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<li><a href="#introduction" id="toc-introduction" class="nav-link active" data-scroll-target="#introduction"><span class="header-section-number">7.1</span> Introduction</a></li>
<li><a href="#samples-and-labels-and-strings" id="toc-samples-and-labels-and-strings" class="nav-link" data-scroll-target="#samples-and-labels-and-strings">7.2 Samples and labels and strings</a></li>
<li><a href="#sec-types-values" id="toc-sec-types-values" class="nav-link" data-scroll-target="#sec-types-values">7.3 Types of values in Python</a></li>
<li><a href="#sec-intro-to-strings" id="toc-sec-intro-to-strings" class="nav-link" data-scroll-target="#sec-intro-to-strings">7.4 String values</a></li>
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<li><a href="#sec-repeating" id="toc-sec-repeating" class="nav-link" data-scroll-target="#sec-repeating">7.6 Repeating elements</a></li>
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<h1 class="title"><span id="sec-sampling-tools" class="quarto-section-identifier"><span class="chapter-number">7</span> <span class="chapter-title">Tools for samples and sampling</span></span></h1>
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<section id="introduction" class="level2" data-number="7.1">
<h2 data-number="7.1" class="anchored" data-anchor-id="introduction"><span class="header-section-number">7.1</span> Introduction</h2>
<p>Now you have some experience with Python, probabilities and resampling, it is time to introduce some useful tools for our experiments and programs.</p>
<div id="nte-sampling_tools" class="callout callout-style-default callout-note callout-titled">
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Note 7.1: Notebook: Sampling tools
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<div class="nb-links">
<p><a class="notebook-link" href="notebooks/sampling_tools.ipynb">Download notebook</a> <a class="interact-button" href="./interact/lab/index.html?path=sampling_tools.ipynb">Interact</a></p>
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<div class="nb-start" name="sampling_tools" title="Sampling tools">
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</section>
<section id="samples-and-labels-and-strings" class="level2" data-number="7.2">
<h2 data-number="7.2" class="anchored" data-anchor-id="samples-and-labels-and-strings">7.2 Samples and labels and strings</h2>
<p>Thus far we have used numbers such as 1 and 0 and 10 to represent the elements we are sampling from. For example, in <a href="resampling_with_code2.html" class="quarto-xref"><span>Chapter 6</span></a>, we were simulating the chance of a particular juror being black, given that 26% of the eligible jurors in the county were black. We used <em>integers</em> for that task, where we started with all the integers from 0 through 99, and asked NumPy to select values at random from those integers. When NumPy selected an integer from 0 through 25, we chose to label the resulting simulated juror as black — there are 26 integers in the range 0 through 25, so there is a 26% chance that any one integer will be in that range. If the integer was from 26 through 99, the simulated juror was white (there are 74 integers in the range 26 through 99).</p>
<p>Here is the process of simulating a single juror, adapted from <a href="resampling_with_code2.html#sec-random-zero-through-ninety-nine" class="quarto-xref"><span>Section 6.3.3</span></a>:</p>
<div class="cell" data-layout-align="center">
<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="co"># Ask Numpy for a random number generator.</span></span>
<span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a>rnd <span class="op">=</span> np.random.default_rng()</span>
<span id="cb1-4"><a href="#cb1-4" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-5"><a href="#cb1-5" aria-hidden="true" tabindex="-1"></a><span class="co"># All the integers from 0 up to, but not including 100.</span></span>
<span id="cb1-6"><a href="#cb1-6" aria-hidden="true" tabindex="-1"></a>zero_thru_99 <span class="op">=</span> np.arange(<span class="dv">100</span>)</span>
<span id="cb1-7"><a href="#cb1-7" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-8"><a href="#cb1-8" aria-hidden="true" tabindex="-1"></a><span class="co"># Get one random numbers from 0 through 99</span></span>
<span id="cb1-9"><a href="#cb1-9" aria-hidden="true" tabindex="-1"></a>a <span class="op">=</span> rnd.choice(zero_thru_99)</span>
<span id="cb1-10"><a href="#cb1-10" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-11"><a href="#cb1-11" aria-hidden="true" tabindex="-1"></a><span class="co"># Show the result</span></span>
<span id="cb1-12"><a href="#cb1-12" aria-hidden="true" tabindex="-1"></a>a</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>np.int64(59)</code></pre>
</div>
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<p>After that, we have to unpack our labeling of 0 through 25 as being “black” and 26 through 99 as being “white”. We might do that like this:</p>
<div class="cell" data-layout-align="center">
<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>this_juror_is_black <span class="op">=</span> a <span class="op"><</span> <span class="dv">26</span></span>
<span id="cb3-2"><a href="#cb3-2" aria-hidden="true" tabindex="-1"></a>this_juror_is_black</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>np.False_</code></pre>
</div>
</div>
<div class="python">
<div class="callout callout-style-default callout-note callout-titled">
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<i class="callout-icon"></i>
