190 lines
5.0 KiB
XML
190 lines
5.0 KiB
XML
<?xml version="1.0" encoding="UTF-8"?>
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<chapter xml:id="ch-Confidence-Intervals" xmlns:xi="http://www.w3.org/2001/XInclude">
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<title>Confidence Intervals</title>
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<introduction>
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<p>
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Text before the first section.
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</p>
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</introduction>
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<section xml:id="sec-CLT" xmlns:xi="http://www.w3.org/2001/XInclude">
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<title>Central Limit Theorem</title>
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<p>
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Text of section.
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</p>
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<exercises xml:id="exercises-CLT">
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<exercise>
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<statement>
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<p>
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Suppose a coin has probability 0.4 of coming up heads, and we flip the coin 100 times.
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Let <m>S</m> be the number of heads.
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Estimate the probability that <m>34 \leq S \leq 44</m>.
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</p>
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</statement>
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</exercise>
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<exercise>
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<statement>
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<p>
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Suppose a fair die is rolled 100 times, and let <m>m</m> be the average value of the rolls.
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Estimate the probability that <m>3.45 \leq m \leq 3.55</m>.
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</p>
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</statement>
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</exercise>
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<exercise>
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<introduction>
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<p>
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The heights of men in the US have a mean of 69 in and a variance of about 9 in<m>^2</m>, and the heights of women in the US have a mean of 63.5 in with a variance of 6.25 in<m>^2</m>.
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</p>
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</introduction>
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<task>
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<statement>
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<p>
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Suppose the heights of 30 men are sampled, and a sample mean <m>m</m> is taken.
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Find the expected value and variance of <m>m</m>.
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</p>
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</statement>
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</task>
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<task>
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<statement>
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<p>
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Estimate the probability that <m>m \geq 69.5</m>.
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</p>
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</statement>
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</task>
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<task>
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<statement>
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<p>
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What if the sampled group was women?
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</p>
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</statement>
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</task>
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</exercise>
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</exercises>
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</section>
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<section xml:id="sec-Confidence-Intervals" xmlns:xi="http://www.w3.org/2001/XInclude">
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<title>Confidence Intervals</title>
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<p>
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Text of section.
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</p>
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<exercises xml:id="exercises-Confidence-Intervals">
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<exercise>
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<statement>
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<p>
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Suppose we flip a coin 100 times and count 60 heads.
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Let <m>p</m> be the (unknown) probability that the coin comes up heads on a flip.
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Give an approximate 95% confidence interval for the value of <m>p</m>.
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</p>
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</statement>
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</exercise>
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<exercise>
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<introduction>
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<p>
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Suppose in a sample of 100 people, 12 are left-handed.
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</p>
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</introduction>
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<task>
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<statement>
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<p>
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Give a 95% confidence interval for the proportion <m>p</m> of left-handed people.
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</p>
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</statement>
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</task>
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<task>
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<statement>
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<p>
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Give a 90% confidence interval.
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</p>
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</statement>
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</task>
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</exercise>
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<exercise>
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<statement>
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<p>
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The weights of five mice are measured and recorded below.
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Give a 95% confidence interval for the sample mean weight of mice.
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(Pretend 5 measurements is large enough for the CLT to apply.)
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</p>
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<table>
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<title></title>
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<tabular>
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<row header="yes">
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<cell halign="center">mouse <m>i</m></cell>
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<cell halign="center">1</cell>
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<cell halign="center">2</cell>
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<cell halign="center">3</cell>
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<cell halign="center">4</cell>
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<cell halign="center">5</cell>
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</row>
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<row>
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<cell halign="center">weight <m>\widetilde{w}_i</m> (g)</cell>
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<cell halign="center">26</cell>
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<cell halign="center">32</cell>
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<cell halign="center">33</cell>
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<cell halign="center">20</cell>
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<cell halign="center">29</cell>
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</row>
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</tabular>
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</table>
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</statement>
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</exercise>
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<exercise>
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<statement>
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<p>
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The heights of five plants are measured and recorded below.
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Give a 95% confidence interval around the sample mean for the heights of the plants.
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(Pretend 5 measurements is large enough for the CLT to apply.)
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</p>
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<table>
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<title></title>
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<tabular>
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<row header="yes">
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<cell halign="center">plant <m>i</m></cell>
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<cell halign="center">1</cell>
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<cell halign="center">2</cell>
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<cell halign="center">3</cell>
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<cell halign="center">4</cell>
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<cell halign="center">5</cell>
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</row>
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<row>
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<cell halign="center">height <m>\widetilde{h}_i</m> (in)</cell>
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<cell halign="center">15</cell>
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<cell halign="center">14</cell>
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<cell halign="center">18</cell>
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<cell halign="center">21</cell>
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<cell halign="center">17</cell>
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</row>
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</tabular>
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</table>
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</statement>
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</exercise>
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</exercises>
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</section>
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</chapter> |