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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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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-binomial-distribution" class="internal"><span class="codenumber">2.1.2</span> <span class="title">The Binomial Distribution</span></a></div></li>
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<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#exercises-Discrete-RVs" class="internal"><span class="codenumber">2.1.5</span> <span class="title">Exercises</span></a></div></li>
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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="ch-Expected-Value.html" class="internal"><span class="codenumber">3</span> <span class="title">Expected Value and Variance</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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<div class="toc-title-box"><a href="ch-Confidence-Intervals.html" class="internal"><span class="codenumber">4</span> <span class="title">Confidence Intervals</span></a></div>
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<li class="toc-item toc-section">
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<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>
|
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</li>
|
||
<li class="toc-item toc-section contains-active">
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||
<div class="toc-title-box"><a href="sec-Example.html" class="internal"><span class="codenumber">4.4</span> <span class="title">Example</span></a></div>
|
||
<ul class="structural toc-item-list active"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Example.html#sec-Example-8" class="internal"><span class="codenumber">4.4</span> <span class="title">Exercises</span></a></div></li></ul>
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</li>
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</ul>
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<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>
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<ul class="structural toc-item-list">
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<li class="toc-item toc-section">
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<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>
|
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<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>
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<li class="toc-item toc-section">
|
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<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>
|
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<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>
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<li class="toc-item toc-section">
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<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>
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<ul class="structural toc-item-list"><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>
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<li class="toc-item toc-section">
|
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<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>
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<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>
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<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>
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<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>
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<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>
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<li class="toc-item toc-section">
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<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>
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<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>
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<main class="ptx-main"><div id="ptx-content" class="ptx-content"><section class="section" id="sec-Example"><h2 class="heading hide-type">
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<span class="type">Section</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4.4</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Example</span>
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</h2>
|
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<div class="para" id="sec-Example-2">Welcome to PreTeXt. Let’s type up some math. According to Pythagoras, <span class="process-math">\(a^2 + b^2 = c^2\text{.}\)</span> But he also thought <span class="process-math">\(\sqrt{2}\)</span> was rational, so maybe we shouldn’t take his word for it. Let’s see a proof by picture.<div class="autopermalink" data-description="Paragraph"><a href="#sec-Example-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<article class="theorem theorem-like" id="thm-Pythagorean-Theorem"><h3 class="heading">
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<span class="type">Theorem</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4.4.1</span><span class="period heading-divison-mark heading-divison-mark__period">.</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Pythagorean Theorem.</span>
|
||
</h3>
|
||
<div class="para logical" id="thm-Pythagorean-Theorem-2-1">
|
||
<div class="para">Suppose a triangle has side lengths <span class="process-math">\(a, b\)</span> and hypotenuse length <span class="process-math">\(c\text{.}\)</span> Then</div>
|
||
<div class="displaymath process-math" id="thm-Pythagorean-Theorem-2-1-3">
|
||
\begin{gather*}
|
