Question 1 A group of investors wants to develop a chain of fast-food restaurant

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Question 1 A group of investors wants to develop a chain of fast-food restaurants. In determining potential costs for each facility, they must consider, among other expenses, the average monthly electric bill. They decide to sample some fast-food restaurants currently operating to estimate the monthly cost of electricity. They want to be 90% con?dent of their results and want the error of the interval estimate to be no more than $100. They estimate that such bills range from $600 to $2,500. How large a sample should they take? Question 2 Suppose a study reports that the average price for a gallon of self-serve regular unleaded gasoline is $3.16. You believe that the ?gure is higher in your area of the country. You decide to test this claim for your part of the United States by randomly calling gasoline stations. Your random survey of 25 stations produces the following prices (all in $). Assume gasoline prices for a region are normally distributed. Do the data you obtained provide enough evidence to reject the claim? Use a 1% level of signi?cance. 3.27 3.29 3.16 3.20 3.37 3.20 3.23 3.19 3.20 3.24 3.16 3.07 3.27 3.09 3.35 3.15 3.23 3.14 3.05 3.35 3.21 3.14 3.14 3.07 3.10

Explanation / Answer

The margin of error for this confidence interval for the mean is 100 = z*SD/sqrt(n).

Solving for n: sqrt n = z*SD/1.645, now square both sides: n = [ z*SD/100 ]^2

The z score for a 90% confidence interval is 1.645.

The standard deviation is not given so an estimate must be used. If the bills range from 600 to 2500 then the range is 2500-600 = 1900. Now there is some decision to be made here as to whether to equate the range to 4*SD or 6*SD. Your textbook may provide guidance for which to choose.

These are based on what is sometimes called "the range rule of thumb" which is a consequence of the empirical rule and Chebychev's theorem.

Suppose you choose 1900 = 6*SD, then SD = 1900/6.

Then n = [ 1.645*(1900/6) / 100 ]^2 = 27.13.

So a sample size of 28 or more should do it. Using 1900/4 for the SD results in a sample size of 62.

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