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<li><a href="#sec-soln-paired-differences" id="toc-sec-soln-paired-differences" class="nav-link active" data-scroll-target="#sec-soln-paired-differences"><span class="header-section-number">A.1</span> Solution for paired differences exercise <span>24.3.1</span></a></li>
<li><a href="#sec-soln-seatbelt-proportions" id="toc-sec-soln-seatbelt-proportions" class="nav-link" data-scroll-target="#sec-soln-seatbelt-proportions"><span class="header-section-number">A.2</span> Solution to seatbelt proportions exercise <span>24.3.2</span></a></li>
<li><a href="#sec-soln-unemployment-percent" id="toc-sec-soln-unemployment-percent" class="nav-link" data-scroll-target="#sec-soln-unemployment-percent"><span class="header-section-number">A.3</span> Solution for unemployment percentage <span>27.8.1</span></a></li>
<li><a href="#sec-soln-battery-lifetime" id="toc-sec-soln-battery-lifetime" class="nav-link" data-scroll-target="#sec-soln-battery-lifetime"><span class="header-section-number">A.4</span> Solution for battery lifetime <span>27.8.2</span></a></li>
<li><a href="#sec-soln-optical-density" id="toc-sec-soln-optical-density" class="nav-link" data-scroll-target="#sec-soln-optical-density"><span class="header-section-number">A.5</span> Solution for optical density <span>27.8.3</span></a></li>
<li><a href="#sec-soln-voter-participation" id="toc-sec-soln-voter-participation" class="nav-link" data-scroll-target="#sec-soln-voter-participation"><span class="header-section-number">A.6</span> Solution for voter participation <span>29.7.1</span></a></li>
<li><a href="#sec-soln-runs-strikeouts" id="toc-sec-soln-runs-strikeouts" class="nav-link" data-scroll-target="#sec-soln-runs-strikeouts"><span class="header-section-number">A.7</span> Solution for association of runs and strikeouts <span>29.7.2</span></a></li>
<li><a href="#sec-soln-runs-strikeouts-r" id="toc-sec-soln-runs-strikeouts-r" class="nav-link" data-scroll-target="#sec-soln-runs-strikeouts-r"><span class="header-section-number">A.8</span> Solution for runs, strikeouts and correlation coefficient <span>29.7.3</span></a></li>
<li><a href="#sec-soln-money-exchange" id="toc-sec-soln-money-exchange" class="nav-link" data-scroll-target="#sec-soln-money-exchange"><span class="header-section-number">A.9</span> Solution for money and exchange rate <span>29.7.4</span></a></li>
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<h1 class="title"><span id="sec-exercise-solutions" class="quarto-section-identifier">Appendix A — Exercise Solutions</span></h1>
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<section id="sec-soln-paired-differences" class="level2" data-number="A.1">
<h2 data-number="A.1" class="anchored" data-anchor-id="sec-soln-paired-differences"><span class="header-section-number">A.1</span> Solution for <a href="testing_measured.html#sec-exr-paired-differences" class="quarto-xref">paired differences exercise <span>24.3.1</span></a></h2>
<p>We suggested that you ignored the pairing of the before and after samples, and that is what we will do here. Then we will extend the treatment to take the pairing into account.</p>
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Note A.1: Notebook: Paired differences solution
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<p><a class="notebook-link" href="notebooks/paired_differences_solution.zip">Download zip with notebook + data file</a> <a class="interact-button" href="./interact/lab/index.html?path=paired_differences_solution.ipynb">Interact</a></p>
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<div class="nb-start" name="paired_differences_solution" title="Paired differences solution">
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<div class="sourceCode cell-code" id="cb1"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> numpy <span class="im">as</span> np</span>
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> pandas <span class="im">as</span> pd</span>
<span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-4"><a href="#cb1-4" aria-hidden="true" tabindex="-1"></a>rnd <span class="op">=</span> np.random.default_rng()</span>
<span id="cb1-5"><a href="#cb1-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-6"><a href="#cb1-6" aria-hidden="true" tabindex="-1"></a>df <span class="op">=</span> pd.read_csv(<span class="st">'data/hamilton.csv'</span>)</span>
<span id="cb1-7"><a href="#cb1-7" aria-hidden="true" tabindex="-1"></a>before <span class="op">=</span> np.array(df[<span class="st">'score_before'</span>])</span>
<span id="cb1-8"><a href="#cb1-8" aria-hidden="true" tabindex="-1"></a>after <span class="op">=</span> np.array(df[<span class="st">'score_after'</span>])</span>
<span id="cb1-9"><a href="#cb1-9" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-10"><a href="#cb1-10" aria-hidden="true" tabindex="-1"></a>observed_diff <span class="op">=</span> np.mean(after) <span class="op">-</span> np.mean(before)</span>
<span id="cb1-11"><a href="#cb1-11" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-12"><a href="#cb1-12" aria-hidden="true" tabindex="-1"></a><span class="co"># Let us start with a permutation test.</span></span>
<span id="cb1-13"><a href="#cb1-13" aria-hidden="true" tabindex="-1"></a>both <span class="op">=</span> np.concatenate([before, after])</span>
<span id="cb1-14"><a href="#cb1-14" aria-hidden="true" tabindex="-1"></a>n_before <span class="op">=</span> <span class="bu">len</span>(before)</span>
