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									Practice Problems for Confidence Intervals

X ± Z σ/n ; X ± t s/n ; p ± Z (p(1-p)/n)

1. A press release issued by our university claims that West Chester students study at least as
much as the national average for students at four year universities. Across the nation, 73 percent
of all students at four year universities study at least four hours per week. Seventy percent of one
hundred randomly selected West Chester students surveyed claimed to study more than four
hours per week. Should the University retract its previous statement? Explain why or why not.

2. A car manufacturer is considering switching to a new brand of brake shoes. To test the shoes,
the manufacturer submits the shoes to a heat test. The higher the failure temperature, the better
the shoe. Past experience has shown that heat failure temperature is normally distributed. A
sample of five shoes failed at the following temperatures: 300, 310, 280, 350 and 260F. If the
old shoes, which are considered reliable fail at an average temperature of 290F, should the
company make the switch? Explain your answer.

3. A government researcher is trying to determine whether or not a new car model averages at
least 30 miles per gallon in the city. If it does, the government will give the manufacturer a $100
tax break per car. From previous studies, the standard deviation of mpg is known to be 5 mpg. If
the average mpg of the new model in a sample of 36 is 29mpg, should the government award the
tax break? Explain why or why not.

4. A company is thinking about offering a lifetime guarantee for its mufflers. The company
figures that so long as its customers own their cars on average for less than ten years the offer
will be profitable. (The warranty would only apply to the original customer.) The following
figures were collected from a random sample of past customers:

years until sale: 2, 3, 6, 9, 7, 20, 2

Assuming the company is extremely conservative when it comes to trying new ideas, what
should the company do and why? Give the CEO a suggestion for improving the study, and
explain why your suggestion will help.

5. In a simple random sample of 36 West Chester adults, 20 favored some type of tax reform.
   Assuming there is little importance to the decision, is it reasonable to assume that the
   majority of all West Chester Adults is in favor of tax reform? Explain why or why not.

6.    A company in the U.S. is considering investing in a new small company in another country.
     One executive is concerned about bankruptcy rates in that country. Unfortunately, the
     country does not keep accurate bankruptcy rates. However, companies that exist are
     required to register with the government each year. A person with some time can follow the
     companies until they disappear from the lists. Usually this means they went bankrupt. The
     company does not have time to do this for every firm that ever existed, so they try a sample.
     Of 36 randomly selected firms they followed through the lists, 4 were bankrupt within 10
     years. In the U.S. approximately 8 percent of all companies go bankrupt within 10 years.
   Does this indicate that the risk of bankruptcy is higher than it is in the U.S.? Explain your

7. Across the nation it takes an average of 4 minutes to process a customer at a drive through fast
food window. In addition, drive through times tend to be normally distributed. At a local
McDunalds the manager thinks his employees are not operating up to national standards.
Because he doesn’t have time to check the speed of every clerk for every customer, it’s difficult
to tell for sure. Based on a suggestion from a colleague, he decides to calculate a confidence
interval. From behind the one way mirror in the booth above the clerks, he times several sales
with the following results:
Customer                1      2       3       4       5       6       7       8
Time (in minutes) 6.00 5.50 8.50 7.00 3.00 7.00 5.00 6.00
Assuming the manager uses a low confidence level, what is the manager likely to conclude after
his study? Based on the same information, do you think his employees are not up to national
standards? Explain your answer.

8. A local company searching for ways to save more money is considering suggesting that its
employees use a “Dial around“ service like “10-10-321, 10-10-220 or Lucky dog to cut down on
the company’s phone bills. The rules and charges for these services vary and are often
complicated, so depending on the length of the call or the time of day, the service may or may not
save money. Nationally the variation in the amount saved per call is around 15 cents. As a test,
the company makes 60 calls with one of the services and finds the average savings on those calls
was 25 cents. While the initial result is encouraging, can we be reasonably sure the service will
save us money? Explain why or why not

9. "More Americans Refuse to Pay Taxes" read the headline in the April 16th edition of the
Philadelphia Examiner. Last year, 5.2 percent of Americans failed to file an income tax form.
This year in a survey of 67 taxpayers, 6 percent had failed to file. Was the Examiner's headline
accurate or misleading? Explain your reasoning

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