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<div class="callout-title-container flex-fill">
Numpy display of single Boolean values
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<div class="callout-body-container callout-body">
<p>You saw in <a href="resampling_method.html#nte-numpy-int-display" class="quarto-xref">Note <span>2.3</span></a> and <a href="resampling_with_code.html#nte-numpy-float-display" class="quarto-xref">Note <span>5.3</span></a> that Numpy has a particular way of displaying single values. <a href="resampling_method.html#nte-numpy-int-display" class="quarto-xref">Note <span>2.3</span></a> was about display of <em>integer</em> (<code>int</code>) values, and <a href="resampling_with_code.html#nte-numpy-float-display" class="quarto-xref">Note <span>5.3</span></a> was about display of <em>floating point</em> (<code>float</code>) values. The display above is Numpy’s version of the Boolean value <code>False</code>. As you’d expect, Numpy shows the single value <code>True</code> as <code>np.True_</code>.</p>
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</div>
<p>This all works as we want it to, but it’s just a little bit difficult to remember the coding (less than 26 means “black”, greater than 25 means “white”). We had to use that coding because we committed ourselves to using random numbers to simulate the outcomes.</p>
<p>However, Python can also store bits of text, called <em>strings</em>. Values that are bits of text can be very useful because the text values can be memorable labels for the entities we are sampling from, in our simulations.</p>
<p>Before we get to strings, let us consider the different types of value we have seen so far.</p>
</section>
<section id="sec-types-values" class="level2" data-number="7.3">
<h2 data-number="7.3" class="anchored" data-anchor-id="sec-types-values">7.3 Types of values in Python</h2>
<div class="python">
<p>You have already come across the idea that Python values can be integers (positive or negative whole numbers), like this:</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>v <span class="op">=</span> <span class="dv">10</span></span>
<span id="cb5-2"><a href="#cb5-2" aria-hidden="true" tabindex="-1"></a>v</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>10</code></pre>
</div>
</div>
<p>Here the variable <code>v</code> holds the value. We can see what type of value <code>v</code> holds by using the <code>type</code> function:</p>
<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="bu">type</span>(v)</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><class 'int'></code></pre>
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</div>
<p>Python can also have <em>floating point</em> values. These are values with a decimal point — numbers that do not have to be integers, but also can be any value between the integers. These floating point values are of type <code>float</code>:</p>
<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>f <span class="op">=</span> <span class="fl">10.1</span></span>
<span id="cb9-2"><a href="#cb9-2" aria-hidden="true" tabindex="-1"></a><span class="bu">type</span>(f)</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><class 'float'></code></pre>
</div>
</div>
<section id="sec-arrays" class="level3" data-number="7.3.1">
<h3 data-number="7.3.1" class="anchored" data-anchor-id="sec-arrays">7.3.1 Numpy arrays</h3>
<p>You have also seen that Numpy contains another type, the <em>array</em>. An array is a value that contains a sequence of values. For example, here is an array of integers:</p>
<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>arr <span class="op">=</span> np.array([<span class="dv">0</span>, <span class="dv">10</span>, <span class="dv">99</span>, <span class="dv">4</span>])</span>
<span id="cb11-2"><a href="#cb11-2" aria-hidden="true" tabindex="-1"></a>arr</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>array([ 0, 10, 99, 4])</code></pre>
</div>
</div>
<p>Notice that this value <code>arr</code> is of type <code>np.ndarray</code>:</p>
<div class="cell" data-layout-align="center">
<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><span class="bu">type</span>(arr)</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><class 'numpy.ndarray'></code></pre>
</div>
</div>
<p>The array has its own internal record of what type of values it holds. This is called the array <code>dtype</code>:</p>
<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>arr.dtype</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>dtype('int64')</code></pre>
</div>
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<p>The array <code>dtype</code> records the type of value stored in the array. All values in the array must be of this type, and all values in the array are therefore of the same type.</p>
<p>The array above contains integers, but we can also make arrays containing floating point values:</p>
<div class="cell" data-layout-align="center">
<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>float_arr <span class="op">=</span> np.array([<span class="fl">0.1</span>, <span class="fl">10.1</span>, <span class="fl">99.0</span>, <span class="fl">4.3</span>])</span>
<span id="cb17-2"><a href="#cb17-2" aria-hidden="true" tabindex="-1"></a>float_arr</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>array([ 0.1, 10.1, 99. , 4.3])</code></pre>
</div>
</div>
<div class="cell" data-layout-align="center">
<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>float_arr.dtype</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>dtype('float64')</code></pre>