<span id="cb1-15"><a href="#cb1-15" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-16"><a href="#cb1-16" aria-hidden="true" tabindex="-1"></a><span class="co"># Samples in the null world.</span></span>
<span id="cb1-17"><a href="#cb1-17" aria-hidden="true" tabindex="-1"></a>n_trials <span class="op">=</span> <span class="dv">10_000</span></span>
<span id="cb1-18"><a href="#cb1-18" aria-hidden="true" tabindex="-1"></a>results <span class="op">=</span> np.zeros(n_trials)</span>
<span id="cb1-19"><a href="#cb1-19" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> i <span class="kw">in</span> <span class="bu">range</span>(n_trials):</span>
<span id="cb1-20"><a href="#cb1-20" aria-hidden="true" tabindex="-1"></a> shuffled <span class="op">=</span> rnd.permuted(both)</span>
<span id="cb1-21"><a href="#cb1-21" aria-hidden="true" tabindex="-1"></a> fake_before <span class="op">=</span> shuffled[:n_before]</span>
<span id="cb1-22"><a href="#cb1-22" aria-hidden="true" tabindex="-1"></a> fake_after <span class="op">=</span> shuffled[n_before:]</span>
<span id="cb1-23"><a href="#cb1-23" aria-hidden="true" tabindex="-1"></a> fake_diff <span class="op">=</span> np.mean(fake_after) <span class="op">-</span> np.mean(fake_before)</span>
<span id="cb1-24"><a href="#cb1-24" aria-hidden="true" tabindex="-1"></a> results[i] <span class="op">=</span> fake_diff</span>
<span id="cb1-25"><a href="#cb1-25" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-26"><a href="#cb1-26" aria-hidden="true" tabindex="-1"></a><span class="co"># We are interested in fake differences that are larger</span></span>
<span id="cb1-27"><a href="#cb1-27" aria-hidden="true" tabindex="-1"></a><span class="co"># in magnitude than the observed difference (hence "abs").</span></span>
<span id="cb1-28"><a href="#cb1-28" aria-hidden="true" tabindex="-1"></a><span class="co"># Here we have no prior hypothesis about which direction the difference</span></span>
<span id="cb1-29"><a href="#cb1-29" aria-hidden="true" tabindex="-1"></a><span class="co"># will go.</span></span>
<span id="cb1-30"><a href="#cb1-30" aria-hidden="true" tabindex="-1"></a>k <span class="op">=</span> np.<span class="bu">sum</span>(np.<span class="bu">abs</span>(results) <span class="op">>=</span> np.<span class="bu">abs</span>(observed_diff))</span>
<span id="cb1-31"><a href="#cb1-31" aria-hidden="true" tabindex="-1"></a>kk <span class="op">=</span> k <span class="op">/</span> n_trials</span>
<span id="cb1-32"><a href="#cb1-32" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="st">'Permutation p null-world abs >= abs observed:'</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>Permutation p null-world abs >= abs observed: 0.2554</code></pre>
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<div class="sourceCode cell-code" id="cb3"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb3-1"><a href="#cb3-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Next a bootstrap test.</span></span>
<span id="cb3-2"><a href="#cb3-2" aria-hidden="true" tabindex="-1"></a>n_after <span class="op">=</span> <span class="bu">len</span>(after) <span class="co"># Of course, in our case, this will be == n_before</span></span>
<span id="cb3-3"><a href="#cb3-3" aria-hidden="true" tabindex="-1"></a>results <span class="op">=</span> np.zeros(n_trials)</span>
<span id="cb3-4"><a href="#cb3-4" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> i <span class="kw">in</span> <span class="bu">range</span>(n_trials):</span>
<span id="cb3-5"><a href="#cb3-5" aria-hidden="true" tabindex="-1"></a> fake_before <span class="op">=</span> rnd.choice(both, size<span class="op">=</span>n_before)</span>
<span id="cb3-6"><a href="#cb3-6" aria-hidden="true" tabindex="-1"></a> fake_after <span class="op">=</span> rnd.choice(both, size<span class="op">=</span>n_after)</span>
<span id="cb3-7"><a href="#cb3-7" aria-hidden="true" tabindex="-1"></a> fake_diff <span class="op">=</span> np.mean(fake_after) <span class="op">-</span> np.mean(fake_before)</span>
<span id="cb3-8"><a href="#cb3-8" aria-hidden="true" tabindex="-1"></a> results[i] <span class="op">=</span> fake_diff</span>
<span id="cb3-9"><a href="#cb3-9" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb3-10"><a href="#cb3-10" aria-hidden="true" tabindex="-1"></a>k <span class="op">=</span> np.<span class="bu">sum</span>(np.<span class="bu">abs</span>(results) <span class="op">>=</span> np.<span class="bu">abs</span>(observed_diff))</span>
<span id="cb3-11"><a href="#cb3-11" aria-hidden="true" tabindex="-1"></a>kk <span class="op">=</span> k <span class="op">/</span> n_trials</span>
<span id="cb3-12"><a href="#cb3-12" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="st">'Bootstrap p null-world abs >= abs observed:'</span>, kk)</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>Bootstrap p null-world abs >= abs observed: 0.2217</code></pre>
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<p>Finally we consider the pairs. Here we <em>do</em> take the pairs into account. We have some reason to think that the patients or cars vary in some substantial way in their level of depression, or their tendency to break down, but we believe that the patients’ <em>response</em> to treatment or the difference between the mechanics is the value of interest.</p>
<p>In that case, we are interested in the <em>differences</em> between the pairs. In the null world, these before / after (mechanic A / mechanic B) differences are random. In the null-world, where there is no difference between before/after or mechanics 1 and 2, we can flip the before / after (A / B) pairs and be in the same world.</p>
