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Math 251, 19 November 2002, Exam III
Name:
.
Instructions. Do each of the following 6 problems. Please justify your answers by
showing all appropriate work. Thank you and good luck!
1. The distribution of heights of 11-year-old girls is normally distributed with a mean of
62 inches and a standard deviation of 2.5 inches.
(a) (2 pts) What height has a percentile rank of 70?
(b) (2 pts) What proportion of 11 year-old-girls are between 60 and 65 inches tall?
(c) (3 pts) In a group of 800 randomly selected 11-year-old girls, how many would you
expect find that are (i) less than 60 inches tall; and (ii) more than 65 inches tall?
2. A farm in Idaho claims that the mean weight of a box of potatoes that they ship is 40
lbs. The quality control person on this farm collected a random sample of 60 boxes and
found a mean weight of 40.2 lbs with a standard deviation of .7 lbs.
(a) (3 pts) If the true mean is exactly 40 lbs, what is the probability of the quality control
person finding a random sample of 60 boxes with a mean weight of 40.2 lbs or more?
(b) (2 pts) If you are the quality control person, would you be satisfied that mean weight
of the boxes you ship is at least 40lbs? Explain.
3. Alaska Airlines has found that 94% of people with tickets will show up for their Friday
afternoon flight from Seattle to Ontario. Suppose that there are 128 passengers holding
tickets for this flight, and the jet can carry 120 passengers, and that the decisions of
passengers to show up are independent of one another.
(a) (2 pts) Verify that the normal approximation of the binomial distribution can be used
for this problem.
(b) (4 pts) What is the probability that more than 120 passengers will show up for the
flight (i.e., not everyone with a ticket will get a seat on the flight)?
4. A recent survey of 65 randomly selected gas stations in California found that the mean
price for unleaded gasoline in the sample was $1.47 per gallon with a sample standard
deviation of $.07.
(a) (2 pts) Why is the central limit theorem important in constructing confidence
intervals from large samples?
(b) (4 pts) Find a 96% confidence interval for the mean price of unleaded gasoline in
California.
(c) (3 pts) Assuming the population standard deviation is .07, what sample size would be
needed to have a margin of error E=.01 with 96% confidence?
5. A recent (November 11-14, 2002) Gallup poll of 1001 adult Americans found that 14%
said fear of war as the most important issue facing America.
(a) (2 pts) What conditions are necessary in order to construct a confidence interval for a
population proportion?
(b) (4 pts) Find a 92% confidence interval for the proportion of adult Americans that
believe fear of war is the most important issue facing America.
(c) (2 pts) Based on the answer to (b), would it be appropriate for the Gallup organization
to state that less than 15% of adult Americans believe fear of war is the most important
issue facing America? Explain.
6. (a) (3 pts) What size of sample is needed by the Gallup organization to estimate a
population proportion within .03 with 99% confidence? In your calculation assume the
worst case scenario that p = .5 and q = .5.
(b) (2 pts) What value should be used for tc when constructing a 95% confidence interval
for the mean of an approximately normal population when using a sample size of 17?
Assume the population standard deviation is not known.
(c) (3 pts) What values of L2 and R2 should be used when constructing a 99%
confidence interval for variance from a normal population given a sample size of 14?