</div>
</div>
</section>
<section id="sec-lists" class="level3" data-number="7.3.2">
<h3 data-number="7.3.2" class="anchored" data-anchor-id="sec-lists">7.3.2 Lists</h3>
<p>We have elided past another Python type, the <em>list</em>. In fact we have already used lists in making arrays. For example, here we make an array with four values:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb21"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb21-1"><a href="#cb21-1" aria-hidden="true" tabindex="-1"></a>np.array([<span class="dv">0</span>, <span class="dv">10</span>, <span class="dv">99</span>, <span class="dv">4</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>array([ 0, 10, 99, 4])</code></pre>
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</div>
<p>We could also write the statement above in two steps:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb23"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb23-1"><a href="#cb23-1" aria-hidden="true" tabindex="-1"></a>my_list <span class="op">=</span> [<span class="dv">0</span>, <span class="dv">10</span>, <span class="dv">99</span>, <span class="dv">4</span>]</span>
<span id="cb23-2"><a href="#cb23-2" aria-hidden="true" tabindex="-1"></a>np.array(my_list)</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>array([ 0, 10, 99, 4])</code></pre>
</div>
</div>
<p>In the first statement — <code>my_list = [0, 10, 99, 4]</code> — we construct a <em>list</em> — a container for the four values. Let’s look at the <code>my_list</code> value:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb25"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb25-1"><a href="#cb25-1" aria-hidden="true" tabindex="-1"></a>my_list</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>[0, 10, 99, 4]</code></pre>
</div>
</div>
<p>Notice that we do not see <code>array</code> in the display — this is not an array but a list:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb27"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb27-1"><a href="#cb27-1" aria-hidden="true" tabindex="-1"></a><span class="bu">type</span>(my_list)</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><class 'list'></code></pre>
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</div>
<p>A list is a basic Python type. We can construct it by using the square brackets notation that you see above; we start with <code>[</code>, then we put the values we want to go in the list, separated by commas, followed by <code>]</code>. Here is another list:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb29"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb29-1"><a href="#cb29-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Creating another list.</span></span>
<span id="cb29-2"><a href="#cb29-2" aria-hidden="true" tabindex="-1"></a>list_2 <span class="op">=</span> [<span class="dv">5</span>, <span class="dv">10</span>, <span class="dv">20</span>]</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<p>As you saw, we have been building arrays by building lists, and then passing the list to the <code>np.array</code> function, to create an array.</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb30"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb30-1"><a href="#cb30-1" aria-hidden="true" tabindex="-1"></a>list_again <span class="op">=</span> [<span class="dv">100</span>, <span class="dv">10</span>, <span class="dv">0</span>]</span>
<span id="cb30-2"><a href="#cb30-2" aria-hidden="true" tabindex="-1"></a>np.array(list_again)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>array([100, 10, 0])</code></pre>
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<p>Of course, we can do this one line, as we have been doing up till now, by constructing the list inside the parentheses of the function. So, the following cell has just the same output as the cell above:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb32"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb32-1"><a href="#cb32-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Constructing the list inside the function brackets.</span></span>
<span id="cb32-2"><a href="#cb32-2" aria-hidden="true" tabindex="-1"></a>np.array([<span class="dv">100</span>, <span class="dv">10</span>, <span class="dv">0</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>array([100, 10, 0])</code></pre>
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<p>Lists are like arrays in that they are values that contain values, but they are unlike arrays in various ways — that we will not go into now. We often use lists to construct sequences into lists to turn them into arrays. For our purposes, and particularly for our calculations, arrays are much more useful and efficient than lists.</p>
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<section id="sec-intro-to-strings" class="level2" data-number="7.4">
<h2 data-number="7.4" class="anchored" data-anchor-id="sec-intro-to-strings">7.4 String values</h2>
<p>So far, all the values you have seen in Python arrays have been numbers. Now we get on to values that are bits of text. These are called <em>strings</em>.</p>
<p>Here is a single Python string value:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb34"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb34-1"><a href="#cb34-1" aria-hidden="true" tabindex="-1"></a>s <span class="op">=</span> <span class="st">"Resampling"</span></span>
<span id="cb34-2"><a href="#cb34-2" aria-hidden="true" tabindex="-1"></a>s</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>'Resampling'</code></pre>
