-reordering-bars"><i class="fa fa-check"></i><b>C.1.2</b> Reordering bars in a barplot</a></li>
<li class="chapter" data-level="C.1.3" data-path="C-appendixC.html"><a href="C-appendixC.html#appendix-money-on-axis"><i class="fa fa-check"></i><b>C.1.3</b> Showing money on an axis</a></li>
<li class="chapter" data-level="C.1.4" data-path="C-appendixC.html"><a href="C-appendixC.html#appendix-changing-values"><i class="fa fa-check"></i><b>C.1.4</b> Changing values inside cells</a></li>
<li class="chapter" data-level="C.1.5" data-path="C-appendixC.html"><a href="C-appendixC.html#appendix-convert-numerical-categorical"><i class="fa fa-check"></i><b>C.1.5</b> Converting a numerical variable to a categorical one</a></li>
<li class="chapter" data-level="C.1.6" data-path="C-appendixC.html"><a href="C-appendixC.html#appendix-prop"><i class="fa fa-check"></i><b>C.1.6</b> Computing proportions</a></li>
<li class="chapter" data-level="C.1.7" data-path="C-appendixC.html"><a href="C-appendixC.html#appendix-commas"><i class="fa fa-check"></i><b>C.1.7</b> Dealing with %, commas, and $</a></li>
</ul></li>
<li class="chapter" data-level="C.2" data-path="C-appendixC.html"><a href="C-appendixC.html#interactive-graphics"><i class="fa fa-check"></i><b>C.2</b> Interactive graphics</a>
<ul>
<li class="chapter" data-level="C.2.1" data-path="C-appendixC.html"><a href="C-appendixC.html#interactive-linegraphs"><i class="fa fa-check"></i><b>C.2.1</b> Interactive linegraphs</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="D" data-path="D-appendixD.html"><a href="D-appendixD.html"><i class="fa fa-check"></i><b>D</b> Learning Check Solutions</a>
<ul>
<li class="chapter" data-level="D.1" data-path="D-appendixD.html"><a href="D-appendixD.html#chapter-1-solutions"><i class="fa fa-check"></i><b>D.1</b> Chapter 1 Solutions</a></li>
</ul></li>
<li class="chapter" data-level="E" data-path="E-appendixE.html"><a href="E-appendixE.html"><i class="fa fa-check"></i><b>E</b> Versions of R Packages Used</a></li>
<li class="chapter" data-level="" data-path="references.html"><a href="references.html"><i class="fa fa-check"></i>References</a></li>
</ul>
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<div id="appendixB" class="section level1" number="13">
<h1><span class="header-section-number">B</span> Inference Examples</h1>
<p>This appendix is designed to provide you with examples of the five basic hypothesis tests and their corresponding confidence intervals. Traditional theory-based methods as well as computational-based methods are presented. <!-- You can also use this appendix as a way to check for understanding of which statistical graphic is most appropriate given the problem set-up. --></p>
<div class="learncheck">
<p>
<strong>Note: This appendix is still under construction. If you would like to contribute, please check us out on GitHub at <a href="https://github.com/moderndive/moderndive_book" class="uri">https://github.com/moderndive/moderndive_book</a>.</strong>
</p>
</div>
<div id="needed-packages-1" class="section level3 unnumbered">
<h3>Needed packages</h3>
<div class="sourceCode" id="cb458"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb458-1"><a href="B-appendixB.html#cb458-1" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(tidyverse)</span>
<span id="cb458-2"><a href="B-appendixB.html#cb458-2" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(infer)</span>
<span id="cb458-3"><a href="B-appendixB.html#cb458-3" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(janitor)</span></code></pre></div>
</div>
<div id="inference-mind-map" class="section level2" number="13.1">
<h2><span class="header-section-number">B.1</span> Inference mind map</h2>
<p>To help you better navigate and choose the appropriate analysis, we’ve created a mind map on <a href="http://coggle.it" class="uri">http://coggle.it</a> available <a href="https://coggle.it/diagram/Vxlydu1akQFeqo6-">here</a> and below.</p>
<div class="figure" style="text-align: center"><span id="fig:infer-map"></span>
<iframe src="https://coggle.it/diagram/Vxlydu1akQFeqo6-" width="200%" height="1200pt">
</iframe>
<p class="caption">
FIGURE B.1: Mind map for Inference.
