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<h2 id="toc-title">Table of contents</h2>
<ul>
<li><a href="#introduction" id="toc-introduction" class="nav-link active" data-scroll-target="#introduction"><span class="header-section-number">21.1</span> Introduction</a>
<ul class="collapse">
<li><a href="#sec-building-strings" id="toc-sec-building-strings" class="nav-link" data-scroll-target="#sec-building-strings"><span class="header-section-number">21.1.1</span> Building strings for labels and messages</a></li>
</ul></li>
<li><a href="#should-a-single-sample-of-counted-data-be-considered-different-from-a-benchmark-universe" id="toc-should-a-single-sample-of-counted-data-be-considered-different-from-a-benchmark-universe" class="nav-link" data-scroll-target="#should-a-single-sample-of-counted-data-be-considered-different-from-a-benchmark-universe"><span class="header-section-number">21.2</span> Should a single sample of counted data be considered different from a benchmark universe?</a>
<ul class="collapse">
<li><a href="#sec-fruitfly" id="toc-sec-fruitfly" class="nav-link" data-scroll-target="#sec-fruitfly"><span class="header-section-number">21.2.1</span> Example: Does irradiation affect the sex ratio in fruit flies?</a></li>
<li><a href="#sec-female-calves" id="toc-sec-female-calves" class="nav-link" data-scroll-target="#sec-female-calves"><span class="header-section-number">21.2.2</span> Example: Does a treatment increase the female calf rate?</a></li>
<li><a href="#sec-contract-poll" id="toc-sec-contract-poll" class="nav-link" data-scroll-target="#sec-contract-poll"><span class="header-section-number">21.2.3</span> Example: A public-opinion poll</a></li>
<li><a href="#example-did-the-trump-clinton-poll-indicate-that-trump-would-win" id="toc-example-did-the-trump-clinton-poll-indicate-that-trump-would-win" class="nav-link" data-scroll-target="#example-did-the-trump-clinton-poll-indicate-that-trump-would-win"><span class="header-section-number">21.2.4</span> Example: Did the Trump-Clinton poll indicate that Trump would win?</a></li>
<li><a href="#sec-cancer-cures" id="toc-sec-cancer-cures" class="nav-link" data-scroll-target="#sec-cancer-cures"><span class="header-section-number">21.2.5</span> Example: Comparison of possible cancer cure to placebo</a></li>
<li><a href="#example-did-attitudes-about-marijuana-change" id="toc-example-did-attitudes-about-marijuana-change" class="nav-link" data-scroll-target="#example-did-attitudes-about-marijuana-change"><span class="header-section-number">21.2.6</span> Example: Did attitudes about marijuana change?</a></li>
<li><a href="#sec-framingham-example" id="toc-sec-framingham-example" class="nav-link" data-scroll-target="#sec-framingham-example"><span class="header-section-number">21.2.7</span> Example: Infarction and cholesterol: Framingham study</a></li>
<li><a href="#sec-pig-rations" id="toc-sec-pig-rations" class="nav-link" data-scroll-target="#sec-pig-rations"><span class="header-section-number">21.2.8</span> Example: Is one pig ration more effective than the other?</a></li>
<li><a href="#example-do-planet-densities-differ" id="toc-example-do-planet-densities-differ" class="nav-link" data-scroll-target="#example-do-planet-densities-differ"><span class="header-section-number">21.2.9</span> Example: Do planet densities differ?</a></li>
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<li><a href="#conclusion" id="toc-conclusion" class="nav-link" data-scroll-target="#conclusion"><span class="header-section-number">21.3</span> Conclusion</a></li>
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<h1 class="title"><span id="sec-testing-counts-one" class="quarto-section-identifier"><span class="chapter-number">21</span> <span class="chapter-title">Hypothesis-Testing with Counted Data, Part 1</span></span></h1>
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<section id="introduction" class="level2" data-number="21.1">
<h2 data-number="21.1" class="anchored" data-anchor-id="introduction"><span class="header-section-number">21.1</span> Introduction</h2>
<p>The first task in inferential statistics is to make one or more <em>point estimates</em> — that is, to make one or more statements about <em>how much</em> there is of something we are interested in — including especially the mean and the dispersion. (That work goes under the label “estimation” and is discussed in <a href="point_estimation.html" class="quarto-xref"><span>Chapter 19</span></a>.) Frequently the next step, after making such quantitative estimation of the universe from which a sample has been drawn, is to consider whether two or more samples are different from each other, or whether the single sample is different from a specified value; this work goes under the label “hypothesis testing.” We ask: Did something happen? Or: Is there a difference between two universes? These are yes-no questions.</p>
<p>In other cases, the next step is to inquire into the reliability of the estimates; this goes under the label “confidence intervals.” (Some writers include assessing reliability under the rubric of estimation, but I judge it better not to do so).</p>
<p>So: Having reviewed how to convert hypothesis-testing problems into statistically testable questions in <a href="framing_questions.html" class="quarto-xref"><span>Chapter 20</span></a>, we now must ask: How does one employ resampling methods to make the statistical test? As is always the case when using resampling techniques, there is no unique series of steps by which to proceed. The crucial criterion in assessing the model is whether it accurately simulates the actual event. With hypothesis-testing problems, any number of models may be correct. Generally speaking, though, the model that makes fullest use of the quantitative information available from the data is the best model.</p>
