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<h2 id="toc-title">Table of contents</h2>
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<li><a href="#sec-pig-rations-measured" id="toc-sec-pig-rations-measured" class="nav-link active" data-scroll-target="#sec-pig-rations-measured"><span class="header-section-number">24.0.1</span> Example: The Pig Rations Still Once Again, Using Measured Data</a></li>
<li><a href="#sec-liquor-public-measured" id="toc-sec-liquor-public-measured" class="nav-link" data-scroll-target="#sec-liquor-public-measured"><span class="header-section-number">24.0.2</span> Example: Is There a Difference in Liquor Prices Between State-Run and Privately-Run Systems?</a></li>
<li><a href="#example-is-there-a-difference-between-treatments-to-prevent-low-birthweights" id="toc-example-is-there-a-difference-between-treatments-to-prevent-low-birthweights" class="nav-link" data-scroll-target="#example-is-there-a-difference-between-treatments-to-prevent-low-birthweights"><span class="header-section-number">24.0.3</span> Example: Is there a difference between treatments to prevent low birthweights?</a></li>
<li><a href="#sec-eg-bootstrap-sampling" id="toc-sec-eg-bootstrap-sampling" class="nav-link" data-scroll-target="#sec-eg-bootstrap-sampling"><span class="header-section-number">24.0.4</span> Example: Bootstrap Sampling with Replacement</a></li>
<li><a href="#example-permutation-sampling-without-replacement" id="toc-example-permutation-sampling-without-replacement" class="nav-link" data-scroll-target="#example-permutation-sampling-without-replacement"><span class="header-section-number">24.0.5</span> Example: Permutation Sampling Without Replacement</a></li>
<li><a href="#differences-among-four-means" id="toc-differences-among-four-means" class="nav-link" data-scroll-target="#differences-among-four-means"><span class="header-section-number">24.1</span> Differences among four means</a>
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<li><a href="#example-differences-among-four-pig-rations" id="toc-example-differences-among-four-pig-rations" class="nav-link" data-scroll-target="#example-differences-among-four-pig-rations"><span class="header-section-number">24.1.1</span> Example: differences among four pig rations</a></li>
</ul></li>
<li><a href="#using-squared-differences" id="toc-using-squared-differences" class="nav-link" data-scroll-target="#using-squared-differences"><span class="header-section-number">24.2</span> Using Squared Differences</a></li>
<li><a href="#exercises" id="toc-exercises" class="nav-link" data-scroll-target="#exercises"><span class="header-section-number">24.3</span> Exercises</a>
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<li><a href="#sec-exr-paired-differences" id="toc-sec-exr-paired-differences" class="nav-link" data-scroll-target="#sec-exr-paired-differences"><span class="header-section-number">24.3.1</span> Exercise: paired differences</a></li>
<li><a href="#sec-exr-seatbelt-proportions" id="toc-sec-exr-seatbelt-proportions" class="nav-link" data-scroll-target="#sec-exr-seatbelt-proportions"><span class="header-section-number">24.3.2</span> Exercise: seatbelt proportions</a></li>
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<h1 class="title"><span id="sec-testing-measured" class="quarto-section-identifier"><span class="chapter-number">24</span> <span class="chapter-title">The Statistics of Hypothesis-Testing With Measured Data</span></span></h1>
</div>
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</div>
</header>
<!---
Discussion here of dispersion - have we discussed this before?
Methods of estimating dispersion.