</p>
</div>
</div>
<div id="one-mean" class="section level2" number="13.2">
<h2><span class="header-section-number">B.2</span> One mean</h2>
<div id="problem-statement" class="section level3" number="13.2.1">
<h3><span class="header-section-number">B.2.1</span> Problem statement</h3>
<p>The National Survey of Family Growth conducted by the
Centers for Disease Control gathers information on family life, marriage and divorce, pregnancy,
infertility, use of contraception, and men’s and women’s health. One of the variables collected on
this survey is the age at first marriage. 5,534 randomly sampled US women between 2006 and 2010 completed the survey. The women sampled here had been married at least once. Do we have evidence that the mean age of first marriage for all US women from 2006 to 2010 is greater than 23 years? <span class="citation">(Tweaked a bit from <a href="#ref-isrs2014" role="doc-biblioref">Diez, Barr, and Çetinkaya-Rundel 2014</a> [Chapter 4])</span></p>
</div>
<div id="competing-hypotheses" class="section level3" number="13.2.2">
<h3><span class="header-section-number">B.2.2</span> Competing hypotheses</h3>
<div id="in-words" class="section level4 unnumbered">
<h4>In words</h4>
<ul>
<li>Null hypothesis: The mean age of first marriage for all US women from 2006 to 2010 is equal to 23 years.</li>
<li>Alternative hypothesis: The mean age of first marriage for all US women from 2006 to 2010 is greater than 23 years.</li>
</ul>
</div>
<div id="in-symbols-with-annotations" class="section level4 unnumbered">
<h4>In symbols (with annotations)</h4>
<ul>
<li><span class="math inline">\(H_0: \mu = \mu_{0}\)</span>, where <span class="math inline">\(\mu\)</span> represents the mean age of first marriage for all US women from 2006 to 2010 and <span class="math inline">\(\mu_0\)</span> is 23.</li>
<li><span class="math inline">\(H_A: \mu > 23\)</span></li>
</ul>
</div>
<div id="set-alpha" class="section level4 unnumbered">
<h4>Set <span class="math inline">\(\alpha\)</span></h4>
<p>It’s important to set the significance level before starting the testing using the data. Let’s set the significance level at 5% here.</p>
</div>
</div>
<div id="exploring-the-sample-data" class="section level3" number="13.2.3">
<h3><span class="header-section-number">B.2.3</span> Exploring the sample data</h3>
<div class="sourceCode" id="cb459"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb459-1"><a href="B-appendixB.html#cb459-1" aria-hidden="true" tabindex="-1"></a>age_at_marriage <span class="ot"><-</span> <span class="fu">read_csv</span>(<span class="st">"https://moderndive.com/data/ageAtMar.csv"</span>)</span></code></pre></div>
<div class="sourceCode" id="cb460"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb460-1"><a href="B-appendixB.html#cb460-1" aria-hidden="true" tabindex="-1"></a>age_summ <span class="ot"><-</span> age_at_marriage <span class="sc">%>%</span></span>
<span id="cb460-2"><a href="B-appendixB.html#cb460-2" aria-hidden="true" tabindex="-1"></a> <span class="fu">summarize</span>(</span>
<span id="cb460-3"><a href="B-appendixB.html#cb460-3" aria-hidden="true" tabindex="-1"></a> <span class="at">sample_size =</span> <span class="fu">n</span>(),</span>
<span id="cb460-4"><a href="B-appendixB.html#cb460-4" aria-hidden="true" tabindex="-1"></a> <span class="at">mean =</span> <span class="fu">mean</span>(age),</span>
<span id="cb460-5"><a href="B-appendixB.html#cb460-5" aria-hidden="true" tabindex="-1"></a> <span class="at">sd =</span> <span class="fu">sd</span>(age),</span>
<span id="cb460-6"><a href="B-appendixB.html#cb460-6" aria-hidden="true" tabindex="-1"></a> <span class="at">minimum =</span> <span class="fu">min</span>(age),</span>
<span id="cb460-7"><a href="B-appendixB.html#cb460-7" aria-hidden="true" tabindex="-1"></a> <span class="at">lower_quartile =</span> <span class="fu">quantile</span>(age, <span class="fl">0.25</span>),</span>
<span id="cb460-8"><a href="B-appendixB.html#cb460-8" aria-hidden="true" tabindex="-1"></a> <span class="at">median =</span> <span class="fu">median</span>(age),</span>
<span id="cb460-9"><a href="B-appendixB.html#cb460-9" aria-hidden="true" tabindex="-1"></a> <span class="at">upper_quartile =</span> <span class="fu">quantile</span>(age, <span class="fl">0.75</span>),</span>