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<p>What is the <code>type</code> of the new bit-of-text value <code>s</code>?</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb36"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb36-1"><a href="#cb36-1" aria-hidden="true" tabindex="-1"></a><span class="bu">type</span>(s)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code><class 'str'></code></pre>
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<p>The Python <code>str</code> value is a bit of text, and therefore consists of a sequence of characters.</p>
<p>As arrays are containers for other things, such as numbers, strings are containers for characters.</p>
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<p>As we can find the number of elements in an array (<a href="resampling_with_code2.html#sec-array-length" class="quarto-xref"><span>Section 6.5</span></a>), we can find the number of characters in a string with the <code>len</code> function:</p>
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<div class="sourceCode cell-code" id="cb38"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb38-1"><a href="#cb38-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Number of characters in s</span></span>
<span id="cb38-2"><a href="#cb38-2" aria-hidden="true" tabindex="-1"></a><span class="bu">len</span>(s)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>10</code></pre>
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<p>As we can <em>index</em> into array values to get individual elements (<a href="resampling_with_code2.html#sec-array-indexing" class="quarto-xref"><span>Section 6.6</span></a>), we can index into string values to get individual characters:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb40"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb40-1"><a href="#cb40-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Get the second character of the string</span></span>
<span id="cb40-2"><a href="#cb40-2" aria-hidden="true" tabindex="-1"></a><span class="co"># Remember, Python's index positions start at 0.</span></span>
<span id="cb40-3"><a href="#cb40-3" aria-hidden="true" tabindex="-1"></a>second_char <span class="op">=</span> s[<span class="dv">1</span>]</span>
<span id="cb40-4"><a href="#cb40-4" aria-hidden="true" tabindex="-1"></a>second_char</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>'e'</code></pre>
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</section>
<section id="strings-in-s" class="level2" data-number="7.5">
<h2 data-number="7.5" class="anchored" data-anchor-id="strings-in-s">7.5 Strings in arrays</h2>
<p>As we can store numbers as elements in arrays, we can also store strings as array elements.</p>
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<div class="sourceCode cell-code" id="cb42"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb42-1"><a href="#cb42-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Just for clarity, make the list first.</span></span>
<span id="cb42-2"><a href="#cb42-2" aria-hidden="true" tabindex="-1"></a><span class="co"># Lists can also contain strings.</span></span>
<span id="cb42-3"><a href="#cb42-3" aria-hidden="true" tabindex="-1"></a>list_of_strings <span class="op">=</span> [<span class="st">'Julian'</span>, <span class="st">'Lincoln'</span>, <span class="st">'Simon'</span>]</span>
<span id="cb42-4"><a href="#cb42-4" aria-hidden="true" tabindex="-1"></a><span class="co"># Then pass the list to np.array to make the array.</span></span>
<span id="cb42-5"><a href="#cb42-5" aria-hidden="true" tabindex="-1"></a>arr_of_strings <span class="op">=</span> np.array(list_of_strings)</span>
<span id="cb42-6"><a href="#cb42-6" aria-hidden="true" tabindex="-1"></a>arr_of_strings</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>array(['Julian', 'Lincoln', 'Simon'], dtype='<U7')</code></pre>
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</div>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb44"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb44-1"><a href="#cb44-1" aria-hidden="true" tabindex="-1"></a><span class="co"># We can also create the list and the array in one line,</span></span>
<span id="cb44-2"><a href="#cb44-2" aria-hidden="true" tabindex="-1"></a><span class="co"># as we have been doing up til now.</span></span>
<span id="cb44-3"><a href="#cb44-3" aria-hidden="true" tabindex="-1"></a>arr_of_strings <span class="op">=</span> np.array([<span class="st">'Julian'</span>, <span class="st">'Lincoln'</span>, <span class="st">'Simon'</span>])</span>
<span id="cb44-4"><a href="#cb44-4" aria-hidden="true" tabindex="-1"></a>arr_of_strings</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>array(['Julian', 'Lincoln', 'Simon'], dtype='<U7')</code></pre>
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<p>Notice the array <code>dtype</code>:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb46"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb46-1"><a href="#cb46-1" aria-hidden="true" tabindex="-1"></a>arr_of_strings.dtype</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>dtype('<U7')</code></pre>
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<p>The <code>U</code> in the <code>dtype</code> tells you that the elements in the array are <a href="https://en.wikipedia.org/wiki/Unicode">Unicode</a> strings (Unicode is a computer representation of text characters). The number after the <code>U</code> gives the maximum number of characters for any string in the array, here set to the length of the longest string when we created the array.</p>
