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<h1 class="heading"><a href="my-great-book.html"><span class="title">Math 1044 Notes</span></a></h1>
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<div class="toc-title-box"><a href="sec-Probability.html" class="internal"><span class="codenumber">1.2</span> <span class="title">Definition of Probability</span></a></div>
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<div class="toc-title-box"><a href="sec-Conditional-Probability.html" class="internal"><span class="codenumber">1.3</span> <span class="title">Conditional Probability</span></a></div>
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<div class="toc-title-box"><a href="ch-Random-Variables.html" class="internal"><span class="codenumber">2</span> <span class="title">Random Variables</span></a></div>
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<div class="toc-title-box"><a href="sec-Discrete-RVs.html" class="internal"><span class="codenumber">2.1</span> <span class="title">Discrete Random Variables</span></a></div>
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<div class="toc-title-box"><a href="sec-Continuous-RVs.html" class="internal"><span class="codenumber">2.2</span> <span class="title">Continuous Random Variables</span></a></div>
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<div class="toc-title-box"><a href="sec-Joint-Distributions.html" class="internal"><span class="codenumber">2.3</span> <span class="title">Joint Distributions</span></a></div>
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<div class="toc-title-box"><a href="sec-Expected-Value.html" class="internal"><span class="codenumber">3.1</span> <span class="title">Expected Value</span></a></div>
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<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-CLT.html#exercises-CLT" class="internal"><span class="codenumber">4.2</span> <span class="title">Exercises</span></a></div></li></ul>
|
||
</li>
|
||
<li class="toc-item toc-section">
|
||
<div class="toc-title-box"><a href="sec-Confidence-Intervals.html" class="internal"><span class="codenumber">4.3</span> <span class="title">Confidence Intervals</span></a></div>
|
||
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Confidence-Intervals.html#exercises-Confidence-Intervals" class="internal"><span class="codenumber">4.3</span> <span class="title">Exercises</span></a></div></li></ul>
|
||
</li>
|
||
</ul>
|
||
</li>
|
||
<li class="toc-item toc-chapter contains-active">
|
||
<div class="toc-title-box"><a href="ch-Hypothesis-Testing.html" class="internal"><span class="codenumber">5</span> <span class="title">Hypothesis Testing</span></a></div>
|
||
<ul class="structural toc-item-list contains-active">
|
||
<li class="toc-item toc-section">
|
||
<div class="toc-title-box"><a href="sec-One-Sample-Tests.html" class="internal"><span class="codenumber">5.1</span> <span class="title">One Sample Tests</span></a></div>
|
||
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-One-Sample-Tests.html#exercises-One-Sample-Tests" class="internal"><span class="codenumber">5.1</span> <span class="title">Exercises</span></a></div></li></ul>
|
||
</li>
|
||
<li class="toc-item toc-section">
|
||
<div class="toc-title-box"><a href="sec-Two-Sample-Tests.html" class="internal"><span class="codenumber">5.2</span> <span class="title">Two Sample Tests</span></a></div>
|
||
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Two-Sample-Tests.html#exercises-Two-Sample-Tests" class="internal"><span class="codenumber">5.2</span> <span class="title">Exercises</span></a></div></li></ul>
|
||
</li>
|
||
<li class="toc-item toc-section contains-active">
|
||
<div class="toc-title-box"><a href="sec-Power.html" class="internal"><span class="codenumber">5.3</span> <span class="title">Power of a Test</span></a></div>
|
||
<ul class="structural toc-item-list active"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Power.html#exercises-Power" class="internal"><span class="codenumber">5.3</span> <span class="title">Exercises</span></a></div></li></ul>
|
||
</li>
|
||
<li class="toc-item toc-section">
|
||
<div class="toc-title-box"><a href="sec-Chi-Squared.html" class="internal"><span class="codenumber">5.4</span> <span class="title"><span class="process-math">\(\chi^2\)</span> Test</span></a></div>
|
||
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Chi-Squared.html#exercises-Chi-Squared" class="internal"><span class="codenumber">5.4</span> <span class="title">Exercises</span></a></div></li></ul>
|
||
</li>
|
||
</ul>
|
||
</li>
|
||
<li class="toc-item toc-chapter">
|
||