<p>Notice that flipping the before / after or A / B in the pair just changes the sign of the difference.</p>
<p>So we will simulate the effect of flipping the values in the pair, by choosing a random sign for the pair, where -1 means pair is flipped, and 1 means pair is in original order. We recalculated the mean difference with these random signs (flips) applied, and these will be our values in the null-world.</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb5"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb5-1"><a href="#cb5-1" aria-hidden="true" tabindex="-1"></a><span class="co"># A test of paired difference with sign flips for the null world.</span></span>
<span id="cb5-2"><a href="#cb5-2" aria-hidden="true" tabindex="-1"></a>differences <span class="op">=</span> after <span class="op">-</span> before</span>
<span id="cb5-3"><a href="#cb5-3" aria-hidden="true" tabindex="-1"></a>observed_mdiff <span class="op">=</span> np.mean(differences)</span>
<span id="cb5-4"><a href="#cb5-4" aria-hidden="true" tabindex="-1"></a>n_both <span class="op">=</span> <span class="bu">len</span>(differences)</span>
<span id="cb5-5"><a href="#cb5-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb5-6"><a href="#cb5-6" aria-hidden="true" tabindex="-1"></a>results <span class="op">=</span> np.zeros(n_trials)</span>
<span id="cb5-7"><a href="#cb5-7" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> i <span class="kw">in</span> <span class="bu">range</span>(n_trials):</span>
<span id="cb5-8"><a href="#cb5-8" aria-hidden="true" tabindex="-1"></a> <span class="co"># Choose random signs to perform random flips of the pairs.</span></span>
<span id="cb5-9"><a href="#cb5-9" aria-hidden="true" tabindex="-1"></a> signs <span class="op">=</span> rnd.choice([<span class="op">-</span><span class="dv">1</span>, <span class="dv">1</span>], size<span class="op">=</span>n_both)</span>
<span id="cb5-10"><a href="#cb5-10" aria-hidden="true" tabindex="-1"></a> <span class="co"># Do flips.</span></span>
<span id="cb5-11"><a href="#cb5-11" aria-hidden="true" tabindex="-1"></a> fake_differences <span class="op">=</span> signs <span class="op">*</span> differences</span>
<span id="cb5-12"><a href="#cb5-12" aria-hidden="true" tabindex="-1"></a> <span class="co"># Calculate mean difference and store result.</span></span>
<span id="cb5-13"><a href="#cb5-13" aria-hidden="true" tabindex="-1"></a> results[i] <span class="op">=</span> np.mean(fake_differences)</span>
<span id="cb5-14"><a href="#cb5-14" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb5-15"><a href="#cb5-15" aria-hidden="true" tabindex="-1"></a>k <span class="op">=</span> np.<span class="bu">sum</span>(np.<span class="bu">abs</span>(results) <span class="op">>=</span> np.<span class="bu">abs</span>(observed_mdiff))</span>
<span id="cb5-16"><a href="#cb5-16" aria-hidden="true" tabindex="-1"></a>kk <span class="op">=</span> k <span class="op">/</span> n_trials</span>
<span id="cb5-17"><a href="#cb5-17" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="st">'Sign-flip p null-world abs >= abs observed:'</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>Sign-flip p null-world abs >= abs observed: 0.0271</code></pre>
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<p>Notice that the sign-flip test, in which we preserve the information about the patients / cars, is much more convincing than the permutation or bootstrap tests above, where we choose to ignore that information.</p>
<p>This can occur when the values within the pairs (rows) are similar to each other, but less similar across different pairs (rows).</p>
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End of notebook: Paired differences solution
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<p><code>paired_differences_solution</code> starts at <a href="#nte-paired_differences_solution" class="quarto-xref">Note <span>A.1</span></a>.</p>
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<section id="sec-soln-seatbelt-proportions" class="level2" data-number="A.2">
<h2 data-number="A.2" class="anchored" data-anchor-id="sec-soln-seatbelt-proportions"><span class="header-section-number">A.2</span> Solution to <a href="testing_measured.html#sec-exr-seatbelt-proportions" class="quarto-xref">seatbelt proportions exercise <span>24.3.2</span></a></h2>
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Note A.2: Notebook: Seatbelt proportion solution
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<div class="nb-links">
<p><a class="notebook-link" href="notebooks/seatbelt_proportion_solution.ipynb">Download notebook</a> <a class="interact-button" href="./interact/lab/index.html?path=seatbelt_proportion_solution.ipynb">Interact</a></p>
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<div class="nb-start" name="seatbelt_proportion_solution" title="Seatbelt proportion solution">
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<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb7"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb7-1"><a href="#cb7-1" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> numpy <span class="im">as</span> np</span>
<span id="cb7-2"><a href="#cb7-2" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> matplotlib.pyplot <span class="im">as</span> plt</span>