-->
<p><a href="testing_counts_1.html" class="quarto-xref"><span>Chapter 21</span></a> and <a href="point_estimation.html" class="quarto-xref"><span>Chapter 19</span></a> discussed testing a hypothesis with data that either arrive in dichotomized (yes-no) form, or come as data in situations where it is convenient to dichotomize. We next consider hypothesis testing using measured data. Conventional statistical practice employs such devices as the “t-test” and “analysis of variance.” In contrast to those complex devices, the resampling method does not differ greatly from what has been discussed in previous chapters.</p>
<section id="sec-pig-rations-measured" class="level3" data-number="24.0.1">
<h3 data-number="24.0.1" class="anchored" data-anchor-id="sec-pig-rations-measured"><span class="header-section-number">24.0.1</span> Example: The Pig Rations Still Once Again, Using Measured Data</h3>
<p><strong>Testing for the difference between means of two equal-sized samples of measured-data observations</strong></p>
<p>Let us now treat the pig-food problem without converting the quantitative data into qualitative data, because a conversion always loses information.</p>
<p>The term “lose information” can be understood intuitively. Consider two sets of three sacks of corn. Set A includes sacks containing, respectively, one pound, two pounds, and three pounds. Set B includes sacks of one pound, two pounds, and a hundred pounds. If we <em>rank</em> the sacks by weight, the two sets can no longer be distinguished. The one-pound and two-pound sacks have ranks one and two in both cases, and their relative places in their sets are the same. But if we know <em>not only</em> that the one-pound sack is the smallest of its set and the three-pound or hundred-pound sack is the largest, but <em>also</em> that the largest sack is three pounds (or a hundred pounds), we have more information about a set than if we only know the ranks of its sacks.</p>
<p>Rank data are also known as “ordinal” data, whereas data measured in (say) pounds are known as “cardinal” data. Even though converting from cardinal (measured) to ordinal (ranked) data loses information, the conversion may increase convenience, and may therefore be worth doing in some cases.</p>
<p><a href="#tbl-pig-ab-rations" class="quarto-xref">Table <span>24.1</span></a> has the measured data for pig rations A and B.</p>
<div class="cell" data-layout-align="center">
<div class="cell-output-display">
<div id="tbl-pig-ab-rations" class="lightable-paper lightable-striped lightable-hover quarto-float quarto-figure quarto-figure-center anchored" style="font-family: "Arial Narrow", arial, helvetica, sans-serif; width: auto !important; margin-left: auto; margin-right: auto;" data-quarto-postprocess="true">
<figure class="quarto-float quarto-float-tbl figure">
<figcaption class="quarto-float-caption-top quarto-float-caption quarto-float-tbl" id="tbl-pig-ab-rations-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
Table 24.1: Measured data for pig rations A and B
</figcaption>
<div aria-describedby="tbl-pig-ab-rations-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
<table class="lightable-paper lightable-striped lightable-hover caption-top table table-sm table-striped small" data-quarto-postprocess="true">
<thead>
<tr class="header">
<th style="text-align: left;" data-quarto-table-cell-role="th">ration</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">weight_gain</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: left;">A</td>
<td style="text-align: right;">31</td>
</tr>
<tr class="even">
<td style="text-align: left;">A</td>
<td style="text-align: right;">34</td>
</tr>
<tr class="odd">
<td style="text-align: left;">A</td>
<td style="text-align: right;">29</td>
</tr>
<tr class="even">
<td style="text-align: left;">A</td>
<td style="text-align: right;">26</td>
</tr>