<div class="toc-title-box"><a href="ch-Linear-Regression.html" class="internal"><span class="codenumber">6</span> <span class="title">Linear Regression</span></a></div>
|
||
<ul class="structural toc-item-list">
|
||
<li class="toc-item toc-section">
|
||
<div class="toc-title-box"><a href="sec-Correlation.html" class="internal"><span class="codenumber">6.1</span> <span class="title">Correlation</span></a></div>
|
||
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Correlation.html#exercises-Correlation" class="internal"><span class="codenumber">6.1</span> <span class="title">Exercises</span></a></div></li></ul>
|
||
</li>
|
||
<li class="toc-item toc-section">
|
||
<div class="toc-title-box"><a href="sec-Linear-Regression.html" class="internal"><span class="codenumber">6.2</span> <span class="title">Linear Regression</span></a></div>
|
||
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Linear-Regression.html#exercises-Linear-Regression" class="internal"><span class="codenumber">6.2</span> <span class="title">Exercises</span></a></div></li></ul>
|
||
</li>
|
||
</ul>
|
||
</li>
|
||
<li class="toc-item toc-backmatter">
|
||
<div class="toc-title-box"><a href="backmatter.html" class="internal"><span class="title">Backmatter</span></a></div>
|
||
<ul class="structural toc-item-list">
|
||
<li class="toc-item toc-appendix">
|
||
<div class="toc-title-box"><a href="backmatter-2.html" class="internal"><span class="codenumber">A</span> <span class="title"><span class="process-math">\(\Phi(z)\)</span> Table</span></a></div>
|
||
<ul class="structural toc-item-list"><li class="toc-item toc-section"><div class="toc-title-box"><a href="app-Phi-table.html" class="internal"><span class="codenumber">A.1</span> <span class="title"><span class="process-math">\(\Phi(z)\)</span> Table of Values</span></a></div></li></ul>
|
||
</li>
|
||
<li class="toc-item toc-appendix">
|
||
<div class="toc-title-box"><a href="backmatter-3.html" class="internal"><span class="codenumber">B</span> <span class="title"><span class="process-math">\(\chi^2\)</span> Critical Values Table</span></a></div>
|
||
<ul class="structural toc-item-list"><li class="toc-item toc-section"><div class="toc-title-box"><a href="app-Chi-squared-table.html" class="internal"><span class="codenumber">B.1</span> <span class="title"><span class="process-math">\(\chi^2\)</span> Critical Values</span></a></div></li></ul>
|
||
</li>
|
||
<li class="toc-item toc-colophon"><div class="toc-title-box"><a href="backmatter-4.html" class="internal"><span class="title">Colophon</span></a></div></li>
|
||
</ul>
|
||
</li>
|
||
</ul></nav></div>
|
||
<main class="ptx-main"><div id="ptx-content" class="ptx-content"><section class="section" id="sec-Power"><h2 class="heading hide-type">
|
||
<span class="type">Section</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">5.3</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Power of a Test</span>
|
||
</h2>
|
||
<div class="para logical" id="sec-Power-2">
|
||
<div class="para">Recall that the power of a test is the probability of rejecting a false null hypothesis. To understand the power, we must know:</div>
|
||
<ol class="decimal" id="sec-Power-2-1">
|
||
<li id="sec-Power-2-1-1">
|
||
<div class="para" id="sec-Power-2-1-1-1">What data would lead us to reject the null hypothesis?<div class="autopermalink" data-description="Paragraph"><a href="#sec-Power-2-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Item 1"><a href="#sec-Power-2-1-1" title="Copy heading and permalink for Item 1" aria-label="Copy heading and permalink for Item 1">🔗</a></div>
|
||
</li>
|
||
<li id="sec-Power-2-1-2">
|
||
<div class="para" id="sec-Power-2-1-2-1">What is the probability of seeing such data?<div class="autopermalink" data-description="Paragraph"><a href="#sec-Power-2-1-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Item 2"><a href="#sec-Power-2-1-2" title="Copy heading and permalink for Item 2" aria-label="Copy heading and permalink for Item 2">🔗</a></div>
|
||
</li>
|
||
</ol>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#sec-Power-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<article class="example example-like" id="sec-Power-3"><h3 class="heading">
|
||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">5.3.1</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||
</h3>
|
||
<div class="para logical" id="sec-Power-3-1-1">
|
||
<div class="para">We find a coin and wonder if it’s fair. Let <span class="process-math">\(\theta\)</span> be the probability of the coin coming up heads on a flip. Let <span class="process-math">\(S\)</span> count the number of heads in 100 flips. Under the null hypothesis that <span class="process-math">\(\theta = 0.5\text{,}\)</span> <span class="process-math">\(S\)</span> is binomial, with:</div>