||
a^2 + b^2 = c^2.
|
||
\end{gather*}
|
||
</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#thm-Pythagorean-Theorem-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
|
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<div class="autopermalink" data-description="Theorem 4.4.1: Pythagorean Theorem"><a href="#thm-Pythagorean-Theorem" title="Copy heading and permalink for Theorem 4.4.1: Pythagorean Theorem" aria-label="Copy heading and permalink for Theorem 4.4.1: Pythagorean Theorem">🔗</a></div></article><details id="thm-Pythagorean-Theorem-3" class="hiddenproof born-hidden-knowl"><summary class="knowl__link"><h3 class="heading"><span class="type">Proof<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3></summary><article class="hiddenproof knowl__content"><div class="image-box" style="width: 50%; margin-left: 25%; margin-right: 25%;">
|
||
<img src="external/./example.png" class="contained" alt="Right triangle with squares appended to each side."><details class="image-description" aria-live="polite"><summary title="details"><span class="icon material-symbols-outlined" aria-hidden="true"></span></summary><div id="thm-Pythagorean-Theorem-3-1-description"><div class="para" id="thm-Pythagorean-Theorem-3-1-2-1">Right triangle with legs labeled <span class="process-math">\(a\)</span> and <span class="process-math">\(b\)</span> and hypotenuse labeled <span class="process-math">\(c\text{.}\)</span> Squares are appended to each side of the triangle.<div class="autopermalink" data-description="Paragraph"><a href="#thm-Pythagorean-Theorem-3-1-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div></div></details>
|
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</div>
|
||
<div class="autopermalink" data-description="Proof 4.4.1.1"><a href="#thm-Pythagorean-Theorem-3" title="Copy heading and permalink for Proof 4.4.1.1" aria-label="Copy heading and permalink for Proof 4.4.1.1">🔗</a></div></article></details><article class="definition definition-like" id="def-rational"><h3 class="heading">
|
||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4.4.2</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||
</h3>
|
||
<div class="para" id="def-rational-1-1">A real number <span class="process-math">\(x \in \R\)</span> is called <dfn class="terminology">rational</dfn> if there are integers <span class="process-math">\(a, b\)</span> so that <span class="process-math">\(x = \frac{a}{b}\text{.}\)</span> Otherwise, <span class="process-math">\(x\)</span> is called <dfn class="terminology">irrational</dfn>, no matter what Pythagoras says.<div class="autopermalink" data-description="Paragraph"><a href="#def-rational-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
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</div>
|
||
<div class="autopermalink" data-description="Definition 4.4.2"><a href="#def-rational" title="Copy heading and permalink for Definition 4.4.2" aria-label="Copy heading and permalink for Definition 4.4.2">🔗</a></div></article><article class="example example-like" id="sec-Example-5"><h3 class="heading">
|
||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4.4.3</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||
</h3>
|
||
<div class="para" id="sec-Example-5-1">
|
||
<span class="process-math">\(\frac{1}{2}\)</span> is rational.<div class="autopermalink" data-description="Paragraph"><a href="#sec-Example-5-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Example 4.4.3"><a href="#sec-Example-5" title="Copy heading and permalink for Example 4.4.3" aria-label="Copy heading and permalink for Example 4.4.3">🔗</a></div></article><article class="example example-like" id="sec-Example-6"><h3 class="heading">
|
||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4.4.4</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||
</h3>
|
||
<div class="para" id="sec-Example-6-1-1">Is <span class="process-math">\(\sqrt{2}\)</span> rational?<div class="autopermalink" data-description="Paragraph"><a href="#sec-Example-6-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="solutions">
|
||
<details id="sec-Example-6-2" class="hint solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Hint</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="hint solution-like knowl__content">
|
||
<div class="para" id="sec-Example-6-2-1">Why does <span class="process-math">\(\pi = \frac{\sqrt{2}}{1}\)</span> <em class="emphasis">not</em> show that <span class="process-math">\(\sqrt{2}\)</span> is rational?<div class="autopermalink" data-description="Paragraph"><a href="#sec-Example-6-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
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</div>
|
||
<div class="autopermalink" data-description="Hint 4.4.4.1"><a href="#sec-Example-6-2" title="Copy heading and permalink for Hint 4.4.4.1" aria-label="Copy heading and permalink for Hint 4.4.4.1">🔗</a></div>
|
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</div></details><details id="sec-Example-6-3" class="hint solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Hint</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="hint solution-like knowl__content">
|
||
<div class="para" id="sec-Example-6-3-1">What would happen if you assumed <span class="process-math">\(\sqrt{2}\)</span> is rational?<div class="autopermalink" data-description="Paragraph"><a href="#sec-Example-6-3-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="autopermalink" data-description="Hint 4.4.4.2"><a href="#sec-Example-6-3" title="Copy heading and permalink for Hint 4.4.4.2" aria-label="Copy heading and permalink for Hint 4.4.4.2">🔗</a></div>
|
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</div></details><details id="sec-Example-6-4" class="answer solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Answer</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="answer solution-like knowl__content">