<span id="cb7-3"><a href="#cb7-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb7-4"><a href="#cb7-4" aria-hidden="true" tabindex="-1"></a>rnd <span class="op">=</span> np.random.default_rng()</span>
<span id="cb7-5"><a href="#cb7-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb7-6"><a href="#cb7-6" aria-hidden="true" tabindex="-1"></a>pittsburgh <span class="op">=</span> np.repeat([<span class="st">'seatbelt'</span>, <span class="st">'none'</span>], [<span class="dv">36</span>, <span class="dv">36</span>])</span>
<span id="cb7-7"><a href="#cb7-7" aria-hidden="true" tabindex="-1"></a>n_pitts <span class="op">=</span> <span class="bu">len</span>(pittsburgh)</span>
<span id="cb7-8"><a href="#cb7-8" aria-hidden="true" tabindex="-1"></a>chicago <span class="op">=</span> np.repeat([<span class="st">'seatbelt'</span>, <span class="st">'none'</span>], [<span class="dv">77</span>, <span class="dv">52</span>])</span>
<span id="cb7-9"><a href="#cb7-9" aria-hidden="true" tabindex="-1"></a>n_chicago <span class="op">=</span> <span class="bu">len</span>(chicago)</span>
<span id="cb7-10"><a href="#cb7-10" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb7-11"><a href="#cb7-11" aria-hidden="true" tabindex="-1"></a>n_trials <span class="op">=</span> <span class="dv">10_000</span></span>
<span id="cb7-12"><a href="#cb7-12" aria-hidden="true" tabindex="-1"></a>results <span class="op">=</span> np.zeros(n_trials)</span>
<span id="cb7-13"><a href="#cb7-13" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb7-14"><a href="#cb7-14" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> i <span class="kw">in</span> <span class="bu">range</span>(n_trials):</span>
<span id="cb7-15"><a href="#cb7-15" aria-hidden="true" tabindex="-1"></a> fake_pitts <span class="op">=</span> rnd.choice(pittsburgh, size<span class="op">=</span>n_pitts)</span>
<span id="cb7-16"><a href="#cb7-16" aria-hidden="true" tabindex="-1"></a> fake_chicago <span class="op">=</span> rnd.choice(chicago, size<span class="op">=</span>n_chicago)</span>
<span id="cb7-17"><a href="#cb7-17" aria-hidden="true" tabindex="-1"></a> fake_p_pitts <span class="op">=</span> np.<span class="bu">sum</span>(fake_pitts <span class="op">==</span> <span class="st">'seatbelt'</span>) <span class="op">/</span> n_pitts</span>
<span id="cb7-18"><a href="#cb7-18" aria-hidden="true" tabindex="-1"></a> fake_p_chicago <span class="op">=</span> np.<span class="bu">sum</span>(fake_chicago <span class="op">==</span> <span class="st">'seatbelt'</span>) <span class="op">/</span> n_chicago</span>
<span id="cb7-19"><a href="#cb7-19" aria-hidden="true" tabindex="-1"></a> fake_p_diff <span class="op">=</span> fake_p_pitts <span class="op">-</span> fake_p_chicago</span>
<span id="cb7-20"><a href="#cb7-20" aria-hidden="true" tabindex="-1"></a> results[i] <span class="op">=</span> fake_p_diff</span>
<span id="cb7-21"><a href="#cb7-21" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb7-22"><a href="#cb7-22" aria-hidden="true" tabindex="-1"></a>plt.hist(results, bins<span class="op">=</span><span class="dv">25</span>)</span>
<span id="cb7-23"><a href="#cb7-23" aria-hidden="true" tabindex="-1"></a>plt.title(<span class="st">'Bootstrap distribution of p differences'</span>)</span>
<span id="cb7-24"><a href="#cb7-24" aria-hidden="true" tabindex="-1"></a>plt.xlabel(<span class="st">'Bootstrap p differences'</span>)</span>
<span id="cb7-25"><a href="#cb7-25" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb7-26"><a href="#cb7-26" aria-hidden="true" tabindex="-1"></a>p_limits <span class="op">=</span> np.quantile(results, [<span class="fl">0.025</span>, <span class="fl">0.975</span>])</span>
<span id="cb7-27"><a href="#cb7-27" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb7-28"><a href="#cb7-28" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="st">'95% percent limits for p differences:'</span>, p_limits)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>95% percent limits for p differences: [-0.24386305 0.04521964]</code></pre>
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<p><img src="exercise_solutions_files/figure-html/unnamed-chunk-5-1.png" class="img-fluid quarto-figure quarto-figure-center figure-img" style="width:70.0%"></p>
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End of notebook: Seatbelt proportion solution
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<p><code>seatbelt_proportion_solution</code> starts at <a href="#nte-seatbelt_proportion_solution" class="quarto-xref">Note <span>A.2</span></a>.</p>
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<section id="sec-soln-unemployment-percent" class="level2" data-number="A.3">
<h2 data-number="A.3" class="anchored" data-anchor-id="sec-soln-unemployment-percent"><span class="header-section-number">A.3</span> Solution for <a href="confidence_2.html#sec-exr-unemployment-percent" class="quarto-xref">unemployment percentage <span>27.8.1</span></a></h2>
<p>In a sample of 200 people, 7 percent are found to be unemployed. Determine a 95 percent confidence interval for the true population proportion.</p>
<div id="nte-unemployment_percent_solution" class="callout callout-style-default callout-note callout-titled">
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Note A.3: Notebook: Unemployment percent solution
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<p><a class="notebook-link" href="notebooks/unemployment_percent_solution.ipynb">Download notebook</a> <a class="interact-button" href="./interact/lab/index.html?path=unemployment_percent_solution.ipynb">Interact</a></p>
