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\hskip 1.5in \textbf{Midterm 2A--Stat 100--Spring 1997}   
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You may use calculators, but not books or notes. 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. 
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1. Suppose $Z$ has standard normal distribution.


\begin{enumerate}
\item 
(6 points) \ 
Find $P[Z>-1.02]$ and $P[Z \geq -1.02]$
\item 
(6 points) \ 
Find $P[-1 < Z < 2]$
\item 
(8 points) \ 
Determine the value of $a$ so that $P[-a<Z<a]=0.95$
\end{enumerate}

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2. Suppose $20\%$ of the children in a town have a certain virus. 
\begin{enumerate}
\item 
(17 points) 
Estimate the probability that, in a random sample of 300 children, 
the number of children with the virus will be between 50 and 70
inclusive. 
\item 
(3 points) 
Briefly justify your approximation procedure. 
\end{enumerate}

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3. Suppose a population of men has average weight 170 pounds 
with standard deviation 15 pounds. 

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Estimate the probability that the average weight of 
a random sample of 400 men from this population 
is greater than 171 pounds. 



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4.  A reporter wants to know the average commuting time to work
in the metropolitan D.C. area.  Suppose the standard deviation
of commuting times is known to be 14 minutes.  The reporter will
collect a random sample of commuting times and wants to be at least  
$95\%$ sure that the sample mean is within 2 minutes of the 
actual mean.  

\hskip .3in How large should the sample be? 

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5.  The campus food service conducts a survey to estimate the
proportion of students who eat breakfast every day.  
Out of 250 students surveyed, 200 eat breakfast every day.
\begin{enumerate} 
\item  (17 points) 
Construct a $99\%$ confidence interval for the proportion of students
who eat breakfast every day.
\item (3 points) 
Would a $95\%$ confidence interval be narrower or wider than 
the $99\%$ confidence interval? 
\end{enumerate}
     

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