<p>When attempting to deduce the characteristics of a universe from sample data, or when asking whether a sample was drawn from a particular universe, a crucial issue is whether a “one-tailed test” or a “two-tailed test” should be applied. That is, in examining the results of our resampling experiment based on the benchmark universe, do we examine both ends of the frequency distribution, or just one? If there is strong reason to believe <em>a priori</em> that the difference between the benchmark (null) universe and the sample will be in a given direction — for example if you hypothesize that the sample mean will be <em>smaller</em> than the mean of the benchmark universe — you should then employ a <em>one-tailed test</em>. If you do <em>not</em> have strong basis for such a prediction, use the <em>two-tailed</em> test. As an example, when a scientist tests a new medication, his/her hypothesis would be that the number of patients who get well will be higher in the treated group than in the control group. Thus, s/he applies the one-tailed test. See the text below for more detail on one- and two-tailed tests.</p>
<p>Some language first:</p>
<p><strong>Hypothesis:</strong> In inferential statistics, a statement or claim about a universe that can be tested and that you wish to investigate.</p>
<p><strong>Testing:</strong> The process of investigating the validity of a hypothesis.</p>
<p><strong>Benchmark (or null) hypothesis:</strong> A particular hypothesis chosen for convenience when testing hypotheses in inferential statistics. For example, we could test the hypothesis that there is <em>no difference</em> between a sample and a given universe, or between two samples, or that a parameter is less than or greater than a certain value. The benchmark universe refers to this hypothesis. (The concept of the benchmark or null hypothesis was discussed in <a href="probability_theory_1b.html" class="quarto-xref"><span>Chapter 9</span></a> and <a href="framing_questions.html" class="quarto-xref"><span>Chapter 20</span></a>.)</p>
<p>Soon we will begin the actual statistical testing of various sorts of hypotheses about samples and populations.</p>
<p>But, before we get there, we will take a short technical detour.</p>
<section id="sec-building-strings" class="level3" data-number="21.1.1">
<h3 data-number="21.1.1" class="anchored" data-anchor-id="sec-building-strings"><span class="header-section-number">21.1.1</span> Building strings for labels and messages</h3>
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Note 21.1: Notebook: Building strings for labels
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<p><a class="notebook-link" href="notebooks/building_strings.Rmd">Download notebook</a> <a class="interact-button" href="./interact/lab/index.html?path=building_strings.ipynb">Interact</a></p>
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</div>
<div class="nb-start" name="building_strings" title="Building strings for labels">
</div>
<p>As the examples in this book proceed, we will use more code techniques to write the examples in a more concise and efficient way.</p>
<p>One task that we often have, is to build up helpful strings to use as labels on plots, or to print out as messages. These strings will often mix numbers and text. For example, we may want to print out a helpful message such as: <code>Simulation using 10000 trials</code>, where the number 10000 in the message comes from some variable, such as <code>n_trials</code>. Let’s set that variable now:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb1"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a>n_trials <span class="ot"><-</span> <span class="dv">10000</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<p>Building the useful string above would involve taking the string <code>'Simulation using '</code>, then appending a string to represent the number 10,000 — as in:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb2"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Convert the number (integer) to a string (character vector) representing the</span></span>
<span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a><span class="co"># number.</span></span>
<span id="cb2-3"><a href="#cb2-3" aria-hidden="true" tabindex="-1"></a><span class="fu">as.character</span>(n_trials)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>[1] "10000"</code></pre>
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<p>Finally we need to append another string to the result — <code>' trials.'</code>. So our task will be to <em>concatenate</em> (stick together) these three strings.</p>
<div class="r">
<p>You can concatenate strings in R with the <code>paste</code> function. We pass <code>paste</code> the strings and values we want it to combine, as arguments. It converts each argument to a string, and concatenates the resulting strings, separating them with a space (by default).</p>
<p>For example, to make a new string that concatenates the strings <code>'resampling'</code>, <code>'is'</code> and <code>'better'</code>, separated by spaces, we could use:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb4"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a><span class="fu">paste</span>(<span class="st">'resampling'</span>, <span class="st">'is'</span>, <span class="st">'better'</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>[1] "resampling is better"</code></pre>
</div>
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<p>If we wanted to insert a number as a string into this sequence, we can send the <em>number</em> as an argument. <code>paste</code> converts the number to a string before concatenation.</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb6"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb6-1"><a href="#cb6-1" aria-hidden="true" tabindex="-1"></a><span class="fu">paste</span>(<span class="st">'resampling'</span>, <span class="st">'is'</span>, <span class="dv">100</span>, <span class="st">'times'</span>, <span class="st">'better'</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>[1] "resampling is 100 times better"</code></pre>
</div>
</div>
<p>To create the string we started this section with, we could write:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb8"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb8-1"><a href="#cb8-1" aria-hidden="true" tabindex="-1"></a><span class="fu">paste</span>(<span class="st">'Simulation using'</span>, n_trials, <span class="st">'trials.'</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>[1] "Simulation using 10000 trials."</code></pre>