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<div class="nb-start" name="unemployment_percent_solution" title="Unemployment percent solution">
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<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb9"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb9-1"><a href="#cb9-1" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> numpy <span class="im">as</span> np</span>
<span id="cb9-2"><a href="#cb9-2" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> matplotlib.pyplot <span class="im">as</span> plt</span>
<span id="cb9-3"><a href="#cb9-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb9-4"><a href="#cb9-4" aria-hidden="true" tabindex="-1"></a>rnd <span class="op">=</span> np.random.default_rng()</span>
<span id="cb9-5"><a href="#cb9-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb9-6"><a href="#cb9-6" aria-hidden="true" tabindex="-1"></a>n_trials <span class="op">=</span> <span class="dv">10_000</span></span>
<span id="cb9-7"><a href="#cb9-7" aria-hidden="true" tabindex="-1"></a>results <span class="op">=</span> np.zeros(n_trials)</span>
<span id="cb9-8"><a href="#cb9-8" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb9-9"><a href="#cb9-9" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> i <span class="kw">in</span> <span class="bu">range</span>(n_trials):</span>
<span id="cb9-10"><a href="#cb9-10" aria-hidden="true" tabindex="-1"></a> fake_people <span class="op">=</span> rnd.choice([<span class="st">'no job'</span>, <span class="st">'job'</span>], size<span class="op">=</span><span class="dv">200</span>, p<span class="op">=</span>[<span class="fl">0.07</span>, <span class="fl">0.93</span>])</span>
<span id="cb9-11"><a href="#cb9-11" aria-hidden="true" tabindex="-1"></a> p_unemployed <span class="op">=</span> np.<span class="bu">sum</span>(fake_people <span class="op">==</span> <span class="st">'no job'</span>) <span class="op">/</span> <span class="dv">200</span></span>
<span id="cb9-12"><a href="#cb9-12" aria-hidden="true" tabindex="-1"></a> results[i] <span class="op">=</span> p_unemployed</span>
<span id="cb9-13"><a href="#cb9-13" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb9-14"><a href="#cb9-14" aria-hidden="true" tabindex="-1"></a>plt.hist(results, bins<span class="op">=</span><span class="dv">25</span>)</span>
<span id="cb9-15"><a href="#cb9-15" aria-hidden="true" tabindex="-1"></a>plt.title(<span class="st">'Bootstrap distribution p unemployed'</span>)</span>
<span id="cb9-16"><a href="#cb9-16" aria-hidden="true" tabindex="-1"></a>plt.xlabel(<span class="st">'Bootstrap p unemployed'</span>)</span>
<span id="cb9-17"><a href="#cb9-17" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb9-18"><a href="#cb9-18" aria-hidden="true" tabindex="-1"></a>p_limits <span class="op">=</span> np.quantile(results, [<span class="fl">0.025</span>, <span class="fl">0.975</span>])</span>
<span id="cb9-19"><a href="#cb9-19" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb9-20"><a href="#cb9-20" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="st">'95% percent limits for p differences:'</span>, p_limits)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>95% percent limits for p differences: [0.035 0.105]</code></pre>
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End of notebook: Unemployment percent solution
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<p><code>unemployment_percent_solution</code> starts at <a href="#nte-unemployment_percent_solution" class="quarto-xref">Note <span>A.3</span></a>.</p>
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<section id="sec-soln-battery-lifetime" class="level2" data-number="A.4">
<h2 data-number="A.4" class="anchored" data-anchor-id="sec-soln-battery-lifetime"><span class="header-section-number">A.4</span> Solution for <a href="confidence_2.html#sec-exr-battery-lifetime" class="quarto-xref">battery lifetime <span>27.8.2</span></a></h2>
<blockquote class="blockquote">
<p>A sample of 20 batteries is tested, and the average lifetime is 28.85 months. Establish a 95 percent confidence interval for the true average value. The sample values (lifetimes in months) are listed below.</p>
</blockquote>
<p>We use the “bootstrap” technique of drawing many bootstrap re-samples with replacement from the original sample, and observing how the re-sample means are distributed.</p>
<div id="nte-battery_lifetime_solution" class="callout callout-style-default callout-note callout-titled">
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Note A.4: Notebook: Battery lifetime solution
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<p><a class="notebook-link" href="notebooks/battery_lifetime_solution.ipynb">Download notebook</a> <a class="interact-button" href="./interact/lab/index.html?path=battery_lifetime_solution.ipynb">Interact</a></p>
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<div class="nb-start" name="battery_lifetime_solution" title="Battery lifetime solution">
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<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb11"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb11-1"><a href="#cb11-1" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> numpy <span class="im">as</span> np</span>
<span id="cb11-2"><a href="#cb11-2" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> matplotlib.pyplot <span class="im">as</span> plt</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>rnd <span class="op">=</span> np.random.default_rng()</span>