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<p>It is easy to run into trouble with Numpy string arrays where the elements have a maximum length, as here. Remember, the <code>dtype</code> of the array tells you what type of element the array can hold. Here the <code>dtype</code> is telling you that the array can hold strings of maximum length 7 characters. Now imagine trying to put a longer string into the array — what do you think would happen?</p>
<p>This happens:</p>
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<div class="sourceCode cell-code" id="cb48"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb48-1"><a href="#cb48-1" aria-hidden="true" tabindex="-1"></a><span class="co"># An array of small strings.</span></span>
<span id="cb48-2"><a href="#cb48-2" aria-hidden="true" tabindex="-1"></a>small_strings <span class="op">=</span> np.array([<span class="st">'six'</span>, <span class="st">'one'</span>, <span class="st">'two'</span>])</span>
<span id="cb48-3"><a href="#cb48-3" aria-hidden="true" tabindex="-1"></a>small_strings.dtype</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>dtype('<U3')</code></pre>
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<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb50"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb50-1"><a href="#cb50-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Set a new value for the first element (first string).</span></span>
<span id="cb50-2"><a href="#cb50-2" aria-hidden="true" tabindex="-1"></a>small_strings[<span class="dv">0</span>] <span class="op">=</span> <span class="st">'seven'</span></span>
<span id="cb50-3"><a href="#cb50-3" aria-hidden="true" tabindex="-1"></a>small_strings</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>array(['sev', 'one', 'two'], dtype='<U3')</code></pre>
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<p>Numpy truncates the new string to match the original maximum length.</p>
<p>For that reason, it is often useful to instruct Numpy that you want to use effectively infinite length strings, by specifying the array <code>dtype</code> as <code>object</code> <em>when you make the array</em>, like this:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb52"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb52-1"><a href="#cb52-1" aria-hidden="true" tabindex="-1"></a><span class="co"># An array of small strings, but this time, tell Numpy</span></span>
<span id="cb52-2"><a href="#cb52-2" aria-hidden="true" tabindex="-1"></a><span class="co"># that the strings should be of effectively infinite length.</span></span>
<span id="cb52-3"><a href="#cb52-3" aria-hidden="true" tabindex="-1"></a>small_strings_better <span class="op">=</span> np.array([<span class="st">'six'</span>, <span class="st">'one'</span>, <span class="st">'two'</span>], dtype<span class="op">=</span><span class="bu">object</span>)</span>
<span id="cb52-4"><a href="#cb52-4" aria-hidden="true" tabindex="-1"></a>small_strings_better</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>array(['six', 'one', 'two'], dtype=object)</code></pre>
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<p>Notice that the code uses a <em>named function argument</em> (<a href="resampling_with_code.html#sec-named-arguments" class="quarto-xref"><span>Section 5.8</span></a>), to specify to <code>np.array</code> that the array elements should be of type <code>object</code>. This type can store any Python value, and so, when the array is storing strings, it will use Python’s own string values as elements, rather than the more efficient but more fragile Unicode strings that Numpy uses by default.</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb54"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb54-1"><a href="#cb54-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Set a new value for the first element in the new array.</span></span>
<span id="cb54-2"><a href="#cb54-2" aria-hidden="true" tabindex="-1"></a>small_strings_better[<span class="dv">0</span>] <span class="op">=</span> <span class="st">'seven'</span></span>
<span id="cb54-3"><a href="#cb54-3" aria-hidden="true" tabindex="-1"></a>small_strings_better</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>array(['seven', 'one', 'two'], dtype=object)</code></pre>
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<p>As for any array, you can select elements with <em>indexing</em>. When you select an element with a given position (index), you get the <em>string</em> at at that position:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb56"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb56-1"><a href="#cb56-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Julian Lincoln Simon's second name.</span></span>
<span id="cb56-2"><a href="#cb56-2" aria-hidden="true" tabindex="-1"></a><span class="co"># (Remember, Python's positions start at 0).</span></span>
<span id="cb56-3"><a href="#cb56-3" aria-hidden="true" tabindex="-1"></a>middle_name <span class="op">=</span> arr_of_strings[<span class="dv">1</span>]</span>
<span id="cb56-4"><a href="#cb56-4" aria-hidden="true" tabindex="-1"></a>middle_name</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.str_('Lincoln')</code></pre>
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Numpy display of single string-type values
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<p><code>np.str_</code> in the output shows us that the result was a Numpy single value of <em>string</em> (<code>str_</code>) type.</p>
