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\begin{document}

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\title{Stat 312: Lecture 7\\
 Confidence Intervals III.}

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\author{Moo K. Chung\\
mchung@stat.wisc.edu}
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\section*{Concepts}
\begin{enumerate}

\item If $n$ is sufficiently large, approximate $100(1-\alpha)\%$
confidence interval for $\mu$ is $\bar x \pm z_{\alpha/2}s/\sqrt{n}$,
 where $s$ is the sample standard deviation.

\item Let $p$ denote the proportion of an individual with a
specified property. $100(1-\alpha)\%$ CI for a {\em population
proportion} $p$ is
$$\hat p \pm z_{\alpha/2}\sqrt{\hat p \hat q/n}.$$
\item One-sided confidence interval:
An upper confidence bound for $\mu$ is
$$\mu < \bar x + z_{\alpha}s/\sqrt{n}$$
and a lower confidence bound for $\mu$ is
$$\mu > \bar x - z_{\alpha}s/\sqrt{n}.$$
\end{enumerate}

\section*{In-class problems}
{\bf Ex.} Toss $n=100$ biased coins with $P(H)=p$. Suppose you
observe 38 heads. Construct 95$\%$ CI of $p$.
\texttt{rbinom(n,1,p)} will generate a Bernoulli random sample of
size $n$ with $P(X_i=1)=p$.
\begin{verbatim}
> X<-rbinom(100,1,0.4)
> X
  [1] 1 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0
 [17] 0 0 1 1 0 1 0 1 1 0 1 0 1 0 1 0
 [33] 1 1 0 0 0 0 1 0 1 0 0 0 0 0 1 0
 [49] 0 1 0 0 0 0 1 1 0 1 0 1 1 0 0 1
 [65] 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 0
 [81] 1 0 1 0 0 1 1 0 0 0 1 0 0 1 1 1
 [97] 0 1 1 0
> sd(X)
[1] 0.4878317
> 0.38+1.96*0.49/sqrt(100)
[1] 0.47604
> 0.38-1.96*0.49/sqrt(100)
[1] 0.28396
\end{verbatim}
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Exercise 7.23. Use Concept 2.\\
When 37 helmets were subjected to a certain impact test, 24
showed damage. Let $p$ denote the proportion of helmets that
would show damage under the test. Find a $99\%$ CI for $p$.

\section*{Self-study problems}
Example 7.8., 7.8. Use Concept 2.
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