<span id="cb460-10"><a href="B-appendixB.html#cb460-10" aria-hidden="true" tabindex="-1"></a> <span class="at">max =</span> <span class="fu">max</span>(age)</span>
<span id="cb460-11"><a href="B-appendixB.html#cb460-11" aria-hidden="true" tabindex="-1"></a> )</span>
<span id="cb460-12"><a href="B-appendixB.html#cb460-12" aria-hidden="true" tabindex="-1"></a><span class="fu">kable</span>(age_summ) <span class="sc">%>%</span></span>
<span id="cb460-13"><a href="B-appendixB.html#cb460-13" aria-hidden="true" tabindex="-1"></a> <span class="fu">kable_styling</span>(</span>
<span id="cb460-14"><a href="B-appendixB.html#cb460-14" aria-hidden="true" tabindex="-1"></a> <span class="at">font_size =</span> <span class="fu">ifelse</span>(<span class="fu">is_latex_output</span>(), <span class="dv">10</span>, <span class="dv">16</span>),</span>
<span id="cb460-15"><a href="B-appendixB.html#cb460-15" aria-hidden="true" tabindex="-1"></a> <span class="at">latex_options =</span> <span class="fu">c</span>(<span class="st">"hold_position"</span>)</span>
<span id="cb460-16"><a href="B-appendixB.html#cb460-16" aria-hidden="true" tabindex="-1"></a> )</span></code></pre></div>
<table class="table" style="font-size: 16px; margin-left: auto; margin-right: auto;">
<thead>
<tr>
<th style="text-align:right;">
sample_size
</th>
<th style="text-align:right;">
mean
</th>
<th style="text-align:right;">
sd
</th>
<th style="text-align:right;">
minimum
</th>
<th style="text-align:right;">
lower_quartile
</th>
<th style="text-align:right;">
median
</th>
<th style="text-align:right;">
upper_quartile
</th>
<th style="text-align:right;">
max
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:right;">
5534
</td>
<td style="text-align:right;">
23.4
</td>
<td style="text-align:right;">
4.72
</td>
<td style="text-align:right;">
10
</td>
<td style="text-align:right;">
20
</td>
<td style="text-align:right;">
23
</td>
<td style="text-align:right;">
26
</td>
<td style="text-align:right;">
43
</td>
</tr>
</tbody>
</table>
<p>The histogram below also shows the distribution of <code>age</code>.</p>
<div class="sourceCode" id="cb461"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb461-1"><a href="B-appendixB.html#cb461-1" aria-hidden="true" tabindex="-1"></a><span class="fu">ggplot</span>(<span class="at">data =</span> age_at_marriage, <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> age)) <span class="sc">+</span></span>
<span id="cb461-2"><a href="B-appendixB.html#cb461-2" aria-hidden="true" tabindex="-1"></a> <span class="fu">geom_histogram</span>(<span class="at">binwidth =</span> <span class="dv">3</span>, <span class="at">color =</span> <span class="st">"white"</span>)</span></code></pre></div>
<p><img src="ModernDive_files/figure-html/hist1b-1.png" width="\textwidth" style="display: block; margin: auto;" /></p>
<p>The observed statistic of interest here is the sample mean:</p>
<div class="sourceCode" id="cb462"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb462-1"><a href="B-appendixB.html#cb462-1" aria-hidden="true" tabindex="-1"></a>x_bar <span class="ot"><-</span> age_at_marriage <span class="sc">%>%</span></span>
<span id="cb462-2"><a href="B-appendixB.html#cb462-2" aria-hidden="true" tabindex="-1"></a> <span class="fu">specify</span>(<span class="at">response =</span> age) <span class="sc">%>%</span></span>
<span id="cb462-3"><a href="B-appendixB.html#cb462-3" aria-hidden="true" tabindex="-1"></a> <span class="fu">calculate</span>(<span class="at">stat =</span> <span class="st">"mean"</span>)</span>
<span id="cb462-4"><a href="B-appendixB.html#cb462-4" aria-hidden="true" tabindex="-1"></a>x_bar</span></code></pre></div>
<pre><code># A tibble: 1 x 1
stat
<dbl>
1 23.4402</code></pre>
<div id="guess-about-statistical-significance" class="section level4 unnumbered">
<h4>Guess about statistical significance</h4>
<p>We are looking to see if the observed sample mean of 23.44 is statistically greater than <span class="math inline">\(\mu_0 = 23\)</span>. They seem to be quite close, but we have a large sample size here. Let’s guess that the large sample size will lead us to reject this practically small difference.</p>