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</div>
<p>We will start using <code>paste</code> from here, to compile messages to print, or labels for plots.</p>
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End of notebook: Building strings for labels
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<p><code>building_strings</code> starts at <a href="#nte-building_strings" class="quarto-xref">Note <span>21.1</span></a>.</p>
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<p>Now we begin with statistical testing for hypotheses on samples and populations.</p>
</section>
</section>
<section id="should-a-single-sample-of-counted-data-be-considered-different-from-a-benchmark-universe" class="level2" data-number="21.2">
<h2 data-number="21.2" class="anchored" data-anchor-id="should-a-single-sample-of-counted-data-be-considered-different-from-a-benchmark-universe"><span class="header-section-number">21.2</span> Should a single sample of counted data be considered different from a benchmark universe?</h2>
<section id="sec-fruitfly" class="level3" data-number="21.2.1">
<h3 data-number="21.2.1" class="anchored" data-anchor-id="sec-fruitfly"><span class="header-section-number">21.2.1</span> Example: Does irradiation affect the sex ratio in fruit flies?</h3>
<p><strong>Where the Benchmark Universe Mean (in this case, the Proportion) is Known, is the Mean (Proportion) of the Population Affected by the Treatment?)</strong></p>
<p>You think you have developed a technique for irradiating the genes of fruit flies so that the sex ratio of the offspring will <em>not</em> be half males and half females. In the first twenty cases you treat, there are fourteen males and six females. Does this experimental result confirm that the irradiation does work?</p>
<p>First convert the scientific question — whether or not the treatment affects the sex distribution — into a probability-statistical question: Is the observed sample likely to have come from a benchmark universe in which the sex ratio is one male to one female? The benchmark (null) hypothesis, then, is that the treatment makes no difference and the sample comes from the one-male-to-one-female universe. Therefore, we investigate <em>how likely a one-to-one universe is to produce a distribution of fourteen or more of just one sex</em>.</p>
<p>A coin has a one-to-one (one out of two) chance of coming up tails. Therefore, we might flip a coin in groups of twenty flips, and count the number of tails in each twenty flips. Or we can use a random number table. The following steps will produce a sound estimate:</p>
<ul>
<li><strong>Step 1.</strong> Let tails = male, heads = female.</li>
<li><strong>Step 2.</strong> Flip twenty coins and count the number of males. If 14 or more males occur, record “yes.” Also, if 6 or fewer males occur, record “yes” because this means we have gotten 14 or more females. Otherwise, record “no.”</li>
<li><strong>Step 3.</strong> Repeat step 2 perhaps 100 times.</li>
<li><strong>Step 4.</strong> Calculate the proportion “yes” in the 100 trials. This proportion estimates the probability that a fruit-fly population with a propensity to produce 50 percent males will by chance produce as many as 14 or as few as 6 males in a sample of 20 flies.</li>
</ul>
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<figcaption class="quarto-float-caption-top quarto-float-caption quarto-float-tbl" id="tbl-fruitfly-trials-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
Table 21.1: Results from 25 random trials for Fruitfly problem
</figcaption>
<div aria-describedby="tbl-fruitfly-trials-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
<table class="lightable-paper lightable-striped lightable-hover caption-top table table-sm table-striped small" data-quarto-postprocess="true">
<thead>
<tr class="header">
<th style="text-align: right;" data-quarto-table-cell-role="th">Trial no</th>
<th style="text-align: right;" data-quarto-table-cell-role="th"># of tails</th>
<th style="text-align: left;" data-quarto-table-cell-role="th">>=14 or <=6</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: right;">1</td>
<td style="text-align: right;">12</td>
<td style="text-align: left;">No</td>
</tr>
<tr class="even">
<td style="text-align: right;">2</td>
<td style="text-align: right;">12</td>
<td style="text-align: left;">No</td>
</tr>
<tr class="odd">
<td style="text-align: right;">3</td>
<td style="text-align: right;">8</td>
<td style="text-align: left;">No</td>
</tr>
<tr class="even">
<td style="text-align: right;">4</td>
<td style="text-align: right;">11</td>
<td style="text-align: left;">No</td>
</tr>
<tr class="odd">
<td style="text-align: right;">5</td>
<td style="text-align: right;">8</td>
<td style="text-align: left;">No</td>
</tr>
<tr class="even">
<td style="text-align: right;">6</td>
<td style="text-align: right;">10</td>
<td style="text-align: left;">No</td>
</tr>
<tr class="odd">
<td style="text-align: right;">7</td>
<td style="text-align: right;">11</td>
<td style="text-align: left;">No</td>
</tr>
<tr class="even">
<td style="text-align: right;">8</td>
<td style="text-align: right;">6</td>
<td style="text-align: left;">Yes</td>
</tr>
<tr class="odd">
<td style="text-align: right;">9</td>
<td style="text-align: right;">6</td>
<td style="text-align: left;">Yes</td>
</tr>
<tr class="even">
<td style="text-align: right;">10</td>
<td style="text-align: right;">10</td>
<td style="text-align: left;">No</td>
</tr>
<tr class="odd">
<td style="text-align: right;">11</td>
<td style="text-align: right;">11</td>
<td style="text-align: left;">No</td>
</tr>
<tr class="even">
<td style="text-align: right;">12</td>
<td style="text-align: right;">12</td>
<td style="text-align: left;">No</td>
</tr>
<tr class="odd">
<td style="text-align: right;">13</td>
<td style="text-align: right;">7</td>
<td style="text-align: left;">No</td>
</tr>
<tr class="even">
<td style="text-align: right;">14</td>
<td style="text-align: right;">15</td>
<td style="text-align: left;">Yes</td>
</tr>
<tr class="odd">
<td style="text-align: right;">15</td>
