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{\LARGE
Statistics \hfill Class Project --- Part 3 of 3 \\
}
\end{center}

\vskip 0.2in

This is the final part of the three-part class project. 
The grading for the course project will
be 80\% correctness, 20\% clarity of presentation. 

\vskip 0.15in

\begin{minipage}{4.5in}
\noindent{\bf Problem 1.} In 1960, a young statistician conducted a
survey of his small hometown. He selected 20 households in his town
at random and asked them how large their household was.
The results of this survey are shown to the right.

\vskip 0.1in 

(a) Using the appropriate technique from Chapter 4, determine the
sample mean $\overline{x}$ and sample standard deviation $s$ for
this sample.

\vskip 0.1in

(b) Determine a 95\% confidence interval for $\mu$, the average
number of persons per household in the town. Express your answer to 
two decimal places.
\end{minipage} \hfill
\begin{minipage}{2in}
\begin{tabular}{|c|c|} \hline
\multicolumn{2}{|c|}{\bf 1960 Town Survey Results} \\ \hline
Size of household & Frequency \\ \hline
1  &  3 \\
2  &  7 \\
3  &  5 \\
4  &  2 \\
5  &  2 \\
6  &  0 \\
7  &  0 \\
8  &  1 \\ \hline
\end{tabular}
\end{minipage}

\vskip 1in

\begin{minipage}{4.5in}
\noindent{\bf Problem 2.} Inspired by these preliminary results, 
the young statistician commissioned a larger survey to determine the 
average size of an {\it American} household. The results of the
simple random sample of 1,164 households yielded the results shown
to the right.

\vskip 0.1in 

(a) Determine the sample mean $\overline{x}$ and sample standard 
deviation $s$ for this sample.

\vskip 0.1in

(b) Determine a 95\% confidence interval for $\mu$, the average
number of persons in a 1960 American household. Express your answer
to two decimal places.
\end{minipage} \hfill
\begin{minipage}{2in}
\begin{tabular}{|c|c|} \hline
\multicolumn{2}{|c|}{\bf 1960 U.S. Survey Results} \\ \hline
Size of household & Frequency \\ \hline
1  &  160 \\
2  &  312 \\
3  &  225 \\
4  &  199 \\
5  &  142 \\
6  &   71 \\
7  &   35 \\
8  &   18 \\
9  &    2 \\ \hline
\end{tabular}
\end{minipage} 

\newpage

\begin{minipage}{5in}
\noindent{\bf Problem 3.} According to the 1960 Census, the average
size of an American family was $\mu = 3.35$ persons and a standard
deviation of $\sigma = 1.82$ persons. However, as the years have 
passed, the (now older) statistican believes that the average size of an
American household has decreased since 1960. The statistician believes
that, in the intervening 37 years, people are getting married later, 
divorce rates are higher, and the birth rate is lower. Accordingly, 
he commissions another survey in 1997; this new survey used a simple
random sample of 636 American households. The results are shown to the right.
\end{minipage} \hfill
\begin{minipage}{2in}
\begin{tabular}{|c|c|} \hline
\multicolumn{2}{|c|}{\bf 1997 U.S. Survey Results} \\ \hline
Size of household & Frequency \\ \hline
1  &  158 \\
2  &  203 \\
3  &  117 \\
4  &   96 \\
5  &   40 \\
6  &   15 \\
7  &    4 \\
8  &    2 \\
9  &    1 \\ \hline
\end{tabular}
\end{minipage}

\vskip 0.2in

In this problem, we will consider the demographic changes in American
households since 1960.

\vskip 0.1in

(a) Determine the sample mean $\overline{x}$ for this sample.

\vskip 0.1in

(b) Formulate the null hypothesis $H_0$ and the alternate hypothesis $H_1$.

\vskip 0.1in

(c) Should a one-tailed or two-tailed test be employed? What is the
critical value that corresponds to a level of significance of $\alpha = 0.01$?

\vskip 0.1in

(d) What is the $z-$value that corresponds to the above value of 
$\overline{x}$?

\vskip 0.1in

(e) Should the null hypothesis be accepted or rejected?
In one sentence, state your conclusion in words.

\vfill

Note: While the above story is ficticious, the data sets in Problems 2
and 3 {\it are} representative of the sizes of American households in 1960
and 1997. For more information, see the U.S. Census Bureau's Web page
at \verb|http://www.census.gov|

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