</div>
</div>
<div id="non-traditional-methods" class="section level3" number="13.2.4">
<h3><span class="header-section-number">B.2.4</span> Non-traditional methods</h3>
<div id="bootstrapping-for-hypothesis-test" class="section level4 unnumbered">
<h4>Bootstrapping for hypothesis test</h4>
<p>In order to look to see if the observed sample mean of 23.44 is statistically greater than <span class="math inline">\(\mu_0 = 23\)</span>, we need to account for the sample size. We also need to determine a process that replicates how the original sample of size 5534 was selected.</p>
<p>We can use the idea of <em>bootstrapping</em> to simulate the population from which the sample came and then generate samples from that simulated population to account for sampling variability. Recall how bootstrapping would apply in this context:</p>
<ol style="list-style-type: decimal">
<li>Sample with replacement from our original sample of 5534 women and repeat this process 10,000 times,</li>
<li>calculate the mean for each of the 10,000 bootstrap samples created in Step 1.,</li>
<li>combine all of these bootstrap statistics calculated in Step 2 into a <code>boot_distn</code> object, and</li>
<li>shift the center of this distribution over to the null value of 23. (This is needed since it will be centered at 23.44 via the process of bootstrapping.)</li>
</ol>
<div class="sourceCode" id="cb464"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb464-1"><a href="B-appendixB.html#cb464-1" aria-hidden="true" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">2018</span>)</span>
<span id="cb464-2"><a href="B-appendixB.html#cb464-2" aria-hidden="true" tabindex="-1"></a>null_distn_one_mean <span class="ot"><-</span> age_at_marriage <span class="sc">%>%</span></span>
<span id="cb464-3"><a href="B-appendixB.html#cb464-3" aria-hidden="true" tabindex="-1"></a> <span class="fu">specify</span>(<span class="at">response =</span> age) <span class="sc">%>%</span></span>
<span id="cb464-4"><a href="B-appendixB.html#cb464-4" aria-hidden="true" tabindex="-1"></a> <span class="fu">hypothesize</span>(<span class="at">null =</span> <span class="st">"point"</span>, <span class="at">mu =</span> <span class="dv">23</span>) <span class="sc">%>%</span></span>
<span id="cb464-5"><a href="B-appendixB.html#cb464-5" aria-hidden="true" tabindex="-1"></a> <span class="fu">generate</span>(<span class="at">reps =</span> <span class="dv">10000</span>) <span class="sc">%>%</span></span>
<span id="cb464-6"><a href="B-appendixB.html#cb464-6" aria-hidden="true" tabindex="-1"></a> <span class="fu">calculate</span>(<span class="at">stat =</span> <span class="st">"mean"</span>)</span></code></pre></div>
<div class="sourceCode" id="cb465"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb465-1"><a href="B-appendixB.html#cb465-1" aria-hidden="true" tabindex="-1"></a>null_distn_one_mean <span class="sc">%>%</span> <span class="fu">visualize</span>()</span></code></pre></div>
<p><img src="ModernDive_files/figure-html/unnamed-chunk-529-1.png" width="\textwidth" style="display: block; margin: auto;" /></p>
<p>We can next use this distribution to observe our <span class="math inline">\(p\)</span>-value. Recall this is a right-tailed test so we will be looking for values that are greater than or equal to 23.44 for our <span class="math inline">\(p\)</span>-value.</p>
<div class="sourceCode" id="cb466"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb466-1"><a href="B-appendixB.html#cb466-1" aria-hidden="true" tabindex="-1"></a>null_distn_one_mean <span class="sc">%>%</span></span>
<span id="cb466-2"><a href="B-appendixB.html#cb466-2" aria-hidden="true" tabindex="-1"></a> <span class="fu">visualize</span>(<span class="at">obs_stat =</span> x_bar, <span class="at">direction =</span> <span class="st">"greater"</span>)</span></code></pre></div>
<p><img src="ModernDive_files/figure-html/unnamed-chunk-530-1.png" width="\textwidth" style="display: block; margin: auto;" /></p>
<div id="calculate-p-value" class="section level5 unnumbered">
<h5>Calculate <span class="math inline">\(p\)</span>-value</h5>