<span id="cb11-5"><a href="#cb11-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb11-6"><a href="#cb11-6" aria-hidden="true" tabindex="-1"></a>lifetimes <span class="op">=</span> np.array([<span class="dv">30</span>, <span class="dv">32</span>, <span class="dv">31</span>, <span class="dv">28</span>, <span class="dv">31</span>, <span class="dv">29</span>, <span class="dv">29</span>, <span class="dv">24</span>, <span class="dv">30</span>, <span class="dv">31</span>,</span>
<span id="cb11-7"><a href="#cb11-7" aria-hidden="true" tabindex="-1"></a> <span class="dv">28</span>, <span class="dv">28</span>, <span class="dv">32</span>, <span class="dv">31</span>, <span class="dv">24</span>, <span class="dv">23</span>, <span class="dv">31</span>, <span class="dv">27</span>, <span class="dv">27</span>, <span class="dv">31</span>])</span>
<span id="cb11-8"><a href="#cb11-8" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb11-9"><a href="#cb11-9" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="st">'Mean is:'</span>, np.mean(lifetimes))</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>Mean is: 28.85</code></pre>
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<div class="sourceCode cell-code" id="cb13"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb13-1"><a href="#cb13-1" aria-hidden="true" tabindex="-1"></a>n_lifetimes <span class="op">=</span> <span class="bu">len</span>(lifetimes)</span>
<span id="cb13-2"><a href="#cb13-2" aria-hidden="true" tabindex="-1"></a>results <span class="op">=</span> np.zeros(n_trials)</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="cf">for</span> i <span class="kw">in</span> <span class="bu">range</span>(n_trials):</span>
<span id="cb13-5"><a href="#cb13-5" aria-hidden="true" tabindex="-1"></a> <span class="co"># Draw 20 lifetimes from "lifetimes, randomly and with replacement.</span></span>
<span id="cb13-6"><a href="#cb13-6" aria-hidden="true" tabindex="-1"></a> fake_lifetimes <span class="op">=</span> rnd.choice(lifetimes, size<span class="op">=</span>n_lifetimes)</span>
<span id="cb13-7"><a href="#cb13-7" aria-hidden="true" tabindex="-1"></a> <span class="co"># Find the average lifetime of the 20.</span></span>
<span id="cb13-8"><a href="#cb13-8" aria-hidden="true" tabindex="-1"></a> fake_mean <span class="op">=</span> np.mean(fake_lifetimes)</span>
<span id="cb13-9"><a href="#cb13-9" aria-hidden="true" tabindex="-1"></a> <span class="co"># Keep score.</span></span>
<span id="cb13-10"><a href="#cb13-10" aria-hidden="true" tabindex="-1"></a> results[i] <span class="op">=</span> fake_mean</span>
<span id="cb13-11"><a href="#cb13-11" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb13-12"><a href="#cb13-12" aria-hidden="true" tabindex="-1"></a>plt.hist(results, bins<span class="op">=</span><span class="dv">25</span>)</span>
<span id="cb13-13"><a href="#cb13-13" aria-hidden="true" tabindex="-1"></a>plt.title(<span class="st">'Bootstrap distribution of mean battery lifetimes'</span>)</span>
<span id="cb13-14"><a href="#cb13-14" aria-hidden="true" tabindex="-1"></a>plt.xlabel(<span class="st">'Bootstrap mean battery lifetime'</span>)</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>mean_limits <span class="op">=</span> np.quantile(results, [<span class="fl">0.025</span>, <span class="fl">0.975</span>])</span>
<span id="cb13-17"><a href="#cb13-17" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb13-18"><a href="#cb13-18" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="st">'95% percent limits for mean lifetimes:'</span>, mean_limits)</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>95% percent limits for mean lifetimes: [27.65 29.95]</code></pre>
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<p><img src="exercise_solutions_files/figure-html/unnamed-chunk-9-1.png" class="img-fluid quarto-figure quarto-figure-center figure-img" style="width:70.0%"></p>
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End of notebook: Battery lifetime solution
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<p><code>battery_lifetime_solution</code> starts at <a href="#nte-battery_lifetime_solution" class="quarto-xref">Note <span>A.4</span></a>.</p>
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<section id="sec-soln-optical-density" class="level2" data-number="A.5">
<h2 data-number="A.5" class="anchored" data-anchor-id="sec-soln-optical-density"><span class="header-section-number">A.5</span> Solution for <a href="confidence_2.html#sec-exr-optical-density" class="quarto-xref">optical density <span>27.8.3</span></a></h2>
<div id="nte-optical_density_solution" class="callout callout-style-default callout-note callout-titled">
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Note A.5: Notebook: Optical density solution
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<div class="nb-links">
<p><a class="notebook-link" href="notebooks/optical_density_solution.ipynb">Download notebook</a> <a class="interact-button" href="./interact/lab/index.html?path=optical_density_solution.ipynb">Interact</a></p>
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<div class="nb-start" name="optical_density_solution" title="Optical density solution">
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<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb15"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb15-1"><a href="#cb15-1" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> numpy <span class="im">as</span> np</span>
