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\begin{document}

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%\date{January 23, 2003}

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\title{Stat 312: Lecture 6\\
 Confidence Intervals II.}

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\author{Moo K. Chung\\
mchung@stat.wisc.edu}
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\section*{Concepts}
\begin{enumerate}
\item  Let $X_i \sim N(\mu,\sigma^2)$ with known $\sigma^2$ and unknown $\mu$. $100(1-\alpha)\%$ confidence interval for $\mu$ is.
$$\hat \mu_L =\bar x - z_{\alpha/2} \cdot \sigma/\sqrt{n},\; \hat \mu_U=\bar x + z_{\alpha/2} \cdot \sigma/\sqrt{n}.$$

\item The sample size is inversely related to the width of confidence interval.
\item {\em Central Limit Theorem.} Let $X_1,\cdots,X_n$ be a random sample with mean $\mu$ and variance $\sigma^2$. For large $n$, $$Z=\frac{\bar X-\mu}{\sigma /\sqrt{n}} \sim N(0,1).$$
\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.
\end{enumerate}

\section*{In-class problems}
Continuing Exercise 6.25, construct $98\%$ confidence interval.\\
Example 7.4. Response time $\sim N(\mu,\sigma^2),\sigma=25$. Find
the sample size $n$ that ensures 95$\%$ CI with a width of 10.\\
Example 7.6. Alternating current (AC) voltage data
\begin{verbatim}
> data(xmp07.06)
> attach(xmp07.06)
> str(xmp07.06)
`data.frame':48 obs. of 1 variable:
 $ C1: int  62 50 53 57 ...
> boxplot(C1)
> mean(C1)
[1] 54.70833
> sd(C1)
[1] 5.230672
>mean(C1)-qnorm(0.975)*sd(C1)/sqrt(length(C1))
[1] 53.2286
>mean(C1)+qnorm(0.975)*sd(C1)/sqrt(length(C1))
[1] 56.18807
\end{verbatim}
{\bf Ex.} Toss $n=100$ biased coins with $P(H)=p$. Suppose you
observe 38 heads. Construct 95$\%$ CI of $p$.
\begin{figure}[t]
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\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
> sqrt(0.38*(1-0.38)/(100-1))
[1] 0.04878317
> 0.38+1.96*0.049/sqrt(100)
[1] 0.389604
> 0.38-1.96*0.049/sqrt(100)
[1] 0.370396
\end{verbatim}


\section*{Self-study problems}
Example 7.8., Exercise 7.13., 7.19., 7.25.\\
In the above coin tossing example, check if $\bar X$ is MLE.
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