<div class="sourceCode" id="cb467"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb467-1"><a href="B-appendixB.html#cb467-1" aria-hidden="true" tabindex="-1"></a>pvalue <span class="ot"><-</span> null_distn_one_mean <span class="sc">%>%</span></span>
<span id="cb467-2"><a href="B-appendixB.html#cb467-2" aria-hidden="true" tabindex="-1"></a> <span class="fu">get_pvalue</span>(<span class="at">obs_stat =</span> x_bar, <span class="at">direction =</span> <span class="st">"greater"</span>)</span>
<span id="cb467-3"><a href="B-appendixB.html#cb467-3" aria-hidden="true" tabindex="-1"></a>pvalue</span></code></pre></div>
<pre><code># A tibble: 1 x 1
p_value
<dbl>
1 0</code></pre>
<p>So our <span class="math inline">\(p\)</span>-value is 0 and we reject the null hypothesis at the 5% level. You can also see this from the histogram above that we are far into the tail of the null distribution.</p>
</div>
</div>
<div id="bootstrapping-for-confidence-interval" class="section level4 unnumbered">
<h4>Bootstrapping for confidence interval</h4>
<p>We can also create a confidence interval for the unknown population parameter <span class="math inline">\(\mu\)</span> using our sample data using <em>bootstrapping</em>. Note that we don’t need to shift this distribution since we want the center of our confidence interval to be our point estimate <span class="math inline">\(\bar{x}_{obs} = 23.44\)</span>.</p>
<div class="sourceCode" id="cb469"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb469-1"><a href="B-appendixB.html#cb469-1" aria-hidden="true" tabindex="-1"></a>boot_distn_one_mean <span class="ot"><-</span> age_at_marriage <span class="sc">%>%</span></span>
<span id="cb469-2"><a href="B-appendixB.html#cb469-2" aria-hidden="true" tabindex="-1"></a> <span class="fu">specify</span>(<span class="at">response =</span> age) <span class="sc">%>%</span></span>
<span id="cb469-3"><a href="B-appendixB.html#cb469-3" aria-hidden="true" tabindex="-1"></a> <span class="fu">generate</span>(<span class="at">reps =</span> <span class="dv">10000</span>) <span class="sc">%>%</span></span>
<span id="cb469-4"><a href="B-appendixB.html#cb469-4" aria-hidden="true" tabindex="-1"></a> <span class="fu">calculate</span>(<span class="at">stat =</span> <span class="st">"mean"</span>)</span></code></pre></div>
<div class="sourceCode" id="cb470"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb470-1"><a href="B-appendixB.html#cb470-1" aria-hidden="true" tabindex="-1"></a>ci <span class="ot"><-</span> boot_distn_one_mean <span class="sc">%>%</span></span>
<span id="cb470-2"><a href="B-appendixB.html#cb470-2" aria-hidden="true" tabindex="-1"></a> <span class="fu">get_ci</span>()</span>
<span id="cb470-3"><a href="B-appendixB.html#cb470-3" aria-hidden="true" tabindex="-1"></a>ci</span></code></pre></div>
<pre><code># A tibble: 1 x 2
lower_ci upper_ci
<dbl> <dbl>
1 23.3148 23.5669</code></pre>
<div class="sourceCode" id="cb472"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb472-1"><a href="B-appendixB.html#cb472-1" aria-hidden="true" tabindex="-1"></a>boot_distn_one_mean <span class="sc">%>%</span></span>
<span id="cb472-2"><a href="B-appendixB.html#cb472-2" aria-hidden="true" tabindex="-1"></a> <span class="fu">visualize</span>(<span class="at">endpoints =</span> ci, <span class="at">direction =</span> <span class="st">"between"</span>)</span></code></pre></div>
<p><img src="ModernDive_files/figure-html/unnamed-chunk-534-1.png" width="\textwidth" style="display: block; margin: auto;" /></p>
<p>We see that 23 is not contained in this confidence interval as a plausible value of <span class="math inline">\(\mu\)</span> (the unknown population mean) and the entire interval is larger than 23. This matches with our hypothesis test results of rejecting the null hypothesis in favor of the alternative (<span class="math inline">\(\mu > 23\)</span>).</p>
<p><strong>Interpretation</strong>: We are 95% confident the true mean age of first marriage for all US women from 2006 to 2010 is between 23.315 and 23.567.</p>
</div>
</div>
<div id="traditional-methods" class="section level3" number="13.2.5">
<h3><span class="header-section-number">B.2.5</span> Traditional methods</h3>
<div id="check-conditions" class="section level4 unnumbered">
<h4>Check conditions</h4>