|
||
<div class="displaymath process-math" id="sec-Power-3-1-1-5">
|
||
\begin{align*}
|
||
\E(S) \amp = (n)(\theta) = (100)(0.5) = 50 \\
|
||
\Var(S) \amp = (n)(\theta)(1 - \theta) = (100)(0.5)(0.5) = 25
|
||
\end{align*}
|
||
</div>
|
||
<div class="para">Using a normal approximation, <span class="process-math">\(S \approx \Norm(50, 25)\text{.}\)</span> In the calculation of the <span class="process-math">\(p\)</span>-value for a 2-tailed test, we will eventually reach the expression:</div>
|
||
<div class="displaymath process-math" id="sec-Power-3-1-1-8">
|
||
\begin{gather*}
|
||
\Pr(Z \geq z) + \Pr(Z \leq -z)
|
||
\end{gather*}
|
||
</div>
|
||
<div class="para">for some <span class="process-math">\(z\)</span>-score. To address the first question above, we can calculate, for a specified significance level <span class="process-math">\(\alpha\text{,}\)</span> the <span class="process-math">\(z\)</span>-score that represents data at the cutoff between accepting and rejecting the null hypothesis:</div>
|
||
<div class="displaymath process-math" id="sec-Power-3-1-1-12">
|
||
\begin{align*}
|
||
\alpha \amp = \Pr(Z \geq z) + \Pr(Z \leq -z) \\
|
||
\amp = \Phi(-z) + (1 - \Phi(z)) \\
|
||
\amp = 2 \Phi(-z) \\
|
||
\Rightarrow \Phi(-z) \amp = \frac{\alpha}{2} \\
|
||
-z \amp = \Phi^{-1}\left(\frac{\alpha}{2}\right)
|
||
\end{align*}
|
||
</div>
|
||
<div class="para">For the significance level <span class="process-math">\(\alpha = 0.05\text{,}\)</span> we’ll have <span class="process-math">\(z = 1.96\text{.}\)</span> Now, we convert back to values of <span class="process-math">\(S\text{,}\)</span> keeping in mind that a continuity correction would be involved:</div>
|
||
<div class="displaymath process-math" id="sec-Power-3-1-1-16">
|
||
\begin{align*}
|
||
\frac{S - 0.5 - 50}{\sqrt{25}} \amp = 1.96 \\
|
||
S \amp = 50.5 + 1.96 \sqrt{25} = 60.3.
|
||
\end{align*}
|
||
</div>
|
||
<div class="para">We can’t flip a fractional number of heads, so it would take a result of 61 or more heads (or, equally extreme in the other direction, 39 heads or fewer) to reject the null hypothesis.</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#sec-Power-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Example 5.3.1"><a href="#sec-Power-3" title="Copy heading and permalink for Example 5.3.1" aria-label="Copy heading and permalink for Example 5.3.1">🔗</a></div></article><div class="para" id="sec-Power-4">To address the second question, we find a obstacle. How can we calculate the probability of seeing data extreme enough data to reject <span class="process-math">\(H_0\)</span> without knowing the true value of <span class="process-math">\(\theta\text{?}\)</span> In fact, the power is not a single number, but a function of <span class="process-math">\(\theta\text{.}\)</span> For each hypothetical value of <span class="process-math">\(\theta\text{,}\)</span> we’ll calculate the probability of seeing extreme enough data to reject <span class="process-math">\(H_0\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#sec-Power-4" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<article class="example example-like" id="sec-Power-5"><h3 class="heading">
|
||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">5.3.2</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||
</h3>
|
||
<div class="para logical" id="sec-Power-5-1-1">
|
||
<div class="para">Continuing the previous example, we know that, if <span class="process-math">\(S \geq 61\)</span> or <span class="process-math">\(S \leq 39\text{,}\)</span> we would reject <span class="process-math">\(H_0\text{.}\)</span> For a given value of <span class="process-math">\(\theta\text{:}\)</span>
|
||
</div>
|
||
<div class="displaymath process-math" data-contains-math-knowls="./knowl/xref/app-Phi-table.html" id="sec-Power-5-1-1-5">
|
||
\begin{align*}
|
||
\E(S) \amp = n\theta = 100\theta \\
|
||
\Var(S) \amp = n\theta(1-\theta) = 100\theta(1-\theta) \\
|
||
\Pr(S \geq 61) + \Pr(S \leq 39) \amp \approx \Pr\left(Z \geq \frac{60.5 - 100\theta}{\sqrt{100\theta(1 - \theta)}}\right) + \Pr\left(Z \leq \frac{39.5 - 100\theta}{\sqrt{100\theta(1 - \theta)}}\right)
|
||
\end{align*}
|
||
</div>
|
||
<div class="para">For example, if the true value of the parameter is <span class="process-math">\(\theta = 0.5\text{:}\)</span>
|
||
</div>
|
||
<div class="displaymath process-math" data-contains-math-knowls="./knowl/xref/app-Phi-table.html" id="sec-Power-5-1-1-7">
|
||
\begin{align*}
|
||