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<p>Notice the output from this cell:</p>
<p>As for numbers, we can compare strings with, for example, the <code>==</code> operator, that asks whether the two strings are equal:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb58"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb58-1"><a href="#cb58-1" aria-hidden="true" tabindex="-1"></a>middle_name <span class="op">==</span> <span class="st">'Lincoln'</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>True</code></pre>
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</section>
<section id="sec-repeating" class="level2" data-number="7.6">
<h2 data-number="7.6" class="anchored" data-anchor-id="sec-repeating">7.6 Repeating elements</h2>
<p>Now let us go back to the problem of selecting black and white jurors.</p>
<p>We started with the strategy of using numbers 0 through 25 to mean “black” jurors, and 26 through 99 to mean “white” jurors. We selected values at random from 0 through 99, and then worked out whether the number meant a “black” juror (was less than 26) or a “white” juror (was greater than 25).</p>
<p>It would be good to use strings instead of numbers to identify the potential jurors. Then we would not have to remember our coding of 0 through 25 and 26 through 99.</p>
<p>If only there was a way to make an array of 100 strings, where 26 of the strings were “black” and 74 were “white”. Then we could select randomly from that array, and it would be immediately obvious that we had a “black” or “white” juror.</p>
<p>Luckily we can do that, by using the <code>np.repeat</code> function to construct the array.</p>
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<p><code>np.repeat</code> is a useful Numpy function that makes a new output array by repeating elements from the input arrays.</p>
<p>More specifically, <code>np.repeat</code> accepts two arguments (inputs), and sends back an array as output. The first input is an array or list or some other sequence for which you want to repeat elements. The second input tells <code>np.repeat</code> how many repeats to do, for each element.</p>
<p>This is most easy to see by examples.</p>
<p>Here we repeat all three elements in the first sequence (in this case a list) by a count (number) given in the second argument.</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb60"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb60-1"><a href="#cb60-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Repeat each element in the first list (or array) 4 times.</span></span>
<span id="cb60-2"><a href="#cb60-2" aria-hidden="true" tabindex="-1"></a>to_repeat <span class="op">=</span> [<span class="dv">1</span>, <span class="dv">2</span>, <span class="dv">3</span>] <span class="co"># First input argument.</span></span>
<span id="cb60-3"><a href="#cb60-3" aria-hidden="true" tabindex="-1"></a>n_repeats <span class="op">=</span> <span class="dv">4</span> <span class="co"># Second input argument.</span></span>
<span id="cb60-4"><a href="#cb60-4" aria-hidden="true" tabindex="-1"></a>np.repeat(to_repeat, n_repeats)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>array([1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3])</code></pre>
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<p>Of course we could also write this without the variables:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb62"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb62-1"><a href="#cb62-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Repeat each element in the first list (or array) 4 times.</span></span>
<span id="cb62-2"><a href="#cb62-2" aria-hidden="true" tabindex="-1"></a>np.repeat([<span class="dv">1</span>, <span class="dv">2</span>, <span class="dv">3</span>], <span class="dv">4</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>array([1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3])</code></pre>
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<p>The second argument (number of repeats) can also be a sequence, such as an array or list, with one element for each element of the first array. In that case, it gives the number of repeats for each element. We will use string elements for this example.</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb64"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb64-1"><a href="#cb64-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Repeat the first element 3 times, the second 2 times, and the third 4 times.</span></span>
<span id="cb64-2"><a href="#cb64-2" aria-hidden="true" tabindex="-1"></a><span class="co"># Second input is number of repeats for each element of the first argument.</span></span>
<span id="cb64-3"><a href="#cb64-3" aria-hidden="true" tabindex="-1"></a>np.repeat([<span class="st">'one'</span>, <span class="st">'two'</span>, <span class="st">'three'</span>], [<span class="dv">3</span>, <span class="dv">2</span>, <span class="dv">4</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>array(['one', 'one', 'one', 'two', 'two', 'three', 'three', 'three',
'three'], dtype='<U5')</code></pre>
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<p>Here we use <code>np.repeat</code> to make up our simulated jury pool.</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb66"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb66-1"><a href="#cb66-1" aria-hidden="true" tabindex="-1"></a><span class="co"># The values that we will repeat to fill up the larger array.</span></span>