<td style="text-align: right;">13</td>
<td style="text-align: left;">No</td>
</tr>
<tr class="even">
<td style="text-align: right;">16</td>
<td style="text-align: right;">9</td>
<td style="text-align: left;">No</td>
</tr>
<tr class="odd">
<td style="text-align: right;">17</td>
<td style="text-align: right;">9</td>
<td style="text-align: left;">No</td>
</tr>
<tr class="even">
<td style="text-align: right;">18</td>
<td style="text-align: right;">10</td>
<td style="text-align: left;">No</td>
</tr>
<tr class="odd">
<td style="text-align: right;">19</td>
<td style="text-align: right;">10</td>
<td style="text-align: left;">No</td>
</tr>
<tr class="even">
<td style="text-align: right;">20</td>
<td style="text-align: right;">9</td>
<td style="text-align: left;">No</td>
</tr>
<tr class="odd">
<td style="text-align: right;">21</td>
<td style="text-align: right;">12</td>
<td style="text-align: left;">No</td>
</tr>
<tr class="even">
<td style="text-align: right;">22</td>
<td style="text-align: right;">11</td>
<td style="text-align: left;">No</td>
</tr>
<tr class="odd">
<td style="text-align: right;">23</td>
<td style="text-align: right;">4</td>
<td style="text-align: left;">Yes</td>
</tr>
<tr class="even">
<td style="text-align: right;">24</td>
<td style="text-align: right;">16</td>
<td style="text-align: left;">Yes</td>
</tr>
<tr class="odd">
<td style="text-align: right;">25</td>
<td style="text-align: right;">7</td>
<td style="text-align: left;">No</td>
</tr>
</tbody>
</table>
</div>
</figure>
</div>
</div>
</div>
<p><a href="#tbl-fruitfly-trials" class="quarto-xref">Table <span>21.1</span></a> shows the results obtained in 25 trials of twenty flips each. In two of the twenty-five trials (8 percent) there were fourteen or more tails, which we call “males,” and in three of the 25 trials (12 percent) there six or fewer tails, meaning there were fourteen or more heads (“females”). We can therefore estimate that, even if the treatment does <em>not</em> affect the sex and the births over a long period really are one to one, five out of twenty-five times (20 percent) we would get fourteen or more of one sex or the other. Therefore, finding fourteen males out of 20 births is not overwhelming evidence that the treatment has any effect, even though the result is suggestive.</p>
<p>How accurate is the estimate? Seventy-five more trials were made, and of the 100 trials nine contained fourteen or more “males” (9 percent), and 8 trials contained fourteen or more “females” (8 percent), a total of 17 percent. So the first twenty-five trials gave a fairly reliable indication. As a matter of fact, analytically-based computation (not explained here) shows that the probability of getting fourteen or more females out of twenty births is .057 and, of course, the same for fourteen or more males from a one-to-one universe, implying a total probability of .114 of getting fourteen or more males <em>or</em> females.</p>
<p>Now let us obtain larger and more accurate simulation samples with the computer. The key step in the R notebook below represents male fruit flies with the string <code>'male'</code> and female fruit flies with the string <code>'female'</code>. The <span class="r"><code>sample</code></span> function is then used to generate 20 of these strings with an equal probability that either string is selected. This simulates randomly choosing 20 fruit flies on the benchmark assumption — the “null hypothesis” — that each fruit fly has an equal chance of being a male or female. Now we want to discover the chances of getting more than 13 (i.e., 14 or more) males or more than 13 females under these conditions. So we use <span class="r"><code>sum</code></span> to count the number of males in each random sample and then store this value in the <code>scores</code> vector of this number for each sample. We repeat these steps 10,000 times.</p>
<p>After ten thousand samples have been drawn, we count (<code>sum</code>) how often there were more than 13 males and then count the number of times there were fewer than 7 males (because if there were fewer than 7 males there must have been more than 13 females). When we add the two results together we have the probability that the results obtained from the sample of irradiated fruit flies would be obtained from a random sample of fruit flies.</p>
<div id="nte-fruit_fly" class="callout callout-style-default callout-note callout-titled">
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Note 21.2: Notebook: Fruit fly simulation
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<p><a class="notebook-link" href="notebooks/fruit_fly.Rmd">Download notebook</a> <a class="interact-button" href="./interact/lab/index.html?path=fruit_fly.ipynb">Interact</a></p>
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</div>
<div class="nb-start" name="fruit_fly" title="Fruit fly simulation">
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<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb10"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb10-1"><a href="#cb10-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Set the number of trials</span></span>
<span id="cb10-2"><a href="#cb10-2" aria-hidden="true" tabindex="-1"></a>n_trials <span class="ot"><-</span> <span class="dv">10000</span></span>
<span id="cb10-3"><a href="#cb10-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb10-4"><a href="#cb10-4" aria-hidden="true" tabindex="-1"></a><span class="co"># set the sample size for each trial</span></span>
<span id="cb10-5"><a href="#cb10-5" aria-hidden="true" tabindex="-1"></a>sample_size <span class="ot"><-</span> <span class="dv">20</span></span>
<span id="cb10-6"><a href="#cb10-6" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb10-7"><a href="#cb10-7" aria-hidden="true" tabindex="-1"></a><span class="co"># An empty array to store the trials</span></span>
<span id="cb10-8"><a href="#cb10-8" aria-hidden="true" tabindex="-1"></a>scores <span class="ot"><-</span> <span class="fu">numeric</span>(n_trials)</span>
<span id="cb10-9"><a href="#cb10-9" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb10-10"><a href="#cb10-10" aria-hidden="true" tabindex="-1"></a><span class="co"># Do 10000 trials</span></span>