\Pr(S \geq 61) + \Pr(S \leq 39) \amp \approx \Pr(Z \geq 1.11) + \Pr(Z \leq -3.12) \\
|
||
\amp \approx (1 - 0.8665) + 0.0009 \\
|
||
\amp = 0.1344
|
||
\end{align*}
|
||
</div>
|
||
<div class="para">If, instead, <span class="process-math">\(\theta = 0.6\text{:}\)</span>
|
||
</div>
|
||
<div class="displaymath process-math" data-contains-math-knowls="./knowl/xref/app-Phi-table.html" id="sec-Power-5-1-1-9">
|
||
\begin{align*}
|
||
\Pr(S \geq 61) + \Pr(S \leq 39) \amp \approx \Pr(Z \geq 0.10) + \Pr(Z \leq -4.18) \\
|
||
\amp \approx (1 - 0.5398) + 0 \\
|
||
\amp = 0.4602
|
||
\end{align*}
|
||
</div>
|
||
<div class="para">(Note: for the <span class="process-math">\(z\)</span>-score <span class="process-math">\(-4.18\text{,}\)</span> which is below the range covered by <a href="app-Phi-table.html" class="internal" title="Section A.1: \Phi(z) Table of Values">Section A.1</a>, we will treat the <span class="process-math">\(\Phi\)</span> value as close enough to 0. For a <span class="process-math">\(z\)</span>-score outside the range on the other side, we would treat the <span class="process-math">\(\Phi\)</span> value as 1. If we needed more precision than that, we would consult an expanded table.)</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#sec-Power-5-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Example 5.3.2"><a href="#sec-Power-5" title="Copy heading and permalink for Example 5.3.2" aria-label="Copy heading and permalink for Example 5.3.2">🔗</a></div></article><div class="para" id="sec-Power-6">We can see in the example that the power of the test was greater when <span class="process-math">\(\theta = 0.6\)</span> than when <span class="process-math">\(\theta = 0.55\text{.}\)</span> The null hypothesis was <span class="process-math">\(\theta = 0.5\text{.}\)</span> It is reasonable that, if the coin is <em class="emphasis">more different</em> from the null hypothesis, then it will have a higher probabiilty of producing data extreme enough to reject the null hypothesis.<div class="autopermalink" data-description="Paragraph"><a href="#sec-Power-6" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<section class="exercises" id="exercises-Power"><h3 class="heading hide-type">
|
||
<span class="type">Exercises</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber"></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Exercises</span>
|
||
</h3>
|
||
<article class="exercise exercise-like" id="exercises-Power-1"><h4 class="heading"><span class="codenumber">1<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||
<div class="para" id="exercises-Power-1-1-1">Suppose we find a coin and wonder whether it’s fair. As a first test, we decide to flip the coin 200 times and count the number of heads, <span class="process-math">\(S\text{.}\)</span> What values of <span class="process-math">\(S\)</span> would be extreme enough to reject the null hypothesis of a fair coin? If the coin actually has a 0.6 probability of coming up heads, what is the power of this test?<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Power-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Exercise 5.3.1"><a href="#exercises-Power-1" title="Copy heading and permalink for Exercise 5.3.1" aria-label="Copy heading and permalink for Exercise 5.3.1">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Power-2"><h4 class="heading"><span class="codenumber">2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||
<div class="para" id="exercises-Power-2-1-1">Suppose we have a coin which we suspect comes up heads more often than a fair coin would. As a first test, we decide to flip the coin 200 times and count the number of heads, <span class="process-math">\(S\text{.}\)</span> What values of <span class="process-math">\(S\)</span> would be extreme enough to reject the null hypothesis of a fair coin? If the coin actually has a 0.6 probability of coming up heads, what is the power of this test?<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Power-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Exercise 5.3.2"><a href="#exercises-Power-2" title="Copy heading and permalink for Exercise 5.3.2" aria-label="Copy heading and permalink for Exercise 5.3.2">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Power-3"><h4 class="heading"><span class="codenumber">3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||
<div class="introduction" id="exercises-Power-3-1"><div class="para" id="exercises-Power-3-1-1">Suppose we find a six-sided die and wonder whether it’s fair. As a first test, we decide to roll the die 100 times and count the number of times it comes up 1. The expected number of 1’s is 50/3, with a variance of 125/9.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Power-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div></div>
