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Adding more exercises throughout

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andyeisenberg
2026-01-01 22:19:38 +00:00
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source/ch-Confidence-Intervals.ptx
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<title>Confidence Intervals</title>
<introduction>
<p> Text before the first section. </p>
<p>
Text before the first section.
</p>
</introduction>
<section xml:id="sec-CLT" xmlns:xi="http://www.w3.org/2001/XInclude">
<title>Central Limit Theorem</title>
<title>Central Limit Theorem</title>
<p> Text of section. </p>
<p>
Text of section.
</p>
<exercises xml:id="exercises-CLT">
<exercise>
<statement>
<p>
Suppose a coin has probability 0.4 of coming up heads, and we flip the coin 100 times.
Let <m>S</m> be the number of heads.
Estimate the probability that <m>34 \leq S \leq 44</m>.
</p>
</statement>
</exercise>
<exercise>
<statement>
<p>
Suppose a fair die is rolled 100 times, and let <m>m</m> be the average value of the rolls.
Estimate the probability that <m>3.45 \leq m \leq 3.55</m>.
</p>
</statement>
</exercise>
<exercise>
<introduction>
<p>
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>.
</p>
</introduction>
<task>
<statement>
<p>
Suppose the heights of 30 men are sampled, and a sample mean <m>m</m> is taken.
Find the expected value and variance of <m>m</m>.
</p>
</statement>
</task>
<task>
<statement>
<p>
Estimate the probability that <m>m \geq 69.5</m>.
</p>
</statement>
</task>
<task>
<statement>
<p>
What if the sampled group was women?
</p>
</statement>
</task>
</exercise>
</exercises>
</section>
<section xml:id="sec-Confidence-Intervals" xmlns:xi="http://www.w3.org/2001/XInclude">
<title>Confidence Intervals</title>
<title>Confidence Intervals</title>
<p> Text of section. </p>
<p>
Text of section.
</p>
<exercises xml:id="exercises-Confidence-Intervals">
<exercise>
<statement>
<p>
Suppose we flip a coin 100 times and count 60 heads.
Let <m>p</m> be the (unknown) probability that the coin comes up heads on a flip.
Give an approximate 95% confidence interval for the value of <m>p</m>.
</p>
</statement>
</exercise>
<exercise>
<introduction>
<p>
Suppose in a sample of 100 people, 12 are left-handed.
</p>
</introduction>
<task>
<statement>
<p>
Give a 95% confidence interval for the proportion <m>p</m> of left-handed people.
</p>
</statement>
</task>
<task>
<statement>
<p>
Give a 90% confidence interval.
</p>
</statement>
</task>
</exercise>
<exercise>
<statement>
<p>
The weights of five mice are measured and recorded below.
Give a 95% confidence interval for the sample mean weight of mice.
(Pretend 5 measurements is large enough for the CLT to apply.)
</p>
<table>
<title></title>
<tabular>
<row header="yes">
<cell halign="center">mouse <m>i</m></cell>
<cell halign="center">1</cell>
<cell halign="center">2</cell>
<cell halign="center">3</cell>
<cell halign="center">4</cell>
<cell halign="center">5</cell>
</row>
<row>
<cell halign="center">weight <m>\widetilde{w}_i</m> (g)</cell>
<cell halign="center">26</cell>
<cell halign="center">32</cell>
<cell halign="center">33</cell>
<cell halign="center">20</cell>
<cell halign="center">29</cell>
</row>
</tabular>
</table>
</statement>
</exercise>
<exercise>
<statement>
<p>
The heights of five plants are measured and recorded below.
Give a 95% confidence interval around the sample mean for the heights of the plants.
(Pretend 5 measurements is large enough for the CLT to apply.)
</p>
<table>
<title></title>
<tabular>
<row header="yes">
<cell halign="center">plant <m>i</m></cell>
<cell halign="center">1</cell>
<cell halign="center">2</cell>
<cell halign="center">3</cell>
<cell halign="center">4</cell>
<cell halign="center">5</cell>
</row>
<row>
<cell halign="center">height <m>\widetilde{h}_i</m> (in)</cell>
<cell halign="center">15</cell>
<cell halign="center">14</cell>
<cell halign="center">18</cell>
<cell halign="center">21</cell>
<cell halign="center">17</cell>
</row>
</tabular>
</table>
</statement>
</exercise>
</exercises>
</section>
</chapter>