<p>Remember that in order to use the shortcut (formula-based, theoretical) approach, we need to check that some conditions are met.</p>
<ol style="list-style-type: decimal">
<li><p><em>Independent observations</em>: The observations are collected independently.</p>
<p>The cases are selected independently through random sampling so this condition is met.</p></li>
<li><p><em>Approximately normal</em>: The distribution of the response variable should be normal or the sample size should be at least 30.</p>
<p>The histogram for the sample above does show some skew.</p></li>
</ol>
<p>The Q-Q plot below also shows some skew.</p>
<div class="sourceCode" id="cb473"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb473-1"><a href="B-appendixB.html#cb473-1" aria-hidden="true" tabindex="-1"></a><span class="fu">ggplot</span>(<span class="at">data =</span> age_at_marriage, <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">sample =</span> age)) <span class="sc">+</span></span>
<span id="cb473-2"><a href="B-appendixB.html#cb473-2" aria-hidden="true" tabindex="-1"></a> <span class="fu">stat_qq</span>()</span></code></pre></div>
<p><img src="ModernDive_files/figure-html/qqplotmean-1.png" width="\textwidth" style="display: block; margin: auto;" /></p>
<p>The sample size here is quite large though (<span class="math inline">\(n = 5534\)</span>) so both conditions are met.</p>
</div>
<div id="test-statistic" class="section level4 unnumbered">
<h4>Test statistic</h4>
<p>The test statistic is a random variable based on the sample data. Here, we want to look at a way to estimate the population mean <span class="math inline">\(\mu\)</span>. A good guess is the sample mean <span class="math inline">\(\bar{X}\)</span>. Recall that this sample mean is actually a random variable that will vary as different samples are (theoretically, would be) collected. We are looking to see how likely is it for us to have observed a sample mean of <span class="math inline">\(\bar{x}_{obs} = 23.44\)</span> or larger assuming that the population mean is 23 (assuming the null hypothesis is true). If the conditions are met and assuming <span class="math inline">\(H_0\)</span> is true, we can “standardize” this original test statistic of <span class="math inline">\(\bar{X}\)</span> into a <span class="math inline">\(T\)</span> statistic that follows a <span class="math inline">\(t\)</span> distribution with degrees of freedom equal to <span class="math inline">\(df = n - 1\)</span>:</p>
<p><span class="math display">\[ T =\dfrac{ \bar{X} - \mu_0}{ S / \sqrt{n} } \sim t (df = n - 1) \]</span></p>
<p>where <span class="math inline">\(S\)</span> represents the standard deviation of the sample and <span class="math inline">\(n\)</span> is the sample size.</p>
<div id="observed-test-statistic" class="section level5 unnumbered">
<h5>Observed test statistic</h5>
<p>While one could compute this observed test statistic by “hand,” the focus here is on the set-up of the problem and in understanding which formula for the test statistic applies. We can use the <code>t_test()</code> function to perform this analysis for us.</p>
<div class="sourceCode" id="cb474"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb474-1"><a href="B-appendixB.html#cb474-1" aria-hidden="true" tabindex="-1"></a>t_test_results <span class="ot"><-</span> age_at_marriage <span class="sc">%>%</span></span>
<span id="cb474-2"><a href="B-appendixB.html#cb474-2" aria-hidden="true" tabindex="-1"></a> <span class="fu">t_test</span>(</span>
<span id="cb474-3"><a href="B-appendixB.html#cb474-3" aria-hidden="true" tabindex="-1"></a> <span class="at">formula =</span> age <span class="sc">~</span> <span class="cn">NULL</span>,</span>
<span id="cb474-4"><a href="B-appendixB.html#cb474-4" aria-hidden="true" tabindex="-1"></a> <span class="at">alternative =</span> <span class="st">"greater"</span>,</span>
<span id="cb474-5"><a href="B-appendixB.html#cb474-5" aria-hidden="true" tabindex="-1"></a> <span class="at">mu =</span> <span class="dv">23</span></span>
<span id="cb474-6"><a href="B-appendixB.html#cb474-6" aria-hidden="true" tabindex="-1"></a> )</span>
<span id="cb474-7"><a href="B-appendixB.html#cb474-7" aria-hidden="true" tabindex="-1"></a>t_test_results</span></code></pre></div>
