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\begin{document}

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%\date{January 23, 2003}

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\title{Stat 312: Lecture 16\\
Two sample test}

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\author{Moo K. Chung\\
mchung@stat.wisc.edu}
\date{November 4, 2004}
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\maketitle \thispagestyle{empty}
\begin{enumerate}

\item Let $X_1,\cdots,X_n$ and $Y_1,\cdots,Y_m$ be two independent
samples with mean $\mu_X$, $\mu_Y$ and variance $\sigma_X^2$,
$\sigma_Y^2$ respectively. $$\mathbb{E}(\bar X - \bar Y) = \mu_X
-\mu_Y. $$
$${\bf Var} (\bar X -\bar Y) = \sigma_X^2/n +
\sigma_Y^2/m.$$

\item Let $X_1,\cdots,X_n$ and $Y_1,\cdots,Y_m$ be two independent
samples with $X_i \sim N(\mu_X,\sigma_X^2)$ and $Y_j \sim N(\mu_Y,
\sigma_Y^2)$. Then we can show that $$Z=\frac{\bar Y - \bar X -
(\mu_X - \mu_Y)}{\sqrt{\sigma_X^2/n + \sigma_Y^2/m}} \sim N(0,1).$$
{\em Proof.} See lecture note.

\item Based on the above statistical assumptions on $X_i$ and $Y_j$,
we are interested in testing
$$H_0: \mu_X = \mu_Y \mbox{ vs. } H_1:\mu_X  \neq \mu_Y.$$

{\em Example 1.} 12 men and 9 women were employed in the experiment.
Let $X_i$ be the reaction time of the $i$-th man and $Y_j$ be the
reaction time of the $j$-th woman. There is a reason to believe that
$X_i \sim N(\mu_X, 0.18)$ and $Y_j \sim N(\mu_Y, 0.14)$. The
following measurements were taken: $\bar x = 3.6$ seconds, $\bar y=
3.8$ seconds. Is there any difference between the reaction times of
men and women?
\begin{verbatim}
> z=(3.6-3.8)/sqrt(0.18/12+0.14/9)
> z
[1] -1.144155
> qnorm(0.025)
[1] -1.959964
\end{verbatim}



\item Let $X_1,\cdots,X_n$ and $Y_1,\cdots,Y_m$ be two independent
samples with mean $\mu_X$, $\mu_Y$ respectively. Assume $n$ and $m$
to be sufficiently {\bf large}. The test statistic for testing
$$H_0: \mu_X = \mu_Y \mbox{ vs. } H_1: \mu_X \neq \mu_Y$$
is
$$Z=\frac{\bar Y - \bar X}{\sqrt{S_X^2/n + S_Y^2/m}} \sim N(0,1).$$


{\em Example 2.} A study was conducted to compare the reaction times
of men and women to a stimulus. 50 men and 65 women were employed in
the experiment. Let $X_i$ be the reaction time of the $i$-th man and
$Y_j$ be the reaction time of the $j$-th woman. The following
measurements were taken: $\bar x = 3.6$ seconds, $\bar y= 3.8$
seconds, $s_X^2=0.18$, $s_Y^2=0.14$. Is there any difference between
the reaction times of men and women?
\begin{verbatim}
> z=(3.6-3.8)/sqrt(0.18/50+0.14/65)
> z
[1] -2.63664

\end{verbatim}


What is the main reason you are getting the different results in
Example 1 and 2?



\end{enumerate}

{\em Review Problems.} Determine the $P$-values in Example 1 and 2.
Example 9.1., 9.4.

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