<tr class="odd">
<td style="text-align: left;">A</td>
<td style="text-align: right;">32</td>
</tr>
<tr class="even">
<td style="text-align: left;">A</td>
<td style="text-align: right;">35</td>
</tr>
<tr class="odd">
<td style="text-align: left;">A</td>
<td style="text-align: right;">38</td>
</tr>
<tr class="even">
<td style="text-align: left;">A</td>
<td style="text-align: right;">34</td>
</tr>
<tr class="odd">
<td style="text-align: left;">A</td>
<td style="text-align: right;">31</td>
</tr>
<tr class="even">
<td style="text-align: left;">A</td>
<td style="text-align: right;">29</td>
</tr>
<tr class="odd">
<td style="text-align: left;">A</td>
<td style="text-align: right;">32</td>
</tr>
<tr class="even">
<td style="text-align: left;">A</td>
<td style="text-align: right;">31</td>
</tr>
<tr class="odd">
<td style="text-align: left;">B</td>
<td style="text-align: right;">26</td>
</tr>
<tr class="even">
<td style="text-align: left;">B</td>
<td style="text-align: right;">24</td>
</tr>
<tr class="odd">
<td style="text-align: left;">B</td>
<td style="text-align: right;">28</td>
</tr>
<tr class="even">
<td style="text-align: left;">B</td>
<td style="text-align: right;">29</td>
</tr>
<tr class="odd">
<td style="text-align: left;">B</td>
<td style="text-align: right;">30</td>
</tr>
<tr class="even">
<td style="text-align: left;">B</td>
<td style="text-align: right;">29</td>
</tr>
<tr class="odd">
<td style="text-align: left;">B</td>
<td style="text-align: right;">31</td>
</tr>
<tr class="even">
<td style="text-align: left;">B</td>
<td style="text-align: right;">29</td>
</tr>
<tr class="odd">
<td style="text-align: left;">B</td>
<td style="text-align: right;">32</td>
</tr>
<tr class="even">
<td style="text-align: left;">B</td>
<td style="text-align: right;">26</td>
</tr>
<tr class="odd">
<td style="text-align: left;">B</td>
<td style="text-align: right;">28</td>
</tr>
<tr class="even">
<td style="text-align: left;">B</td>
<td style="text-align: right;">32</td>
</tr>
</tbody>
</table>
</div>
</figure>
</div>
</div>
</div>
<p>We begin a measured-data procedure by noting that if the two pig foods are the same, then each of the observed weight gains came from the <em>same benchmark universe</em>. This is the basic tactic in our statistical strategy. That is, if the two foods came from the same universe, <em>our best guess about the composition of that universe is that it includes weight gains just like the twenty-four we have observed</em>, and in the same proportions, because that is all the information that we have about the universe; this is the bootstrap method. Since ours is (by definition) a sample from an infinite (or at least, a very large) universe of possible weight gains, we assume that there are <em>many</em> weight gains in the universe just like the ones we have observed, in the same proportion as we have observed them. For example, we assume that 2/24 of the universe is composed of 34-pound weight gains, as seen in <a href="#fig-rations-hist" class="quarto-xref">Figure <span>24.1</span></a>:</p>
<div class="cell" data-layout-align="center">
<div class="cell-output-display">
<div id="fig-rations-hist" class="quarto-float quarto-figure quarto-figure-center anchored" data-fig-align="center">
<figure class="quarto-float quarto-float-fig figure">
<div aria-describedby="fig-rations-hist-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
<img src="testing_measured_files/figure-html/fig-rations-hist-1.png" class="img-fluid quarto-figure quarto-figure-center figure-img" style="width:70.0%">
</div>
<figcaption class="quarto-float-caption-bottom quarto-float-caption quarto-float-fig" id="fig-rations-hist-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
Figure 24.1: Histogram of weight gain
</figcaption>
</figure>
</div>
</div>
</div>