|
||
<div class="para" id="sec-Example-6-4-1">No!<div class="autopermalink" data-description="Paragraph"><a href="#sec-Example-6-4-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
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</div>
|
||
<div class="autopermalink" data-description="Answer 4.4.4.1"><a href="#sec-Example-6-4" title="Copy heading and permalink for Answer 4.4.4.1" aria-label="Copy heading and permalink for Answer 4.4.4.1">🔗</a></div>
|
||
</div></details><details id="sec-Example-6-5" class="solution solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Solution</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="solution solution-like knowl__content">
|
||
<div class="para logical" id="sec-Example-6-5-1">
|
||
<div class="para">Suppose that <span class="process-math">\(\sqrt{2} = \frac{a}{b}\)</span> for relatively prime integers <span class="process-math">\(a, b\text{.}\)</span> Then:</div>
|
||
<div class="displaymath process-math" id="sec-Example-6-5-1-3">
|
||
\begin{align*}
|
||
2 \amp = \frac{a^2}{b^2}\\
|
||
2b^2 \amp = a^2
|
||
\end{align*}
|
||
</div>
|
||
<div class="para">But, after taking the prime factorization of both sides, the left side has an odd number of prime factors and the right side has an even number. This is a contradiction, so <span class="process-math">\(\sqrt{2}\)</span> is irrational.</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#sec-Example-6-5-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
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</div>
|
||
<div class="autopermalink" data-description="Solution 4.4.4.1"><a href="#sec-Example-6-5" title="Copy heading and permalink for Solution 4.4.4.1" aria-label="Copy heading and permalink for Solution 4.4.4.1">🔗</a></div>
|
||
</div></details>
|
||
</div>
|
||
<div class="autopermalink" data-description="Example 4.4.4"><a href="#sec-Example-6" title="Copy heading and permalink for Example 4.4.4" aria-label="Copy heading and permalink for Example 4.4.4">🔗</a></div></article><details id="sec-Example-7" class="exercise exercise-like born-hidden-knowl"><summary class="knowl__link"><h3 class="heading">
|
||
<span class="type">Exercise</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4.4.5</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||
</h3></summary><article class="exercise exercise-like knowl__content"><div class="para" id="sec-Example-7-1-1">Prove that <span class="process-math">\(\sqrt{3}\)</span> is irrational.<div class="autopermalink" data-description="Paragraph"><a href="#sec-Example-7-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="autopermalink" data-description="Exercise 4.4.5"><a href="#sec-Example-7" title="Copy heading and permalink for Exercise 4.4.5" aria-label="Copy heading and permalink for Exercise 4.4.5">🔗</a></div></article></details><section class="exercises" id="sec-Example-8"><h3 class="heading hide-type">
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<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="sec-Example-8-1"><h4 class="heading"><span class="codenumber">1<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||
<div class="para" id="sec-Example-8-1-1">An exercise.<div class="autopermalink" data-description="Paragraph"><a href="#sec-Example-8-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="autopermalink" data-description="Exercise 4.4.1"><a href="#sec-Example-8-1" title="Copy heading and permalink for Exercise 4.4.1" aria-label="Copy heading and permalink for Exercise 4.4.1">🔗</a></div></article><article class="exercise exercise-like" id="sec-Example-8-2"><h4 class="heading"><span class="codenumber">2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||
<div class="para" id="sec-Example-8-2-1">An exercise.<div class="autopermalink" data-description="Paragraph"><a href="#sec-Example-8-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="autopermalink" data-description="Exercise 4.4.2"><a href="#sec-Example-8-2" title="Copy heading and permalink for Exercise 4.4.2" aria-label="Copy heading and permalink for Exercise 4.4.2">🔗</a></div></article><article class="exercise exercise-like" id="sec-Example-8-3"><h4 class="heading"><span class="codenumber">3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||
<div class="para" id="sec-Example-8-3-1">An exercise.<div class="autopermalink" data-description="Paragraph"><a href="#sec-Example-8-3-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="autopermalink" data-description="Exercise 4.4.3"><a href="#sec-Example-8-3" title="Copy heading and permalink for Exercise 4.4.3" aria-label="Copy heading and permalink for Exercise 4.4.3">🔗</a></div></article><article class="exercise exercise-like" id="sec-Example-8-4"><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="sec-Example-8-4-1">An exercise.<div class="autopermalink" data-description="Paragraph"><a href="#sec-Example-8-4-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="autopermalink" data-description="Exercise 4.4.4"><a href="#sec-Example-8-4" title="Copy heading and permalink for Exercise 4.4.4" aria-label="Copy heading and permalink for Exercise 4.4.4">🔗</a></div></article><div class="autopermalink" data-description="Exercises 4.4"><a href="#sec-Example-8" title="Copy heading and permalink for Exercises 4.4" aria-label="Copy heading and permalink for Exercises 4.4">🔗</a></div></section><div class="autopermalink" data-description="Section 4.4: Example"><a href="#sec-Example" title="Copy heading and permalink for Section 4.4: Example" aria-label="Copy heading and permalink for Section 4.4: Example">🔗</a></div></section></div>
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