Determine the Sample Size Necessary for Estimating the Population Mean within a Specified Margin of Error
-The principal at Riverside High School would like to estimate the mean length of time each day that it takes all the buses to arrive and unload the students. How large a sample is needed if the principal would like to assert with 90% confidence that the sample mean is off by, at most, 7 minutes. Assume = 14 minutes.
A) 11
B) 10
C) 12
D) 13

Question

Quizplus · Accepted Answer

The sample size needed is given by the formula:n = (Z^2 * s^2) / E^2where Z is the z-score for the desired confidence level, s is the sample standard deviation, and E is the margin of error.In this case, the confidence level is 90%, so the z-score is 1.645. The sample standard deviation is 14 minutes, and the margin of error is 7 minutes. Plugging these values into the formula, we get:n = (1.645^2 * 14^2) / 7^2= 11.06Since we cannot have a fraction of a sample, we round up to the nearest whole number, which is 11. Therefore, the principal needs a sample size of 11 to estimate the mean length of time each day that it takes all the buses to arrive and unload the students with 90% confidence that the sample mean is off by, at most, 7 minutes.