<pre><code># A tibble: 1 x 6
statistic t_df p_value alternative lower_ci upper_ci
<dbl> <dbl> <dbl> <chr> <dbl> <dbl>
1 6.93570 5533 2.25216e-12 greater 23.3358 Inf</code></pre>
<p>We see here that the <span class="math inline">\(t_{obs}\)</span> value is 6.936.</p>
</div>
</div>
<div id="compute-p-value" class="section level4 unnumbered">
<h4>Compute <span class="math inline">\(p\)</span>-value</h4>
<p>The <span class="math inline">\(p\)</span>-value—the probability of observing an <span class="math inline">\(t_{obs}\)</span> value of 6.936 or more in our null distribution of a <span class="math inline">\(t\)</span> with 5533 degrees of freedom—is essentially 0.</p>
</div>
<div id="state-conclusion" class="section level4 unnumbered">
<h4>State conclusion</h4>
<p>We, therefore, have sufficient evidence to reject the null hypothesis. Our initial guess that our observed sample mean was statistically greater than the hypothesized mean has supporting evidence here. Based on this sample, we have evidence that the mean age of first marriage for all US women from 2006 to 2010 is greater than 23 years.</p>
</div>
<div id="confidence-interval-1" class="section level4 unnumbered">
<h4>Confidence interval</h4>
<div class="sourceCode" id="cb476"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb476-1"><a href="B-appendixB.html#cb476-1" aria-hidden="true" tabindex="-1"></a><span class="fu">t.test</span>(</span>
<span id="cb476-2"><a href="B-appendixB.html#cb476-2" aria-hidden="true" tabindex="-1"></a> <span class="at">x =</span> age_at_marriage<span class="sc">$</span>age,</span>
<span id="cb476-3"><a href="B-appendixB.html#cb476-3" aria-hidden="true" tabindex="-1"></a> <span class="at">alternative =</span> <span class="st">"two.sided"</span>,</span>
<span id="cb476-4"><a href="B-appendixB.html#cb476-4" aria-hidden="true" tabindex="-1"></a> <span class="at">mu =</span> <span class="dv">23</span></span>
<span id="cb476-5"><a href="B-appendixB.html#cb476-5" aria-hidden="true" tabindex="-1"></a>)<span class="sc">$</span>conf</span></code></pre></div>
<pre><code>[1] 23.3 23.6
attr(,"conf.level")
[1] 0.95</code></pre>
</div>
</div>
<div id="comparing-results" class="section level3" number="13.2.6">
<h3><span class="header-section-number">B.2.6</span> Comparing results</h3>
<p>Observing the bootstrap distribution that were created, it makes quite a bit of sense that the results are so similar for traditional and non-traditional methods in terms of the <span class="math inline">\(p\)</span>-value and the confidence interval since these distributions look very similar to normal distributions. The conditions also being met (the large sample size was the driver here) leads us to better guess that using any of the methods whether they are traditional (formula-based) or non-traditional (computational-based) will lead to similar results.</p>
</div>
</div>
<div id="one-proportion" class="section level2" number="13.3">
<h2><span class="header-section-number">B.3</span> One proportion</h2>
<div id="problem-statement-1" class="section level3" number="13.3.1">
<h3><span class="header-section-number">B.3.1</span> Problem statement</h3>
<p>The CEO of a large electric utility claims that 80 percent of his 1,000,000 customers are satisfied with the service they receive. To test this claim, the local newspaper surveyed 100 customers, using simple random sampling. 73 were satisfied and the remaining were unsatisfied. Based on these findings from the sample, can we reject the CEO’s hypothesis that 80% of the customers are satisfied? [Tweaked a bit from <a href="http://stattrek.com/hypothesis-test/proportion.aspx?Tutorial=AP" class="uri">http://stattrek.com/hypothesis-test/proportion.aspx?Tutorial=AP</a>]</p>
</div>
<div id="competing-hypotheses-1" class="section level3" number="13.3.2">
<h3><span class="header-section-number">B.3.2</span> Competing hypotheses</h3>
<div id="in-words-1" class="section level4 unnumbered">
<h4>In words</h4>
<ul>
<li>Null hypothesis: The proportion of all customers of the large electric utility satisfied with service they receive is equal 0.80.</li>