<p>We recognize, of course, that weight gains other than the exact ones we observed certainly would occur in repeated experiments. And if we thought it reasonable to do so, we could assume that the “distribution” of the weight gains would follow a regular “smooth” shape such as <a href="#fig-rations-ideal" class="quarto-xref">Figure <span>24.2</span></a>. But deciding just how to draw <a href="#fig-rations-ideal" class="quarto-xref">Figure <span>24.2</span></a> from the data in <a href="#fig-rations-hist" class="quarto-xref">Figure <span>24.1</span></a> requires that we make arbitrary assumptions about unknown conditions. And if we were to draw <a href="#fig-rations-ideal" class="quarto-xref">Figure <span>24.2</span></a> in a form that would be sufficiently regular for conventional mathematical analysis, we might have to make some <em>very strong</em> assumptions going far beyond the observed data.</p>
<div class="cell" data-layout-align="center">
<div class="cell-output-display">
<div id="fig-rations-ideal" class="quarto-float quarto-figure quarto-figure-center anchored" data-fig-align="center">
<figure class="quarto-float quarto-float-fig figure">
<div aria-describedby="fig-rations-ideal-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
<img src="testing_measured_files/figure-html/fig-rations-ideal-1.png" class="img-fluid quarto-figure quarto-figure-center figure-img" style="width:70.0%">
</div>
<figcaption class="quarto-float-caption-bottom quarto-float-caption quarto-float-fig" id="fig-rations-ideal-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
Figure 24.2: Idealized distribution for weight gain
</figcaption>
</figure>
</div>
</div>
</div>
<p>Drawing a smooth curve such as <a href="#fig-rations-ideal" class="quarto-xref">Figure <span>24.2</span></a> from the raw data in <a href="#fig-rations-hist" class="quarto-xref">Figure <span>24.1</span></a> might be satisfactory — if done with wisdom and good judgment. But there is no necessity to draw such a smooth curve, in this case or in most cases. We can proceed by assuming simply that the benchmark universe — the universe to which we shall compare our samples, conventionally called the “null” or “hypothetical” universe — is composed only of elements similar to the observations we have in hand. We thereby lose no efficiency and avoid making unsound assumptions.</p>
<p>To carry out our procedure in practice: 1) Write down each of the twenty-four weight gains on a blank index card. We then have one card each for 31, 34, 29, 26, and so on. 2) Shuffle the twenty-four cards thoroughly, and pick one card. 3) Record the weight gain, and replace the card. (Recall that we are treating the weight gains as if they come from an infinite universe — that is, as if the probability of selecting any amount is the same no matter which others are selected randomly. Another way to say this is to state that each selection is independent of each other selection. If we did not replace the card before selecting the next weight gain, the selections would no longer be independent. See <a href="probability_theory_2_compound.html" class="quarto-xref"><span>Chapter 11</span></a> for further discussion of this issue.) 4) Repeat this process until you have made two sets of 12 observations. 5) Call the first hand “food A” and the second hand “food B.” Determine the average weight gain for the two hands, and record it as in <a href="#tbl-eg-pigs-measured" class="quarto-xref">Table <span>24.2</span></a>. Repeat this procedure many times.</p>
<p>In operational steps:</p>
<ul>
<li><strong>Step 1.</strong> Write down each observed weight gain on a card, e.g. 31, 34, 29 …</li>
<li><strong>Step 2.</strong> Shuffle and deal a card.</li>
<li><strong>Step 3.</strong> Record the weight and replace the card.</li>
<li><strong>Step 4.</strong> Repeat steps 2 and 3 eleven more times; call this group A.</li>
<li><strong>Step 5.</strong> Repeat steps 2-3 another twelve times; call this group B.</li>