<span id="cb66-2"><a href="#cb66-2" aria-hidden="true" tabindex="-1"></a><span class="co"># Use a list to store the sequence of values.</span></span>
<span id="cb66-3"><a href="#cb66-3" aria-hidden="true" tabindex="-1"></a>juror_types <span class="op">=</span> [<span class="st">'black'</span>, <span class="st">'white'</span>]</span>
<span id="cb66-4"><a href="#cb66-4" aria-hidden="true" tabindex="-1"></a><span class="co"># The number of times we want to repeat "black" and "white".</span></span>
<span id="cb66-5"><a href="#cb66-5" aria-hidden="true" tabindex="-1"></a><span class="co"># Use a list to store the sequence of values.</span></span>
<span id="cb66-6"><a href="#cb66-6" aria-hidden="true" tabindex="-1"></a>repeat_nos <span class="op">=</span> [<span class="dv">26</span>, <span class="dv">74</span>]</span>
<span id="cb66-7"><a href="#cb66-7" aria-hidden="true" tabindex="-1"></a><span class="co"># Repeat "black" 26 times and "white" 74 times.</span></span>
<span id="cb66-8"><a href="#cb66-8" aria-hidden="true" tabindex="-1"></a><span class="co"># We have passed two lists here, but we could also have passed</span></span>
<span id="cb66-9"><a href="#cb66-9" aria-hidden="true" tabindex="-1"></a><span class="co"># arrays - the Numpy repeat function converts the lists to arrays</span></span>
<span id="cb66-10"><a href="#cb66-10" aria-hidden="true" tabindex="-1"></a><span class="co"># before it builds the repeats.</span></span>
<span id="cb66-11"><a href="#cb66-11" aria-hidden="true" tabindex="-1"></a>jury_pool <span class="op">=</span> np.repeat(juror_types, repeat_nos)</span>
<span id="cb66-12"><a href="#cb66-12" aria-hidden="true" tabindex="-1"></a><span class="co"># Show the result</span></span>
<span id="cb66-13"><a href="#cb66-13" aria-hidden="true" tabindex="-1"></a>jury_pool</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>array(['black', 'black', 'black', 'black', 'black', 'black', 'black',
'black', 'black', 'black', 'black', 'black', 'black', 'black',
'black', 'black', 'black', 'black', 'black', 'black', 'black',
'black', 'black', 'black', 'black', 'black', 'white', 'white',
'white', 'white', 'white', 'white', 'white', 'white', 'white',
'white', 'white', 'white', 'white', 'white', 'white', 'white',
'white', 'white', 'white', 'white', 'white', 'white', 'white',
'white', 'white', 'white', 'white', 'white', 'white', 'white',
'white', 'white', 'white', 'white', 'white', 'white', 'white',
'white', 'white', 'white', 'white', 'white', 'white', 'white',
'white', 'white', 'white', 'white', 'white', 'white', 'white',
'white', 'white', 'white', 'white', 'white', 'white', 'white',
'white', 'white', 'white', 'white', 'white', 'white', 'white',
'white', 'white', 'white', 'white', 'white', 'white', 'white',
'white', 'white'], dtype='<U5')</code></pre>
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<p>We can use this array of repeats of strings, to sample from. The result is easier to grasp, because we are using the string labels, instead of numbers:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb68"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb68-1"><a href="#cb68-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Select one juror at random from the black / white pool.</span></span>
<span id="cb68-2"><a href="#cb68-2" aria-hidden="true" tabindex="-1"></a>one_juror <span class="op">=</span> rnd.choice(jury_pool)</span>
<span id="cb68-3"><a href="#cb68-3" aria-hidden="true" tabindex="-1"></a>one_juror</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.str_('white')</code></pre>
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<p>We can select our full jury of 12 jurors, and see the results in a more obvious form:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb70"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb70-1"><a href="#cb70-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Select 12 jurors at random from the black / white pool.</span></span>
<span id="cb70-2"><a href="#cb70-2" aria-hidden="true" tabindex="-1"></a>one_jury <span class="op">=</span> rnd.choice(jury_pool, <span class="dv">12</span>)</span>
<span id="cb70-3"><a href="#cb70-3" aria-hidden="true" tabindex="-1"></a>one_jury</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>array(['white', 'white', 'white', 'white', 'black', 'white', 'black',
'white', 'white', 'black', 'black', 'white'], dtype='<U5')</code></pre>
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<i class="callout-icon"></i>
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Using the <code>size</code> argument to <code>rnd.choice</code>
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<p>In the code above, we have specified the <em>size</em> of the sample we want (12) with the second argument to <code>rnd.choice</code>. As you saw in <a href="resampling_with_code.html#sec-named-arguments" class="quarto-xref"><span>Section 5.8</span></a>, we can also give names to the function arguments, in this case, to make it clearer what we mean by “12” in the code above. In fact, from now on, that is what we will do; we will specify the <em>size</em> of our sample by using the <em>name</em> for the function argument to <code>rnd.choice</code> — <code>size</code> — like this:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb72"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb72-1"><a href="#cb72-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Select 12 jurors at random from the black / white pool.</span></span>
<span id="cb72-2"><a href="#cb72-2" aria-hidden="true" tabindex="-1"></a><span class="co"># Specify the sample size using the "size" named argument.</span></span>
