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\hskip 1.5in \textbf{Midterm 2B--Stat 100--Spring 1997}   
\vskip .2in  
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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\renewcommand{\labelenumi}{\alph{enumi}.}
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1. Let $Z$ be a standard normal random variable. 


\begin{enumerate}
\item 
(6 points) \ 
Find $P[-0.9 < Z < -0.5]$
\item
(6 points) \ 
Find $P[Z < -2.01]$ and 
$P[Z>-2.01]$ 
\item (8 points) \ 
Determine the value of $a$ so that $P[-a<Z<a]=0.668$
\end{enumerate}

\vspace{20pt}

2. Suppose we toss a fair coin 100 times. Let $X$ be the
number of times we observe heads. 

\begin{enumerate}
\item (4 points) 
What is the exact distribution of $X$? 
(Give a brief, complete description.) 
\item (12 points) 
Use an appropriate 
approximation to estimate $P[45 < X < 60]$. 
\item 
(4 points) 
Justify your use of this approximation. 
\end{enumerate}



\vspace{20pt}

3. Suppose a random variable $X$ has mean  20 and 
standard deviation  5.  Consider the distribution of 
$\overline  X$ for random samples of size 10. 
\begin{enumerate} 
\item 
(8 points) 
What are the mean and standard deviation of $\overline  X$?  
\item 
(6 points) 
Either explain why $\overline  X$ has approximately normal 
distribution, or give an assumption about the distribution 
of $X$ which implies $\overline  X$ has a normal distribution.
\item 
(6 points)
On the basis of your explanation or assumption above, 
give an estimate for 
 P[$\overline X > 22]$.  
\end{enumerate} 

\vspace{20pt}

4.  The U.S. Transportation Department will randomly 
sample traffic reports to estimate the
proportion of accidents involving people over the age of
70.  The Department has no advance estimate of this proportion. 

\hskip .3in  How many reports
should the department select to be at least $97\%$ confident that the
estimate is within 0.01 of the true proportion?
	 
\vspace{20pt}

     
5.  Cholesterol levels are tested for a random sample of 
 300 women over the age of 30. The 
sum of all of the observed levels for the sample group is 52,800.  
The sample
standard deviation of the observations is 15.  

\hskip .3in Construct a $90\%$
confidence interval for the actual average cholesterol level 
of women over the
age of 30.     

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