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\begin{document}

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%\date{January 23, 2003}

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\title{Stat 312: Lecture 14\\
Two-sample tests}

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\author{Moo K. Chung\\
mchung@stat.wisc.edu}
%\date{Feb 20, 2003}
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\maketitle
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\section*{Concepts}
\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 $$Z=\frac{\bar X - \bar Y}{\sqrt{\sigma_X^2/n +
\sigma_Y^2/m}} \sim N(0,1).$$

\item Assume $n$ and $m$ to be {\bf large}. The test statistic for
testing
$$H_0: \mu_X = \mu_Y \mbox{ vs. } H_1: \mu_X \neq \mu_Y$$
$$Z=\frac{\bar X - \bar Y}{\sqrt{\sigma_X^2/n + \sigma_Y^2/m}} \sim N(0,1).$$

\end{enumerate}
\section*{In-class problems}

{\em Example 1.} A study was conducted to compare the reaction
times of men and women to a stimulus. 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
$Y_j \sim N(\mu_Y, 0.14)$. We want to test if the population
means are the same. 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
\end{verbatim}
{\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
> qnorm(0.025)
[1] -1.959964
\end{verbatim}

What is the main reason you are getting the different results in
Example 1 and 2?
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%\caption{$t_{1000}}
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\section*{Self-study problems}
Determine the $P$-value in Example 1 and 2.
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