<span id="cb10-11"><a href="#cb10-11" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> (i <span class="cf">in</span> <span class="dv">1</span><span class="sc">:</span>n_trials) {</span>
<span id="cb10-12"><a href="#cb10-12" aria-hidden="true" tabindex="-1"></a> <span class="co"># Generate 20 simulated fruit flies, where each has an equal chance of being</span></span>
<span id="cb10-13"><a href="#cb10-13" aria-hidden="true" tabindex="-1"></a> <span class="co"># male or female</span></span>
<span id="cb10-14"><a href="#cb10-14" aria-hidden="true" tabindex="-1"></a> a <span class="ot"><-</span> <span class="fu">sample</span>(<span class="fu">c</span>(<span class="st">'male'</span>, <span class="st">'female'</span>),</span>
<span id="cb10-15"><a href="#cb10-15" aria-hidden="true" tabindex="-1"></a> <span class="at">size=</span>sample_size,</span>
<span id="cb10-16"><a href="#cb10-16" aria-hidden="true" tabindex="-1"></a> <span class="at">prob=</span><span class="fu">c</span>(<span class="fl">0.5</span>, <span class="fl">0.5</span>),</span>
<span id="cb10-17"><a href="#cb10-17" aria-hidden="true" tabindex="-1"></a> <span class="at">replace=</span><span class="cn">TRUE</span>)</span>
<span id="cb10-18"><a href="#cb10-18" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb10-19"><a href="#cb10-19" aria-hidden="true" tabindex="-1"></a> <span class="co"># count the number of males in the sample</span></span>
<span id="cb10-20"><a href="#cb10-20" aria-hidden="true" tabindex="-1"></a> b <span class="ot"><-</span> <span class="fu">sum</span>(a <span class="sc">==</span> <span class="st">'male'</span>)</span>
<span id="cb10-21"><a href="#cb10-21" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb10-22"><a href="#cb10-22" aria-hidden="true" tabindex="-1"></a> <span class="co"># store the result of this trial</span></span>
<span id="cb10-23"><a href="#cb10-23" aria-hidden="true" tabindex="-1"></a> scores[i] <span class="ot"><-</span> b</span>
<span id="cb10-24"><a href="#cb10-24" aria-hidden="true" tabindex="-1"></a>}</span>
<span id="cb10-25"><a href="#cb10-25" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb10-26"><a href="#cb10-26" aria-hidden="true" tabindex="-1"></a><span class="co"># Produce a histogram of the trial results</span></span>
<span id="cb10-27"><a href="#cb10-27" aria-hidden="true" tabindex="-1"></a>title_of_plot <span class="ot"><-</span> <span class="fu">paste</span>(<span class="st">"Number of males in"</span>, n_trials,</span>
<span id="cb10-28"><a href="#cb10-28" aria-hidden="true" tabindex="-1"></a> <span class="st">"samples of"</span>, sample_size,</span>
<span id="cb10-29"><a href="#cb10-29" aria-hidden="true" tabindex="-1"></a> <span class="st">"simulated fruit flies"</span>)</span>
<span id="cb10-30"><a href="#cb10-30" aria-hidden="true" tabindex="-1"></a><span class="fu">hist</span>(scores, <span class="at">xlab=</span><span class="st">'Number of Males'</span>, <span class="at">main=</span>title_of_plot)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<p>In the histogram above, we see that in about 12 percent of the trials, the number of males was 14 or more, or 6 or fewer. Or instead of reading the results from the histogram, we can calculate the result by tacking on the following commands to the above program:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb11"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb11-1"><a href="#cb11-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Determine the number of trials in which we had 14 or more males.</span></span>
<span id="cb11-2"><a href="#cb11-2" aria-hidden="true" tabindex="-1"></a>j <span class="ot"><-</span> <span class="fu">sum</span>(scores <span class="sc">>=</span> <span class="dv">14</span>)</span>
<span id="cb11-3"><a href="#cb11-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb11-4"><a href="#cb11-4" aria-hidden="true" tabindex="-1"></a><span class="co"># Determine the number of trials in which we had 6 or fewer males.</span></span>
<span id="cb11-5"><a href="#cb11-5" aria-hidden="true" tabindex="-1"></a>k <span class="ot"><-</span> <span class="fu">sum</span>(scores <span class="sc"><=</span> <span class="dv">6</span>)</span>
<span id="cb11-6"><a href="#cb11-6" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb11-7"><a href="#cb11-7" aria-hidden="true" tabindex="-1"></a><span class="co"># Add the two results together.</span></span>
<span id="cb11-8"><a href="#cb11-8" aria-hidden="true" tabindex="-1"></a>m <span class="ot"><-</span> j <span class="sc">+</span> k</span>
<span id="cb11-9"><a href="#cb11-9" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb11-10"><a href="#cb11-10" aria-hidden="true" tabindex="-1"></a><span class="co"># Convert to a proportion.</span></span>
<span id="cb11-11"><a href="#cb11-11" aria-hidden="true" tabindex="-1"></a>mm <span class="ot"><-</span> m<span class="sc">/</span>n_trials</span>
<span id="cb11-12"><a href="#cb11-12" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb11-13"><a href="#cb11-13" aria-hidden="true" tabindex="-1"></a><span class="co"># Print the results.</span></span>
<span id="cb11-14"><a href="#cb11-14" aria-hidden="true" tabindex="-1"></a><span class="fu">print</span>(mm)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>[1] 0.121</code></pre>
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End of notebook: Fruit fly simulation
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<p><code>fruit_fly</code> starts at <a href="#nte-fruit_fly" class="quarto-xref">Note <span>21.2</span></a>.</p>
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<p>Notice that the strength of the evidence for the effectiveness of the radiation treatment depends upon the original question: whether or not the treatment had <em>any</em> effect on the sex of the fruit fly, which is a two-tailed question. If there were reason to believe at the start that the treatment could increase <em>only</em> the number of <em>males</em>, then we would focus our attention on the result that in only two of the twenty-five trials were fourteen or more males. There would then be only a 2/25 = 0.08 probability of getting the observed results by chance if the treatment really has no effect, rather than the weaker odds against obtaining fourteen or more of <em>either</em> males or females.</p>
