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\hskip 1.5in \textbf{Midterm 2A--Stat 100--Spring 2001}   
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You may use a calculator and one page of notes. 
Each problem is worth 20 points.
 Different parts of a problem have equal weight unless 
otherwise indicated.  
 Put a box around the final answer to a question. 
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1. 
 In a certain city, 20\% of the students play sports. 
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(a) In a random sample of 20 students from this city, 
what is the probability at least 5 play sports? 
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(b) 
In a random sample of 200 students from this city, approximately  
what is the probability at most 45  play sports? 
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 2.
Assume that the standard deviation of the amount of copper 
precipitate from a chemical experiment is 5.0 grams. 
How many times should the experiment be repeated if one wants 
to be 99\% sure that the true mean amount of the precipitate 
is estimated within 1.5 grams? 
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3. 
A software company takes on average 150 days to complete a project, with a 
standard deviation of 20 days. 
For a random sample of 
100 projects, let 
 $\overline X$ denote the average time to complete the project. 

(a) What is the mean  of $\overline X$? 
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(b) What is the standard deviation  of $\overline X$? 
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(c) Estimate the probability that $\overline X$ lies between 145 and
155. \newline 
(d) Briefly justify your answer to (c). 
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4.
A refrigerator company made 3,612 service calls last year.  As part of
a profitability analysis, the company wishes to estimate the 
mean number $\mu$ of man-hours expended by company personnel per service call. 
A random sample of 125 service calls yielded $\overline x = 1.54$ 
man-hours and $s = 1.20$ man-hours, where $x$ denotes the man-hours expended
in a service call.
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(a) (15 points) Construct a 95\% confidence interval for $\mu$. 
(Do not do any arithmetic or calculator work: just plug the 
appropriate numbers into the appropriate formula.) 
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(b) (5 points) 
If you had constructed a 90\% confidence interval, would it
have been wider or narrower then the one in part (a)?

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5. For each part, answer TRUE or FALSE. 
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(a) It is reasonable to use the normal approximation for the 
Binomial(n,p) distribution when n=100 and p=.01 . 
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(b) The sample variance $S^2$ is an unbiased estimator of the 
population variance $\sigma^2$, because $E(S^2)=\sigma^2$. 
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(c) If $X$ is a random variable with mean 0 and standard
 deviation 1, then
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Prob($|X|< 1.96$) = .95 . 
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(d) When a 95\% confidence interval for $\mu$ is computed from sample data, 
the meaning is the following: if the same procedure is applied to many 
samples, then for about 95\% of the samples, the interval computed
will actually contain $\mu$.

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\begin{tabular}{|c||c|c|c|c|c|c|} \hline 
$x$ & 1 & 2 & 3 & 4 & 5 & 6  \\ \hline
$f(x)$ & 0.2 & 0.3 & 0.1 & 0.2 & 0.1 & 0.1 \\ \hline
\end{tabular}
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