|
||
<article class="task exercise-like" id="exercises-Power-3-2"><h5 class="heading"><span class="codenumber">(a)</span></h5>
|
||
<div class="para" id="exercises-Power-3-2-1-1">Using a normal approximation, what is the smallest number of 1’s greater than 50/3 that would be extreme enough to reject the null hypothesis of a fair die?<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Power-3-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Task 5.3.3.a"><a href="#exercises-Power-3-2" title="Copy heading and permalink for Task 5.3.3.a" aria-label="Copy heading and permalink for Task 5.3.3.a">🔗</a></div></article><article class="task exercise-like" id="exercises-Power-3-3"><h5 class="heading"><span class="codenumber">(b)</span></h5>
|
||
<div class="para" id="exercises-Power-3-3-1-1">Using a normal approximation, what is the greatest number of 1’s less than 50/3 that would be extreme enough to reject the null hypothesis of a fair die?<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Power-3-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Task 5.3.3.b"><a href="#exercises-Power-3-3" title="Copy heading and permalink for Task 5.3.3.b" aria-label="Copy heading and permalink for Task 5.3.3.b">🔗</a></div></article><article class="task exercise-like" id="exercises-Power-3-4"><h5 class="heading"><span class="codenumber">(c)</span></h5>
|
||
<div class="para" id="exercises-Power-3-4-1-1">Suppose that this die is weighted so that it rolls a 1 with probability 0.2. What would be the power of our test?<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Power-3-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
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<div class="autopermalink" data-description="Task 5.3.3.c"><a href="#exercises-Power-3-4" title="Copy heading and permalink for Task 5.3.3.c" aria-label="Copy heading and permalink for Task 5.3.3.c">🔗</a></div></article><article class="task exercise-like" id="exercises-Power-3-5"><h5 class="heading"><span class="codenumber">(d)</span></h5>
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<div class="para" id="exercises-Power-3-5-1-1">Suppose we roll the die 100 times and see 23 1’s. Use the maximum likelihood value for the probability of rolling a 1 to calculate the power of the test.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Power-3-5-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Task 5.3.3.d"><a href="#exercises-Power-3-5" title="Copy heading and permalink for Task 5.3.3.d" aria-label="Copy heading and permalink for Task 5.3.3.d">🔗</a></div></article><div class="autopermalink" data-description="Exercise 5.3.3"><a href="#exercises-Power-3" title="Copy heading and permalink for Exercise 5.3.3" aria-label="Copy heading and permalink for Exercise 5.3.3">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Power-4"><h4 class="heading"><span class="codenumber">4<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
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<div class="para" id="exercises-Power-4-1-1">Suppose a particular plant when grown outdoors has an average height of 39 in with a variance of 20 in<span class="process-math">\(^2\text{.}\)</span> We suspect that growing this plant in a greenhouse will increase its height. We take the average height of a sample of 50 plants grown in a greenhouse. What is the minimum average height of this sample that would be extreme enough to reject the null hypothesis of equal means at the <span class="process-math">\(p = 0.05\)</span> significance level? If the plants, when grown in a greenhouse, would truly have an average height of 41 in, what is the power of our test?<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Power-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Exercise 5.3.4"><a href="#exercises-Power-4" title="Copy heading and permalink for Exercise 5.3.4" aria-label="Copy heading and permalink for Exercise 5.3.4">🔗</a></div></article><div class="autopermalink" data-description="Exercises 5.3"><a href="#exercises-Power" title="Copy heading and permalink for Exercises 5.3" aria-label="Copy heading and permalink for Exercises 5.3">🔗</a></div></section><div class="autopermalink" data-description="Section 5.3: Power of a Test"><a href="#sec-Power" title="Copy heading and permalink for Section 5.3: Power of a Test" aria-label="Copy heading and permalink for Section 5.3: Power of a Test">🔗</a></div></section></div>
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