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\hskip 1.5in \textbf{Midterm 2B--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 large southern city, 40\% of the 
people have Type O blood. 
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(a) Out of 20 randomly selected donors from this city, 
what is  the probability that at least 5 have Type 0 blood?
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(b) Out of 100 randomly selected donors from this city, 
approximately what is the probability that at most 43 have Type O
blood? 
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 2. 
From a random sample of 85 third grade students in a large school
district, the mean and standard deviation of reading comprehension scores
of the California Test of Basic Skills (CTBS) are found to be 71 and 27
respectively. 
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(a) (15 points)
 Based on this sample, construct a 95\% confidence interval
for the mean reading comprehension score for all third grade students in
the school district. (Do not do any arithmetic or calculator work: just 
plug the appropriate numbers into the appropriate formula.) 
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(b) (5 points)  Would a 99\% confidence interval be narrower or
wider?
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3.  
A company makes an average profit of 
\$320 per customer, with a standard deviation of \$50  
dollars. Let 
$\overline X$ be the average profit made on 
 a random sample of 10,000 customers.
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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 
\$319 and \$320 .
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d) Briefly justify your answer to (c). 
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4.
 A sample survey of the kindergarten children in a large city 
is being planned, to estimate
the mean number of older siblings a kindergarten child has.
  It is desired to estimate this mean 
so that with 
a 95\% level of confidence, the estimate is within 
 0.1 of the actual mean.  
  A reasonable planning value for the 
population standard deviation $\sigma$
 is 0.7. 

  What sample size is
needed for this estimate of the mean number of older siblings?
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5. Answer True or False for each part. 
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(a) $z_{.05}=1.96$ .
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(b) If a random variable $X$ has a normal distribution, then 
the random variable $2-5X$ also has a normal distribution. 
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(c) Let $\mu$ denote the average number of cars owned by a household 
in College Park. Suppose that a researcher takes a survey and from 
the survey data computes $(1.2, 1.6)$ as 
 a 95\% confidence interval for 
the population mean $\mu$. Then  with probability equal 
to .95, it is true that $1.2<\mu <1.6$. 
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(d) 
The sample average is a consistent estimator of the population 
mean, by the Law of Large Numbers: 
for sufficiently large sample size, the sample average 
is very close to the population mean, with probability close to 1. 

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