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Split sections in Ch 4

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2026-02-01 09:44:40 -05:00
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source/ch-Confidence-Intervals.ptx
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@@ -9,182 +9,9 @@
</p>
</introduction>
<section xml:id="sec-CLT" xmlns:xi="http://www.w3.org/2001/XInclude">
<title>Central Limit Theorem</title>
<xi:include href="./sec-Likelihood.ptx" />
<xi:include href="./sec-CLT.ptx" />
<xi:include href="./sec-Confidence-Intervals.ptx" />
<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>
<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>
source/docinfo.ptx
+2
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@@ -36,6 +36,8 @@
\DeclareMathOperator{\E}{E}
\DeclareMathOperator{\Var}{Var}
\DeclareMathOperator{\Cov}{Cov}
<!-- to indicate an estimator -->
\newcommand{\est}{\widehat}
</macros>
<!-- If you put any latex-image elements you can include preambles -->
source/sec-CLT.ptx
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@@ -0,0 +1,64 @@
<section xml:id="sec-CLT" xmlns:xi="http://www.w3.org/2001/XInclude">
<title>Central Limit Theorem</title>
<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>
source/sec-Confidence-Intervals.ptx
+113
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<section xml:id="sec-Confidence-Intervals" xmlns:xi="http://www.w3.org/2001/XInclude">
<title>Confidence Intervals</title>
<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>
source/sec-Likelihood.ptx
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@@ -0,0 +1,60 @@
<section xml:id="sec-Likelihood" xmlns:xi="http://www.w3.org/2001/XInclude">
<title>Likelihood</title>
<p>
So far, we've been concerned with probability theory.
Starting with a probability distribution and some parameter values, we've tried to answer questions like: What's the probability of seeing certain experimental results? Statistics is concerned with going in the other direction: Upon seeing the experimental results, can we determine the type of underlying probability distribution? Can we determine its parameters?
</p>
<definition xml:id="def-estimator">
<statement>
<p>
An <term>estimator</term> is a value of a parameter computed from a sample of data.
</p>
</statement>
</definition>
<example>
<p>
Suppose we find a coin on the street and don't know whether or not it's fair.
We want to know the probability <m>p</m> of the coin coming up heads.
We might, for example, flip the coin <m>n</m> times and count the number <m>k</m> of heads.
Then, we'll estimate <m>p = \frac{k}{n}</m>.
We'll refer to this as a <term>common sense</term> estimator.
(Other distributions and parameter types will have different notions of "common sense".)
</p>
</example>
<p>
An estimator is, itself, a random variable: it produces a numerical value based on the results of an experiment.
We'll use notation like <m>\est{p}</m> for a random variable which is an estimator for a parameter <m>p</m>.
(Similarly, <m>\est{\lambda}</m> would denote an estimator for a parameter called <m>\lambda</m>.)
</p>
<definition xml:id="def-unbiased">
<statement>
<p>
An estimator <m>\est{p}</m> is called <term>unbiased</term> if <m>\E(\est{p}) = p</m>.
</p>
</statement>
</definition>
<example>
<statement>
<p>
Suppose we have a coin with parameter <m>p</m>, which we'll flip <m>n</m> times and count the number <m>k</m> of heads.
We use the unbiased estimator <m>\est{p} = \frac{k}{n}</m>.
In this case, notice that <m>k \sim \Bin(n, p)</m>, so we know <m>\E(k) = np</m>, although we don't know the value of <m>p</m>.
(We probably do know the value of <m>n</m>; after all, we're flipping the coin!) Now:
<md>
<mrow> \E(\est{p}) = \E\left(\frac{k}{n}\right) = \frac{1}{n} \cdot \E(k) = \frac{1}{n} \cdot np = p. </mrow>
</md>
It's worth pausing for a moment to be appropriately impressed with ourselves.
We still don't know the true value of <m>p</m>.
But we managed to show that our common sense method of estimating <m>p</m> gives, on average, the correct value.
</p>
</statement>
</example>
</section>