<span id="cb72-3"><a href="#cb72-3" aria-hidden="true" tabindex="-1"></a>one_jury <span class="op">=</span> rnd.choice(jury_pool, size<span class="op">=</span><span class="dv">12</span>)</span>
<span id="cb72-4"><a href="#cb72-4" aria-hidden="true" tabindex="-1"></a>one_jury</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>array(['black', 'white', 'white', 'white', 'black', 'white', 'black',
'white', 'white', 'white', 'white', 'white'], dtype='<U5')</code></pre>
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<p>We can use <code>==</code> on the array to get <code>True</code> values where the juror was “black” and <code>False</code> values otherwise:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb74"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb74-1"><a href="#cb74-1" aria-hidden="true" tabindex="-1"></a>are_black <span class="op">=</span> one_jury <span class="op">==</span> <span class="st">'black'</span></span>
<span id="cb74-2"><a href="#cb74-2" aria-hidden="true" tabindex="-1"></a>are_black</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>array([ True, False, False, False, True, False, True, False, False,
False, False, False])</code></pre>
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<p>Finally, we can <code>np.sum</code> to find the number of black jurors (<a href="resampling_with_code.html#sec-count-with-sum" class="quarto-xref"><span>Section 5.15</span></a>):</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb76"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb76-1"><a href="#cb76-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Number of black jurors in this simulated jury.</span></span>
<span id="cb76-2"><a href="#cb76-2" aria-hidden="true" tabindex="-1"></a>n_black <span class="op">=</span> np.<span class="bu">sum</span>(are_black)</span>
<span id="cb76-3"><a href="#cb76-3" aria-hidden="true" tabindex="-1"></a>n_black</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.int64(3)</code></pre>
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<p>Putting that all together, this is our new procedure to select one jury and count the number of black jurors:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb78"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb78-1"><a href="#cb78-1" aria-hidden="true" tabindex="-1"></a>one_jury <span class="op">=</span> rnd.choice(jury_pool, size<span class="op">=</span><span class="dv">12</span>)</span>
<span id="cb78-2"><a href="#cb78-2" aria-hidden="true" tabindex="-1"></a>are_black <span class="op">=</span> one_jury <span class="op">==</span> <span class="st">'black'</span></span>
<span id="cb78-3"><a href="#cb78-3" aria-hidden="true" tabindex="-1"></a>n_black <span class="op">=</span> np.<span class="bu">sum</span>(are_black)</span>
<span id="cb78-4"><a href="#cb78-4" aria-hidden="true" tabindex="-1"></a>n_black</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>np.int64(3)</code></pre>
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<p>Or we can be even more compact by putting several statements together into one line:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb80"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb80-1"><a href="#cb80-1" aria-hidden="true" tabindex="-1"></a><span class="co"># The same as above, but on one line.</span></span>
<span id="cb80-2"><a href="#cb80-2" aria-hidden="true" tabindex="-1"></a>n_black <span class="op">=</span> np.<span class="bu">sum</span>(rnd.choice(jury_pool, size<span class="op">=</span><span class="dv">12</span>) <span class="op">==</span> <span class="st">'black'</span>)</span>
<span id="cb80-3"><a href="#cb80-3" aria-hidden="true" tabindex="-1"></a>n_black</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>np.int64(1)</code></pre>
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<section id="resampling-with-and-without-replacement" class="level2" data-number="7.7">
<h2 data-number="7.7" class="anchored" data-anchor-id="resampling-with-and-without-replacement">7.7 Resampling with and without replacement</h2>
<p>Now let us return to the details of Robert Swain’s case, that you first saw in <a href="resampling_with_code2.html" class="quarto-xref"><span>Chapter 6</span></a>.</p>
<p>We looked at the composition of Robert Swain’s 12-person jury — but in fact, by law, that does not have to be representative of the eligible jurors. The 12-person jury is drawn from a jury <em>panel</em>, of 100 people, and this should, in turn, be drawn from the population of all eligible jurors in the county, consisting, at the time, of “all male citizens in the community over 21 who are reputed to be honest, intelligent men and are esteemed for their integrity, good character and sound judgment.” So, unless there was some bias against black jurors, we might expect the 100-person jury panel to be a plausibly random sample of the eligible jurors, of whom 26% were black. See <a href="https://supreme.justia.com/cases/federal/us/380/202">the Supreme Court case judgement</a> for details.</p>
<p>In fact, in Robert Swain’s trial, there were 8 black members in the 100-person jury panel. We will leave it to you to adapt the simulation from <a href="resampling_with_code2.html" class="quarto-xref"><span>Chapter 6</span></a> to ask the question — is 8% surprising as a random sample from a population with 26% black people?</p>
<p>But we have a different question: given that 8 out of 100 of the jury panel were black, is it surprising that none of the 12-person jury were black? As usual, we can answer that question with simulation.</p>