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\begin{document}

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%\date{January 23, 2003}

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\title{Stat 312: Lecture 13\\
Testing on Population Means}

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\author{Moo K. Chung\\
mchung@stat.wisc.edu}
\date{October 26, 2004}
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\begin{enumerate}

\item Since we can minimize both type I and type II errors
simultaneously in constructing a test procedure, we will control for
fixed type I error (level $\alpha$) and make type II error as small
as possible. The corresponding test procedure is called a {\em level
$\alpha$ test}.

\item Testing mean $\mu$ with known variance $\sigma^2$.
$$H_0: \mu=\mu_0 \mbox{ vs. } H_1: \mu < \mu_0$$
Test statistic: $z=\frac{\bar x -\mu_0}{\sigma/\sqrt{n}}$, which is
a pivot under $H_0$. A {\em pivot} is a statistic whose distribution
is independent of population paramters. Rejection region for level
$\alpha$ test: $z \leq -z_{\alpha}.$

Ex. Intelligence quotient (IQ) is a number used to express the
relative intelligence of a person. An average person has the IQ of
100. Assume IQ of a dog follows $X_i \sim N(\mu, 10^2)$. The IQs of
10 dogs are measured: 30, 25, 70, 110, 40, 80, 50, 60, 100, 60. We
want to test if dogs are as smart as people by testing
$$H_0: \mu = 100 \mbox{ vs. } H_1: \mu < 100$$
at level $\alpha = 0.05$.
\begin{verbatim}
> x<-c(30, 25, 70, 110, 40, 80, 50, 60,
100, 60)
> mean(x)
[1] 62.5
> z<-(mean(x)-100)/(10/sqrt(10))
> z
[1] -11.85854
> qnorm(0.05)
[1] -1.644854
> qnorm(0.025)
[1] -1.959964
\end{verbatim}

Since $z < -z_{0.05} = -1.64$, we reject $H_0$ at $\alpha=0.05$
level.

\item Testing mean $\mu$ with unknown variance $\sigma^2$.
$$H_0: \mu=\mu_0 \mbox{ vs. } H_1: \mu < \mu_0$$ Test statistic:
$t=\frac{\bar x -\mu_0}{s/\sqrt{n}}.$ Rejection region for level
$\alpha$ test: $z \leq -t_{\alpha,n-1}.$


Ex. IQ of a dog, $X_i \sim N(\mu, \sigma^2),$ where $\sigma$ is
unknown. Test
$$H_0: \mu = 100 \mbox{ vs. } H_1: \mu < 100$$
at level $\alpha = 0.05$.
\begin{verbatim}
> t=(mean(x)-100)/(sd(x)/sqrt(10))
> t
[1] -4.205156
> qt(0.05,9)
[1] -1.833113
\end{verbatim}
A simpler method is to use command {\tt t.test}.
\begin{verbatim}
>help(t.test)
...
t.test(x,alternative=c("two.sided", "less",
       "greater"),conf.level = 0.95)
...
> t.test(x,mu=100,alternative="less",
 conf.level=0.95)
        One Sample t-test
 data:  x t = -4.2036, df = 9,
 p-value = 0.001147
 alternative hypothesis: true mean is
 less than 100
 95 percent confidence interval:
    -Inf 78.8531
 sample estimates: mean of x 62.5
\end{verbatim}
%\begin{figure}[t]
%\centering
%\renewcommand{\baselinestretch}{1}
%\includegraphics[scale=0.5]{chi.eps}
%\caption{$t_{1000}}
%\end{figure}


\end{enumerate}

\section*{Review problems}
Example 8.6., 8.7., 8.8., 8.9.

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