<p>Therefore, whether you decide to figure the odds of just fourteen or more males (what is called a “one-tail test”) or the odds for fourteen or more males <em>plus</em> fourteen or more females (a “two-tail test”), depends upon your advance knowledge of the subject. If you have no reason to believe that the treatment will have an effect <em>only</em> in the direction of creating more males and if you figure the odds for the one-tail test anyway, then you will be kidding yourself. Theory comes to bear here. If you have a strong hypothesis, deduced from a strong theory, that there will be more males, then you should figure one-tail odds, but if you have no such theory you should figure the weaker two-tail odds.<a href="#fn1" class="footnote-ref" id="fnref1" role="doc-noteref"><sup>1</sup></a></p>
<p>In the case of the next problem concerning calves, we shall see that a one-tail test is appropriate because we have no interest in producing more male calves. Before leaving this example, let us review our intellectual strategy in handling the problem. First we observe a result (14 males in 20 flies) which differs from the proportion of the benchmark population (50 percent males). Because we have treated this sample with irradiation and observed a result that differs from the untreated benchmark-population’s mean, we speculate that the irradiation caused the sample to differ from the untreated population. We wish to check on whether this speculation is correct.</p>
<p>When asking whether this speculation is correct, we are implicitly asking whether future irradiation would also produce a proportion of males higher than 50 percent. That is, we are implicitly asking whether irradiated flies would produce more samples with male proportions as high as 14/20 than would occur by chance in the absence of irradiation.</p>
<p>If samples as far away as 14/20 from the benchmark population mean of 10/20 would occur frequently by chance, then we would not be impressed with that experimental evidence as proof that irradiation does affect the sex ratio. Hence we set up a model that will tell us the frequency with which samples of 14 or more males out of 20 births would be observed by chance. Carrying out the resampling procedure tells us that perhaps a tenth of the time such samples would be observed by chance. That is not extremely frequent, but it is not infrequent either. Hence we would probably conclude that the evidence is provocative enough to justify further experimentation, but not so strong that we should immediately believe in the truth of this speculation.</p>
<p>The logic of attaching meaning to the probabilistic outcome of a test of a hypothesis is discussed in <a href="significance.html" class="quarto-xref"><span>Chapter 22</span></a>. There also is more about the concept of the level of significance in <a href="significance.html" class="quarto-xref"><span>Chapter 22</span></a>.</p>
<p>Because of the great importance of this sort of case, which brings out the basic principles particularly clearly, let us consider another example:</p>
</section>
<section id="sec-female-calves" class="level3" data-number="21.2.2">
<h3 data-number="21.2.2" class="anchored" data-anchor-id="sec-female-calves"><span class="header-section-number">21.2.2</span> Example: Does a treatment increase the female calf rate?</h3>
<p><strong>What is the probability that among 10 calves born, 9 or more will be female?</strong></p>
<p>Let’s consider this question in the context of a set of queries for performing statistical inference that will be discussed further in <a href="testing_procedures.html" class="quarto-xref"><span>Chapter 25</span></a>.</p>
<p><strong><em>The question</em></strong>: (From <span class="citation" data-cites="hodges1970basic">Hodges Jr and Lehmann (<a href="references.html#ref-hodges1970basic" role="doc-biblioref">1970</a>)</span>): Female calves are more valuable than males. A bio-engineer claims to be able to cause more females to be born than the expected 50 percent rate. He conducts his procedure, and nine females are born out of the next 10 pregnancies among the treated cows. Should you believe his claim? That is, what is the probability of a result this (or more) surprising occurring by chance if his procedure has no effect? In this problem, we assume that on average 100 of 206 births are female, in contrast to the 50-50 benchmark universe in the previous problem.</p>
<p><strong><em>What is the purpose of the work?</em></strong>: Female calves are more valuable than male calves.</p>
<p><strong><em>Statistical inference?</em></strong>: Yes.</p>
<p><strong><em>Confidence interval or Test of hypothesis?</em></strong>: Test of hypothesis.</p>
<p><strong><em>Will you state the costs and benefits of various outcomes, or a loss function?</em></strong>: Yes. One need only say that the benefits are very large, and if the results are promising, it is worth gathering more data to confirm results.</p>
<p><strong><em>How many samples of data are part of the hypothesis test?</em></strong>: One.</p>
<p><strong><em>What is the size of the first sample about which you wish to make significance statements?</em></strong>: Ten.</p>
<p><strong><em>What comparison(s) to make?</em></strong>: Compare the sample to the benchmark universe.</p>
<p><strong><em>What is the benchmark universe</em></strong>: <strong>that embodies the null hypothesis?</strong> 100/206 female.</p>
<p><strong><em>Which symbols for the observed entities?</em></strong>: Balls in bucket, or numbers.</p>
<p><strong><em>What values or ranges of values?</em></strong>: We could write numbers 1 through 206 on pieces of paper, and take numbers 1-100 as “male” and 101-206 as “female”. Or we could use some other mechanism to give us a 100/206 chance of any one calf being female.</p>
<p><strong><em>Finite or infinite universe?</em></strong>: Infinite.</p>