<li>Alternative hypothesis: The proportion of all customers of the large electric utility satisfied with service they receive is different from 0.80.</li>
</ul>
</div>
<div id="in-symbols-with-annotations-1" class="section level4 unnumbered">
<h4>In symbols (with annotations)</h4>
<ul>
<li><span class="math inline">\(H_0: \pi = p_{0}\)</span>, where <span class="math inline">\(\pi\)</span> represents the proportion of all customers of the large electric utility satisfied with service they receive and <span class="math inline">\(p_0\)</span> is 0.8.</li>
<li><span class="math inline">\(H_A: \pi \ne 0.8\)</span></li>
</ul>
</div>
<div id="set-alpha-1" class="section level4 unnumbered">
<h4>Set <span class="math inline">\(\alpha\)</span></h4>
<p>It’s important to set the significance level before starting the testing using the data. Let’s set the significance level at 5% here.</p>
</div>
</div>
<div id="exploring-the-sample-data-1" class="section level3" number="13.3.3">
<h3><span class="header-section-number">B.3.3</span> Exploring the sample data</h3>
<div class="sourceCode" id="cb478"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb478-1"><a href="B-appendixB.html#cb478-1" aria-hidden="true" tabindex="-1"></a>elec <span class="ot"><-</span> <span class="fu">c</span>(<span class="fu">rep</span>(<span class="st">"satisfied"</span>, <span class="dv">73</span>), <span class="fu">rep</span>(<span class="st">"unsatisfied"</span>, <span class="dv">27</span>)) <span class="sc">%>%</span></span>
<span id="cb478-2"><a href="B-appendixB.html#cb478-2" aria-hidden="true" tabindex="-1"></a> <span class="fu">enframe</span>() <span class="sc">%>%</span></span>
<span id="cb478-3"><a href="B-appendixB.html#cb478-3" aria-hidden="true" tabindex="-1"></a> <span class="fu">rename</span>(<span class="at">satisfy =</span> value)</span></code></pre></div>
<p>The bar graph below also shows the distribution of <code>satisfy</code>.</p>
<div class="sourceCode" id="cb479"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb479-1"><a href="B-appendixB.html#cb479-1" aria-hidden="true" tabindex="-1"></a><span class="fu">ggplot</span>(<span class="at">data =</span> elec, <span class="fu">aes</span>(<span class="at">x =</span> satisfy)) <span class="sc">+</span></span>
<span id="cb479-2"><a href="B-appendixB.html#cb479-2" aria-hidden="true" tabindex="-1"></a> <span class="fu">geom_bar</span>()</span></code></pre></div>
<p><img src="ModernDive_files/figure-html/bar-1.png" width="\textwidth" style="display: block; margin: auto;" /></p>
<p>The observed statistic is computed as</p>
<div class="sourceCode" id="cb480"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb480-1"><a href="B-appendixB.html#cb480-1" aria-hidden="true" tabindex="-1"></a>p_hat <span class="ot"><-</span> elec <span class="sc">%>%</span></span>
<span id="cb480-2"><a href="B-appendixB.html#cb480-2" aria-hidden="true" tabindex="-1"></a> <span class="fu">specify</span>(<span class="at">response =</span> satisfy, <span class="at">success =</span> <span class="st">"satisfied"</span>) <span class="sc">%>%</span></span>
<span id="cb480-3"><a href="B-appendixB.html#cb480-3" aria-hidden="true" tabindex="-1"></a> <span class="fu">calculate</span>(<span class="at">stat =</span> <span class="st">"prop"</span>)</span>
<span id="cb480-4"><a href="B-appendixB.html#cb480-4" aria-hidden="true" tabindex="-1"></a>p_hat</span></code></pre></div>
<pre><code># A tibble: 1 x 1
stat
<dbl>
1 0.73</code></pre>
<div id="guess-about-statistical-significance-1" class="section level4 unnumbered">
<h4>Guess about statistical significance</h4>
<p>We are looking to see if the sample proportion of 0.73 is statistically different from <span class="math inline">\(p_0 = 0.8\)</span> based on this sample. They seem to be quite close, and our sample size is not huge here (<span class="math inline">\(n = 100\)</span>). Let’s guess that we do not have evidence to reject the null hypothesis.</p>
</div>
</div>
<div id="non-traditional-methods-1" class="section level3" number="13.3.4">
<h3><span class="header-section-number">B.3.4</span> Non-traditional methods</h3>
<div id="simulation-for-hypothesis-test" class="section level4 unnumbered">
<h4>Simulation for hypothesis test</h4>