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{\bf Nonparametric Statistics Homework III.}\\
Due date: Nov. 12. 2:30pm. Worth $5\%$ of your final grade.
\begin{enumerate}

\item Gross sales before and after a training program is given by\\
Sales person: \hspace{0.1cm}1\hspace{0.3cm}  2 \hspace{0.3cm}3 \hspace{0.3cm}4 \hspace{0.3cm}5 \hspace{0.3cm}6\\
Sales before: 90 83 105 97 110 78\\
Sales after : \hspace{0.1cm}97 80 110 93 123 84.\\
Determine if the training program is effective using the sign
test. What is the P-value?

\item Answer Problem 1 using the signed-rank test. What
assumptions are you making? Compare your result with Problem 1.

\item Using the Mann-Whitney test, check if the women's height are
related to age in a pairwise fashion. There are six possible
pairwise comparisons. For example, you need to compare 20-29 vs.
30-39 and 20-29 vs. 40-49 etc. As this problem illustrates, the Mann-Whitney test is
not suitable for testing more than two samples.\\
20-29: 63.75 68.25 62.25 67.25\\
30-39: 64.75 67.5 64.75 66.5\\
40-49: 68.5 64.25 64.5 66\\
50-59: 65.25 64.75 67.5

\item Using the Kruskal-Wallis test, check if the women's heights
are related to age from Problem 3. Compare your result with Problem 3.\\

\item Using the Kolmogorov-Smirnov two-sample test, show that the
t statistic with large degrees of freedom can be approximated by
the standard normal distribution. How large should the degrees of
freedom be ? Hint: Do simulations on a large sample.

\item (Optional) In 2-way contigency table (size $m \times n$), show that testing equality is equivalent to testing indepence. You need to show $H_0: p_{1j} =p_{2j}=\cdots=p_{mj}$ for all $j$ is equivalent to $H_0': p_{ij}=p_{i \cdot}p_{\cdot j}$ (see my lecture note for notation).

\end{enumerate}

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