<p><strong><em>Which sample(s) do you wish to compare to which, or to the null universe (and perhaps to the alternative universe)?</em></strong>: Ten calves.</p>
<p><strong><em>What procedure to produce the sample entities?</em></strong>: Sampling with replacement.</p>
<p><strong><em>Simple (single step) or complex (multiple “if” drawings)?</em></strong>: Can think of it either way.</p>
<p><strong><em>What to record as the outcome of each resample trial?</em></strong>: The proportion (or number) of females.</p>
<p><strong><em>What is the criterion to be used in the test?</em></strong>: The probability that in a sample of ten calves, nine (or more) females would be drawn by chance from the benchmark universe of 100/206 females.</p>
<p><strong><em>“One tail” or “two tail” test?</em></strong>: One tail, because the farmer is only interested in females. Finding a large proportion of males would not be of interest; it would not cause rejecting the null hypothesis.</p>
<p>The actual computation of probability may be done in several ways, as discussed earlier for four children and for ten cows. Conventional methods are discussed for comparison in <a href="testing_procedures.html" class="quarto-xref"><span>Chapter 25</span></a>. Here is the resampling solution in R.</p>
<div id="nte-female_calves" class="callout callout-style-default callout-note callout-titled">
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Note 21.3: Notebook: Female calf numbers simulation
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<p><a class="notebook-link" href="notebooks/female_calves.Rmd">Download notebook</a> <a class="interact-button" href="./interact/lab/index.html?path=female_calves.ipynb">Interact</a></p>
</div>
</div>
</div>
<div class="nb-start" name="female_calves" title="Female calf numbers simulation">
</div>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb13"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb13-1"><a href="#cb13-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Set the number of trials.</span></span>
<span id="cb13-2"><a href="#cb13-2" aria-hidden="true" tabindex="-1"></a>n_trials <span class="ot"><-</span> <span class="dv">10000</span></span>
<span id="cb13-3"><a href="#cb13-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb13-4"><a href="#cb13-4" aria-hidden="true" tabindex="-1"></a><span class="co"># Set the size of each sample.</span></span>
<span id="cb13-5"><a href="#cb13-5" aria-hidden="true" tabindex="-1"></a>sample_size <span class="ot"><-</span> <span class="dv">10</span></span>
<span id="cb13-6"><a href="#cb13-6" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb13-7"><a href="#cb13-7" aria-hidden="true" tabindex="-1"></a><span class="co"># Probability of any one calf being female.</span></span>
<span id="cb13-8"><a href="#cb13-8" aria-hidden="true" tabindex="-1"></a>p_female <span class="ot"><-</span> <span class="dv">100</span> <span class="sc">/</span> <span class="dv">206</span></span>
<span id="cb13-9"><a href="#cb13-9" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb13-10"><a href="#cb13-10" aria-hidden="true" tabindex="-1"></a><span class="co"># A vector to store the results.</span></span>
<span id="cb13-11"><a href="#cb13-11" aria-hidden="true" tabindex="-1"></a>scores <span class="ot"><-</span> <span class="fu">numeric</span>(n_trials)</span>
<span id="cb13-12"><a href="#cb13-12" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb13-13"><a href="#cb13-13" aria-hidden="true" tabindex="-1"></a><span class="co"># for 10000 repeats.</span></span>
<span id="cb13-14"><a href="#cb13-14" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> (i <span class="cf">in</span> <span class="dv">1</span><span class="sc">:</span>n_trials) {</span>
<span id="cb13-15"><a href="#cb13-15" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb13-16"><a href="#cb13-16" aria-hidden="true" tabindex="-1"></a> a <span class="ot"><-</span> <span class="fu">sample</span>(<span class="fu">c</span>(<span class="st">'female'</span>, <span class="st">'male'</span>),</span>
<span id="cb13-17"><a href="#cb13-17" aria-hidden="true" tabindex="-1"></a> <span class="at">size=</span>sample_size,</span>
<span id="cb13-18"><a href="#cb13-18" aria-hidden="true" tabindex="-1"></a> <span class="at">prob=</span><span class="fu">c</span>(p_female, <span class="dv">1</span> <span class="sc">-</span> p_female),</span>
<span id="cb13-19"><a href="#cb13-19" aria-hidden="true" tabindex="-1"></a> <span class="at">replace=</span><span class="cn">TRUE</span>)</span>
<span id="cb13-20"><a href="#cb13-20" aria-hidden="true" tabindex="-1"></a> b <span class="ot"><-</span> <span class="fu">sum</span>(a <span class="sc">==</span> <span class="st">'female'</span>)</span>
<span id="cb13-21"><a href="#cb13-21" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb13-22"><a href="#cb13-22" aria-hidden="true" tabindex="-1"></a> <span class="co"># Store the result of the current trial.</span></span>
<span id="cb13-23"><a href="#cb13-23" aria-hidden="true" tabindex="-1"></a> scores[i] <span class="ot"><-</span> b</span>
<span id="cb13-24"><a href="#cb13-24" aria-hidden="true" tabindex="-1"></a>}</span>
<span id="cb13-25"><a href="#cb13-25" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb13-26"><a href="#cb13-26" aria-hidden="true" tabindex="-1"></a><span class="co"># Plot a histogram of the scores.</span></span>
<span id="cb13-27"><a href="#cb13-27" aria-hidden="true" tabindex="-1"></a>title_of_plot <span class="ot"><-</span> <span class="fu">paste</span>(<span class="st">"Number of females in"</span>, n_trials,</span>
<span id="cb13-28"><a href="#cb13-28" aria-hidden="true" tabindex="-1"></a> <span class="st">"samples of"</span>, sample_size,</span>
<span id="cb13-29"><a href="#cb13-29" aria-hidden="true" tabindex="-1"></a> <span class="st">"simulated calves"</span>)</span>
<span id="cb13-30"><a href="#cb13-30" aria-hidden="true" tabindex="-1"></a><span class="fu">hist</span>(scores, <span class="at">xlab =</span> <span class="st">'Number of Females'</span>, <span class="at">main=</span>title_of_plot)</span>