<li><strong>Step 6.</strong> Calculate the mean weight gain of each group.</li>
<li><strong>Step 7.</strong> Subtract the mean of group A from the mean of group B and record. If larger (more positive) than 3.16 (the difference between the observed means) or more negative than -3.16, record “more.” Otherwise record “less.”</li>
<li><strong>Step 8.</strong> Repeat this procedure perhaps fifty times, and calculate the proportion “more.” This estimates the probability sought.</li>
</ul>
<p>In none of the first ten simulated trials did the difference in the means of the random hands exceed the observed difference (3.16 pounds, in the top line in the table) between foods A and B. (The difference between group <em>totals</em> tells the same story and is faster, requiring no division calculations.)</p>
<p>In the old days before a computer was easily available, I (JLS) would quit making trials at such a point, confident that a difference in means as great as observed is not likely to happen by chance. (Using the convenient “multiplication rule” described in <a href="probability_theory_1b.html" class="quarto-xref"><span>Chapter 9</span></a>, we can estimate the probability of such an occurrence happening by chance in 10 successive trials as <span class="math inline">\(\frac{1}{2} * \frac{1}{2} * \frac{1}{2} ... = \frac{1}{2}^{10} = 1/1024
\approx .001\)</span> = .1 percent, a small chance indeed.) Nevertheless, let us press on to do 50 trials.</p>
<div id="tbl-eg-pigs-measured" class="quarto-float quarto-figure quarto-figure-center anchored">
<figure class="quarto-float quarto-float-tbl figure">
<figcaption class="quarto-float-caption-top quarto-float-caption quarto-float-tbl" id="tbl-eg-pigs-measured-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
Table 24.2: Results of fifty random samples for measured pig rations
</figcaption>
<div aria-describedby="tbl-eg-pigs-measured-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
<table class="caption-top table">
<colgroup>
<col style="width: 16%">
<col style="width: 18%">
<col style="width: 18%">
<col style="width: 1%">
<col style="width: 17%">
<col style="width: 1%">
<col style="width: 17%">
<col style="width: 1%">
</colgroup>
<thead>
<tr class="header">
<th>Trial #</th>
<th>Mean of first 12 observations (first hand)</th>
<th colspan="2">Mean of second 12 observations (second hand)</th>
<th colspan="2">Difference</th>
<th colspan="2">Greater or less than observed difference?</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>Observed</td>
<td>382 / 12=31.83</td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
<tr class="even">
<td>1</td>
<td>368 / 12=30.67</td>
<td>357 / 12=29.75</td>
<td colspan="2">.87</td>
<td colspan="2">Less</td>
<td></td>
</tr>
<tr class="odd">
<td>2</td>
<td>364 / 12=30.33</td>
<td>361 / 12=30.08</td>
<td colspan="2">.25</td>
<td colspan="2">Less</td>
<td></td>
</tr>
<tr class="even">
<td>3</td>
<td>352 / 12=29.33</td>
<td>373 / 12=31.08</td>
<td colspan="2">(1.75)</td>
<td colspan="2">Less</td>
<td></td>
</tr>
<tr class="odd">
<td>4</td>
<td>378 / 12=31.50</td>
<td>347 / 12=28.92</td>
<td colspan="2">2.58</td>
<td colspan="2">Less</td>
<td></td>
</tr>
<tr class="even">
<td>5</td>
<td>365 / 12=30.42</td>
<td>360 / 12=30.00</td>
<td colspan="2">.42</td>
<td colspan="2">Less</td>
<td></td>
</tr>
<tr class="odd">
<td>6</td>
<td>352 / 12=29.33</td>
<td>373 / 12=31.08</td>
<td colspan="2">(1.75)</td>
<td colspan="2">Less</td>
<td></td>
</tr>
<tr class="even">
<td>7</td>
<td>355 / 12=29.58</td>
<td>370 / 12=30.83</td>
<td colspan="2">(1.25)</td>
<td colspan="2">Less</td>
<td></td>
</tr>
<tr class="odd">
<td>8</td>
<td>366 / 12=30.50</td>
<td>359 / 12=29.92</td>
<td colspan="2">.58</td>
<td colspan="2">Less</td>
<td></td>
</tr>
<tr class="even">
<td>9</td>
<td>360 / 12=30.00</td>
<td>365 / 12=30.42</td>
