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{\bf Nonparametric Statistics Homework II.}\\
Due date: Two weeks from today. Worth $5\%$ of your final grade.
Computer printout should be included.
\begin{enumerate}

\item The binary representation of integers $1,2,3,4,\cdots,$ is
given by $1,10,11,100,\cdots$. Is the binary sequence $11011100
\cdots$ random? Test it based on the binary representation of 10
numbers between 1 and 10. Use the total number of runs as a test
statistic. What is your P-value?

\item From Problem 1, use the length of the longest run as a test
statistic. What is your P-value? (R or Splus may have a built-in
function for the length of the longest run test. Feel free to use
it. Matlab does not have a built-in function as far as I know).

\item Suppose that 20 out of 50 females and 10 out of 40 males are
depressed in a sample. Determine whether the frequency of
depression is related to sex.

\item We are interested in investigating the proportions of four
blood types(ABO system). Somehow we believe that 34$\%$ of people
have blood type A, 15$\%$ blood type B, 23$\%$ blood type AB, and
28$\%$ blood type O. We go out to campus and collect a sample of
100 students, and find the following: A: 12, B: 56, AB: 2, O: 30.
What do you conclude? State the null hypothesis and compute
P-value.

\item Morton counted the occurrence of various words in several
works by Jane Austen: chapters 1 and 3 of Sense and Sensibility,
chapters 1,2 and 3 of Emma, chapters 1 and 6 of Sanditon (written
by Austen).\\

\hspace{4cm}            a $\;$  an this that with without\\
Sense and Sensibility      147     25 \hspace{0.3cm}32 \hspace{0.2cm}94\hspace{0.3cm} 59\hspace{0.5cm} 18\\
Emma  \hspace{2.5cm}                   186 26 \hspace{0.2cm}39 \hspace{0.2cm}105\hspace{0.2cm} 74\hspace{0.5cm} 10\\
Sanditon    \hspace{2cm}        147 25 \hspace{0.3cm}32 \hspace{0.2cm}94 \hspace{0.4cm}59 \hspace{0.5cm}18\\
Examine the consistency of Austen's usage of words from book to
book. State the null hypothesis. What is your P-value?

\item (Optional problem) You are not required to solve it but if
you solve it, you get bonus points. From Problem 1, test the
randomness based on the binary representation of 10,000 integers
between 1 and 10,000. The test based on the total number of run
would be easier. What is your P-value?

\end{enumerate}




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