<span id="cb13-31"><a href="#cb13-31" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb13-32"><a href="#cb13-32" aria-hidden="true" tabindex="-1"></a><span class="co"># Count the number of scores that were greater than or equal to 9.</span></span>
<span id="cb13-33"><a href="#cb13-33" aria-hidden="true" tabindex="-1"></a>k <span class="ot"><-</span> <span class="fu">sum</span>(scores <span class="sc">>=</span> <span class="dv">9</span>)</span>
<span id="cb13-34"><a href="#cb13-34" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb13-35"><a href="#cb13-35" aria-hidden="true" tabindex="-1"></a><span class="co"># Express as a proportion.</span></span>
<span id="cb13-36"><a href="#cb13-36" aria-hidden="true" tabindex="-1"></a>kk <span class="ot"><-</span> k <span class="sc">/</span> n_trials</span>
<span id="cb13-37"><a href="#cb13-37" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb13-38"><a href="#cb13-38" aria-hidden="true" tabindex="-1"></a><span class="co"># Show the proportion.</span></span>
<span id="cb13-39"><a href="#cb13-39" aria-hidden="true" tabindex="-1"></a><span class="fu">message</span>(<span class="st">"Probability of 9 or 10 females occurring by chance: "</span>, kk)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>Probability of 9 or 10 females occurring by chance: 0.0106</code></pre>
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<p>We read from the result in variable <code>kk</code> that the probability of 9 or 10 females occurring by chance is a bit more than one percent.</p>
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End of notebook: Female calf numbers simulation
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<p><code>female_calves</code> starts at <a href="#nte-female_calves" class="quarto-xref">Note <span>21.3</span></a>.</p>
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</section>
<section id="sec-contract-poll" class="level3" data-number="21.2.3">
<h3 data-number="21.2.3" class="anchored" data-anchor-id="sec-contract-poll"><span class="header-section-number">21.2.3</span> Example: A public-opinion poll</h3>
<p><strong>Is the Proportion of a Population Greater Than a Given Value?</strong></p>
<p>A municipal official wants to determine whether a majority of the town’s residents are for or against the awarding of a high-speed internet contract, and she asks you to take a poll. You judge that the voter registration records are a fair representation of the universe in which the politician was interested, and you therefore decided to interview a random selection of registered voters. Of a sample of fifty people who expressed opinions, thirty said “yes” they were for the plan and twenty said “no,” they were against it. How conclusively do the results show that the people in town want this internet contract?</p>
<p>Now comes some necessary subtle thinking in the interpretation of what seems like a simple problem. Notice that our aim in the analysis is to avoid the mistake of saying that the town favors the plan when in fact it does not favor the plan. Our chance of making this mistake is greatest when the voters are evenly split, so we choose as the benchmark (null) hypothesis that 50 percent of the town does not want the plan. This statement really means that “50 percent or more do not want the plan.” We could assess the probability of obtaining our result from a population that is split (say) 52-48 against, but such a probability would necessarily be even smaller, and we are primarily interested in assessing the maximum probability of being wrong. If the maximum probability of error turns out to be inconsequential, then we need not worry about less likely errors.</p>
<p>This problem is very much like the one-group fruit fly irradiation problem above. The only difference is that now we are comparing the observed sample against an arbitrary value of 50 percent (because that is the break-point in a situation where the majority decides) whereas in <a href="#sec-fruitfly" class="quarto-xref"><span>Section 21.2.1</span></a> we compared the observed sample against the normal population proportion (also 50 percent, because that is the normal proportion of males). But it really does not matter <em>why</em> we are comparing the observed sample to the figure of 50 percent; the procedure is the same in both cases. (Please notice that there is nothing special about the 50 percent figure; the same procedure would be followed for 20 percent or 85 percent.)</p>
<p>In brief, we a) take two pieces of paper, write “Yes” on one and “No” on the other, put them in a bucket b) draw a piece of paper from the bucket, record whether it was “Yes” or “No”, replace, and repeat 50 times c) count the number of “yeses” and “noes” in the fifty draws, c) repeat for perhaps a hundred trials, then d) count the proportion of the trials in which a 50-50 universe would produce thirty or more “yes” answers.</p>
<p>In operational steps, the procedure is as follows:</p>
<ul>
<li><strong>Step 1.</strong> “1-5” = no, “6-0” = yes.</li>
<li><strong>Step 2.</strong> In 50 random numbers, count the “yeses,” and record “false positive” if 30 or more “yeses.”</li>
<li><strong>Step 3.</strong> Repeat step 2 perhaps 100 times.</li>
<li><strong>Step 4.</strong> Calculate the proportion of experimental trials showing “false positive.” This estimates the probability that as many as 30 “yeses” would be observed by chance in a sample of 50 people <em>if</em> half (or more) are really against the plan.</li>
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Table 21.2: Results from 20 random trials for contract poll problem
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<th style="text-align: right;" data-quarto-table-cell-role="th">Trial no</th>
<th style="text-align: right;" data-quarto-table-cell-role="th"># of "Noes"</th>
<th style="text-align: right;" data-quarto-table-cell-role="th"># of "Yeses"</th>
<th style="text-align: left;" data-quarto-table-cell-role="th">>= 30 "Yeses"</th>
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<td style="text-align: right;">1</td>
<td style="text-align: right;">25</td>
<td style="text-align: right;">25</td>
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<tr class="even">
<td style="text-align: right;">2</td>