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\hskip 1.5in \textbf{Midterm 3A--Stat 100--Spring 1997}   
\vskip .2in  
You may use calculators, and ONE page of notes (writing allowed  
both sides). Each problem is 
worth 20 points. Different parts of a problem have equal weight unless 
otherwise indicated.  Do not spend too much time on any one problem. 
 Put a box around the final answer to a question. 
\vskip .1in 

\renewcommand{\labelenumi}{\alph{enumi}.}
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1.
It is claimed that more than 60\% of students majoring in biology
are female. In a random sample of 400 biology majors, 250 students were
found to be
female. We want to see whether this result provides strong evidence 
in favor of the claim.  
\begin{enumerate}
\item  
Formulate the null and the alternative hypotheses. 
\item   
Define the appropriate test statistic.  
\item  
Give the form of the rejection region for level of significance $\alpha$. 
\item 
For the above survey-result, calculate the P-value, and 
 determine all the levels 
$\alpha$ at which the null hypothesis would be retained.
\end{enumerate}

\vspace{.3in}

2.
Consider the following random sample of  5 observations from a certain
population: 
\begin{center}
13, 12, 11, 9, 10
\end{center}
For this data, the sample mean is 10 and the sample 
standard deviation is 1.5 .

\begin{enumerate}
\item (10 points) 
Give a 95\%-confidence interval
for the population standard deviation $\sigma$. 
\item (3 points) 
Give an assumption which justifies your computation of that 
confidence interval. 
\item (4 points) 
From this sample, what is the best point estimate 
for the population standard deviation?  
\item (3 points) 
Would a 98\% confidence interval be contained in the interval you 
computed? Briefly explain. 
\end{enumerate}

\vspace{.3in}


3. 
A government administrator wants to know if there is any difference in
the quality of training given to new air traffic controllers at 2
different training centers.  The exam results for the trainees of the
2 locations are as follows: 
\vspace{.1in} 

\hspace{.2in} \hspace{2in} New York \hspace{.5in} Chicago

\hspace{.2in} number of trainees  \hspace{.9in}  60 \hspace{1in} 50

\hspace{.2in} exam mean	\hspace{1.43in}    78  \hspace{1in} 75

\hspace{.2in} exam standard deviation \hspace{.55in}   6  \hspace{.95in}  12

\vspace{.1in}

Test the claim that the quality of training is different at the two
centers at a $5\%$ significance level.  In your response,
\begin{enumerate}
\item
Determine the null and alternative hypotheses.
\item
Define the appropriate test statistic. 
\item
Determine the rejection region.
\item
Determine the value of the test statistic.
\item
Decide if the data support the claim that quality of training is
different at the 2 sites.
\end{enumerate}

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**************THERE ARE MORE PROBLEMS ON THE BACK SIDE***************
\pagebreak 

\hspace{6in} .  
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\hspace{2.5in} IN THE PROBLEMS BELOW: 

\vspace{.2in}

If your answer involves a complicated formula, you need not take 
the time to evaluate it on your calculator; just write it out with 
the appropriate numbers inserted. 

\vspace{.2in}

\hspace{2.5in} ------------------------------------

\vspace{.5in} 
4. Five randomly selected
 people are placed on a vegetarian diet for a month.  The
weight of the five patients is recorded at the beginning and the end
of the month.
\vspace{.1in} 

\hspace{1.3in}  1 \hspace{.37in}2 \hspace{.37in}3 \hspace{.37in}4 \hspace{.37in}5 \hspace{.37in}

\hspace{.2in} Beginning \hspace{.2in} 165 \hspace{.2in}172 \hspace{.2in}130
\hspace{.2in}195 \hspace{.2in}180 

\hspace{.2in} End \hspace{.62in} 160 \hspace{.2in}162 \hspace{.2in}128
\hspace{.2in}185 \hspace{.2in}162 

\vspace{.1in}

Find a $90\%$ confidence interval for the mean amount of weight loss during
the month for people on the vegetarian diet.  


(If your answer involves a complicated formula, you need not take 
the time to evaluate it on your calculator; just write it out with 
the appropriate numbers inserted. 

\vspace{.4in}


5. The KeepOn trucking firm wishes to decide which route between two 
depots is quicker. Five drivers are randomly selected to drive route A 
and five drivers are randomly selected to drive route B, and the 
following sample statistics are obtained for hours of driving time. 

\vspace{.1in} 
\hspace{1.2in} Mean \hspace{.3in} Standard deviation 

\hspace{.2in} Route A \hspace{.4in} 25 \hspace{.7in} 10.0 

\hspace{.2in} Route B \hspace{.4in} 29\hspace{.8in} 8.0 

\vspace{.1in}
\begin{enumerate}
\item (15 points) 
Let $\mu _A$ be the mean travel time over route A and 
let $\mu _B$ be the mean travel time over route B. 
Compute a $95$\% confidence interval for $\mu_A - \mu _B$. 
\item (5 points) 
Give assumptions and/or computations which justify 
your procedure for computing that confidence interval. 
  \end{enumerate}

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