<span id="cb15-2"><a href="#cb15-2" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> matplotlib.pyplot <span class="im">as</span> plt</span>
<span id="cb15-3"><a href="#cb15-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb15-4"><a href="#cb15-4" aria-hidden="true" tabindex="-1"></a>rnd <span class="op">=</span> np.random.default_rng()</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<p>Suppose we have 10 measurements of Optical Density on a batch of HIV negative control samples:</p>
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<div class="sourceCode cell-code" id="cb16"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb16-1"><a href="#cb16-1" aria-hidden="true" tabindex="-1"></a>density <span class="op">=</span> np.array(</span>
<span id="cb16-2"><a href="#cb16-2" aria-hidden="true" tabindex="-1"></a> [<span class="fl">.02</span>, <span class="fl">.026</span>, <span class="fl">.023</span>, <span class="fl">.017</span>, <span class="fl">.022</span>, <span class="fl">.019</span>, <span class="fl">.018</span>, <span class="fl">.018</span>, <span class="fl">.017</span>, <span class="fl">.022</span>])</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<p>Derive a 95 percent confidence interval for the sample mean. Are there enough measurements to produce a satisfactory answer?</p>
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<div class="sourceCode cell-code" id="cb17"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb17-1"><a href="#cb17-1" aria-hidden="true" tabindex="-1"></a>n_density <span class="op">=</span> <span class="bu">len</span>(density)</span>
<span id="cb17-2"><a href="#cb17-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb17-3"><a href="#cb17-3" aria-hidden="true" tabindex="-1"></a>n_trials <span class="op">=</span> <span class="dv">10_000</span></span>
<span id="cb17-4"><a href="#cb17-4" aria-hidden="true" tabindex="-1"></a>results <span class="op">=</span> np.zeros(n_trials)</span>
<span id="cb17-5"><a href="#cb17-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb17-6"><a href="#cb17-6" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> i <span class="kw">in</span> <span class="bu">range</span>(n_trials):</span>
<span id="cb17-7"><a href="#cb17-7" aria-hidden="true" tabindex="-1"></a> fake_density <span class="op">=</span> rnd.choice(density, size<span class="op">=</span>n_density)</span>
<span id="cb17-8"><a href="#cb17-8" aria-hidden="true" tabindex="-1"></a> results[i] <span class="op">=</span> np.mean(fake_density)</span>
<span id="cb17-9"><a href="#cb17-9" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb17-10"><a href="#cb17-10" aria-hidden="true" tabindex="-1"></a>plt.hist(results, bins<span class="op">=</span><span class="dv">25</span>)</span>
<span id="cb17-11"><a href="#cb17-11" aria-hidden="true" tabindex="-1"></a>plt.title(<span class="st">'Bootstrap distribution of density means'</span>)</span>
<span id="cb17-12"><a href="#cb17-12" aria-hidden="true" tabindex="-1"></a>plt.xlabel(<span class="st">'Bootstrap density means'</span>)</span>
<span id="cb17-13"><a href="#cb17-13" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb17-14"><a href="#cb17-14" aria-hidden="true" tabindex="-1"></a>mean_limits <span class="op">=</span> np.quantile(results, [<span class="fl">0.025</span>, <span class="fl">0.975</span>])</span>
<span id="cb17-15"><a href="#cb17-15" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb17-16"><a href="#cb17-16" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="st">'95% percent limits for density mean:'</span>, mean_limits)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>95% percent limits for density mean: [0.0185 0.022 ]</code></pre>
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End of notebook: Optical density solution
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<p><code>optical_density_solution</code> starts at <a href="#nte-optical_density_solution" class="quarto-xref">Note <span>A.5</span></a>.</p>
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<section id="sec-soln-voter-participation" class="level2" data-number="A.6">
<h2 data-number="A.6" class="anchored" data-anchor-id="sec-soln-voter-participation"><span class="header-section-number">A.6</span> Solution for <a href="correlation_causation.html#sec-exr-voter-participation" class="quarto-xref">voter participation <span>29.7.1</span></a></h2>
<p>The observed correlation coefficient between voter participation and spread is moderate and negative. Is this more negative that what might occur by chance, if no correlation exists in some underlying population, from which this sample was taken?</p>
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<li>Create two groups of paper cards: 25 with participation rates, and 25 with the spread values. Arrange the cards in pairs in accordance with the table, and compute the correlation coefficient between the shuffled participation and spread variables.</li>
<li>Shuffle one of the sets, say that with participation, and compute correlation between shuffled participation and spread.</li>
<li>Repeat step 2 many, say 1000, times. Compute the proportion of the trials in which correlation was at least as negative as that for the original data.</li>
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Note A.6: Notebook: Voter participation in 1844 election