<td colspan="2">(.42)</td>
<td colspan="2">Less</td>
<td></td>
</tr>
<tr class="odd">
<td>10</td>
<td>355 / 12=29.58</td>
<td>370 / 12=30.83</td>
<td colspan="2">(1.25)</td>
<td colspan="2">Less</td>
<td></td>
</tr>
<tr class="even">
<td>11</td>
<td>359 / 12=29.92</td>
<td>366 / 12=30.50</td>
<td colspan="2">(.58)</td>
<td colspan="2">Less</td>
<td></td>
</tr>
<tr class="odd">
<td>12</td>
<td>369 / 12=30.75</td>
<td>356 / 12=29.67</td>
<td colspan="2">1.08</td>
<td colspan="2">”</td>
<td></td>
</tr>
<tr class="even">
<td>13</td>
<td>360 / 12=30.00</td>
<td>365 / 12=30.42</td>
<td colspan="2">(.42)</td>
<td colspan="2">Less</td>
<td></td>
</tr>
<tr class="odd">
<td>14</td>
<td>377 / 12=31.42</td>
<td>348 / 12=29.00</td>
<td colspan="2">2.42</td>
<td colspan="2">Less</td>
<td></td>
</tr>
<tr class="even">
<td>15</td>
<td>365 / 12=30.42</td>
<td>360 / 12=30.00</td>
<td colspan="2">.42</td>
<td colspan="2">Less</td>
<td></td>
</tr>
<tr class="odd">
<td>16</td>
<td>364 / 12=30.33</td>
<td>361 / 12=30.08</td>
<td colspan="2">.25</td>
<td colspan="2">Less</td>
<td></td>
</tr>
<tr class="even">
<td>17</td>
<td>363 / 12=30.25</td>
<td>362 / 12=30.17</td>
<td colspan="2">.08</td>
<td colspan="2">Less</td>
<td></td>
</tr>
<tr class="odd">
<td>18</td>
<td>365 / 12=30.42</td>
<td>360 / 12=30.00</td>
<td colspan="2">.42</td>
<td colspan="2">Less</td>
<td></td>
</tr>
<tr class="even">
<td>19</td>
<td>369 / 12=30.75</td>
<td>356 / 12=29.67</td>
<td colspan="2">1.08</td>
<td colspan="2">Less</td>
<td></td>
</tr>
<tr class="odd">
<td>20</td>
<td>369 / 12=30.75</td>
<td>356 / 12=29.67</td>
<td colspan="2">1.08</td>
<td colspan="2">Less</td>
<td></td>
</tr>
<tr class="even">
<td>21</td>
<td>369 / 12=30.75</td>
<td>356 / 12=29.67</td>
<td colspan="2">1.08</td>
<td colspan="2">Less</td>
<td></td>
</tr>
<tr class="odd">
<td>22</td>
<td>364 / 12=30.33</td>
<td>361 / 12=30.08</td>
<td colspan="2">.25</td>
<td colspan="2">Less</td>
<td></td>
</tr>
<tr class="even">
<td>23</td>
<td>363 / 12=30.25</td>
<td>362 / 12=30.17</td>
<td colspan="2">.08</td>
<td colspan="2">Less</td>
<td></td>
</tr>
<tr class="odd">
<td>24</td>
<td>363 / 12=30.25</td>
<td>362 / 12=30.17</td>
<td colspan="2">.08</td>
<td colspan="2">Less</td>
<td></td>
</tr>
<tr class="even">
<td>25</td>
<td>364 / 12=30.33</td>
<td>361 / 12=30.08</td>
<td colspan="2">.25</td>
<td colspan="2">Less</td>
<td></td>
</tr>
<tr class="odd">
<td>26</td>
<td>359 / 12=29.92</td>
<td>366 / 12=30.50</td>
<td colspan="2">(.58)</td>
<td colspan="2">Less</td>
<td></td>
</tr>
<tr class="even">
<td>27</td>
<td>362 / 12=30.17</td>
<td>363 / 12=30.25</td>
<td colspan="2">(.08)</td>
<td colspan="2">Less</td>
<td></td>
</tr>
<tr class="odd">
<td>28</td>
<td>362 / 12=30.17</td>
<td>363 / 12=30.25</td>
<td colspan="2">(.08)</td>
<td colspan="2">Less</td>
<td></td>
</tr>
<tr class="even">
<td>29</td>
<td>373 / 12=31.08</td>
<td>352 / 12=29.33</td>
<td colspan="2">1.75</td>
<td colspan="2">Less</td>
<td></td>
</tr>
<tr class="odd">
<td>30</td>
<td>367 / 12=30.58</td>
<td>358 / 12=29.83</td>
<td colspan="2">.75</td>
<td colspan="2">Less</td>
<td></td>
</tr>
<tr class="even">
<td>31</td>
<td>376 / 12=31.33</td>
<td>349 / 12=29.08</td>
<td colspan="2">2.25</td>
<td colspan="2">Less</td>
<td></td>
</tr>
<tr class="odd">
<td>32</td>
<td>365 / 12=30.42</td>
<td>360 / 12=30.00</td>
<td colspan="2">.42</td>
<td colspan="2">Less</td>
<td></td>
</tr>
<tr class="even">
<td>33</td>
<td>357 / 12=29.75</td>
<td>368 / 12=30.67</td>
<td colspan="2">(1.42)</td>
<td colspan="2">Less</td>
<td></td>
</tr>
<tr class="odd">
<td>34</td>
<td>349 / 12=29.08</td>
<td>376 / 12=31.33</td>
<td colspan="2">2.25</td>
<td colspan="2">Less</td>
<td></td>
</tr>
<tr class="even">
<td>35</td>
<td>356 / 12=29.67</td>
<td>396 / 12=30.75</td>
<td colspan="2">(1.08)</td>
<td colspan="2">Less</td>
<td></td>
</tr>
<tr class="odd">
<td>36</td>