Question 1200613: A consumer advocacy group is doing a large study on car rental practices. Among other things, the consumer group would like to estimate the mean monthly mileage,u, of cars rented in the U.S. over the past year. The consumer group plans to choose a random sample of monthly U.S. rental car mileages and then estimate u using the mean of the sample.
Using the value 750 miles per month as the standard deviation of monthly U.S. rental car mileages from the past year, what is the minimum sample size needed in order for the consumer group to be 95% confident that its estimate is within 150 miles per month of u?
Carry your intermediate computations to at least three decimal places. Write your answer as a whole number (and make sure that it is the minimum whole number that satisfies the requirements).
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! standard dviation is 750 miles.
standad error is equal to standad deviation divided by square root of sample size = 750 / sqrt(n).
critical z-score a 95% confidence interval is plus or minus 1.959963986.
z-score formula is z = (x-m) / s
z is the z-score
x is the raw score
m is the mean
plus or minus (x-m) is the margin of error, which you want to be no greater than 150 miles.
s is the standard error.
with the high critical z-score, you get 1.959963986 = 150 / (750 / sqrt(n))
multiply both sides of the equation by (750 / sqrt(n)) to get:\
1.959963986 * 750 / sqrt(n) = 150.
multiply both sides of this equation by sqrt(n) and divide both sides of this equation by 150 to get:
1.959963986 * 750 / 150 = sqrt(n).
solve for sqrt(n) to get sqrt(n) = 9.79981993.
solve for n to get n = 96.03647866.
when n is equal to or greater than that, your margin of error should be equal to or less than 150.
to test this out, assume n equals that.
your z-score formula becomes 1.959963986 = (x-m) / s
your standard error will be 750 / sqrt(96.03647866) = 76.53201848.
your critical z-score formula on the high side will be:
1.959963986 = (x-m) / 76.53201848.
solve for (x-m) to get (x-m) = 1.959963986 * 76.53201848 = 150.
your margin of error will be plus or minus 150.
since the sample size can't be a fraction, then your sample size will have to be greater than or equal to 97.
your standard error now becomes 750 / sqrt(97) = 76.15096239.
z-score formula of z = (x-m) / s now becomes 1.959963986 = (x-m) / 76.15096239.
solve for (x-m) to get (x-m) = 1.959963986 * 76.15096239 = 149.2531438.
with a sample size of 97 or greater, the margin of error will be within 150 miles.
your mean can be anything, as long as the standard deviation is equal to 750 and the sample size is greater than or equal to 97.
when the sample size is equal to 97, the standard error will be equal to 76.15096239.
here's what it looks like on a graph with a mean of 1000. and a sample size of 97.
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