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<p><a class="notebook-link" href="notebooks/voter_participation_solution.zip">Download zip with notebook + data file</a> <a class="interact-button" href="./interact/lab/index.html?path=voter_participation_solution.ipynb">Interact</a></p>
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<div class="nb-start" name="voter_participation_solution" title="Voter participation in 1844 election">
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<div class="sourceCode cell-code" id="cb19"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb19-1"><a href="#cb19-1" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> numpy <span class="im">as</span> np</span>
<span id="cb19-2"><a href="#cb19-2" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> pandas <span class="im">as</span> pd</span>
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<span id="cb19-4"><a href="#cb19-4" aria-hidden="true" tabindex="-1"></a>rnd <span class="op">=</span> np.random.default_rng()</span>
<span id="cb19-5"><a href="#cb19-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb19-6"><a href="#cb19-6" aria-hidden="true" tabindex="-1"></a>voter_df <span class="op">=</span> pd.read_csv(<span class="st">'data/election_1844.csv'</span>)</span>
<span id="cb19-7"><a href="#cb19-7" aria-hidden="true" tabindex="-1"></a>participation <span class="op">=</span> np.array(voter_df[<span class="st">'Participation'</span>])</span>
<span id="cb19-8"><a href="#cb19-8" aria-hidden="true" tabindex="-1"></a>spread <span class="op">=</span> np.array(voter_df[<span class="st">'Spread'</span>])</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<div class="sourceCode cell-code" id="cb20"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb20-1"><a href="#cb20-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Compute correlation. It's -0.425.</span></span>
<span id="cb20-2"><a href="#cb20-2" aria-hidden="true" tabindex="-1"></a>actual_r <span class="op">=</span> np.corrcoef(participation, spread)[<span class="dv">0</span>, <span class="dv">1</span>]</span>
<span id="cb20-3"><a href="#cb20-3" aria-hidden="true" tabindex="-1"></a>actual_r</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>np.float64(-0.4249067562318352)</code></pre>
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<div class="sourceCode cell-code" id="cb22"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb22-1"><a href="#cb22-1" aria-hidden="true" tabindex="-1"></a>n_trials <span class="op">=</span> <span class="dv">10_000</span></span>
<span id="cb22-2"><a href="#cb22-2" aria-hidden="true" tabindex="-1"></a>results <span class="op">=</span> np.zeros(n_trials)</span>
<span id="cb22-3"><a href="#cb22-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb22-4"><a href="#cb22-4" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> i <span class="kw">in</span> <span class="bu">range</span>(n_trials):</span>
<span id="cb22-5"><a href="#cb22-5" aria-hidden="true" tabindex="-1"></a> <span class="co"># Shuffle the participation rates.</span></span>
<span id="cb22-6"><a href="#cb22-6" aria-hidden="true" tabindex="-1"></a> shuffled <span class="op">=</span> rnd.permuted(participation)</span>
<span id="cb22-7"><a href="#cb22-7" aria-hidden="true" tabindex="-1"></a> <span class="co"># Compute re-sampled correlation.</span></span>
<span id="cb22-8"><a href="#cb22-8" aria-hidden="true" tabindex="-1"></a> fake_r <span class="op">=</span> np.corrcoef(shuffled, spread)[<span class="dv">0</span>, <span class="dv">1</span>]</span>
<span id="cb22-9"><a href="#cb22-9" aria-hidden="true" tabindex="-1"></a> <span class="co"># Keep the value in the results.</span></span>
<span id="cb22-10"><a href="#cb22-10" aria-hidden="true" tabindex="-1"></a> results[i] <span class="op">=</span> fake_r</span>
<span id="cb22-11"><a href="#cb22-11" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb22-12"><a href="#cb22-12" aria-hidden="true" tabindex="-1"></a>plt.hist(results, bins<span class="op">=</span><span class="dv">25</span>)</span>
<span id="cb22-13"><a href="#cb22-13" aria-hidden="true" tabindex="-1"></a>plt.title(<span class="st">'Distribution of shuffled correlations'</span>)</span>
<span id="cb22-14"><a href="#cb22-14" aria-hidden="true" tabindex="-1"></a>plt.xlabel(<span class="st">'Correlation with shuffled participation'</span>)</span>
<span id="cb22-15"><a href="#cb22-15" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb22-16"><a href="#cb22-16" aria-hidden="true" tabindex="-1"></a><span class="co"># Count the trials when result <= observed.</span></span>
<span id="cb22-17"><a href="#cb22-17" aria-hidden="true" tabindex="-1"></a>k <span class="op">=</span> np.<span class="bu">sum</span>(results <span class="op"><=</span> actual_r)</span>
<span id="cb22-18"><a href="#cb22-18" aria-hidden="true" tabindex="-1"></a><span class="co"># Compute the proportion of such trials.</span></span>
<span id="cb22-19"><a href="#cb22-19" aria-hidden="true" tabindex="-1"></a>kk <span class="op">=</span> k <span class="op">/</span> n_trials</span>
<span id="cb22-20"><a href="#cb22-20" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb22-21"><a href="#cb22-21" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="st">'Proportion of shuffled r <= observed:'</span>, np.<span class="bu">round</span>(kk, <span class="dv">2</span>))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>Proportion of shuffled r <= observed: 0.03</code></pre>
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End of notebook: Voter participation in 1844 election