Suppose you wish to estimate a population mean based on a sample of n observations. What sample size is required if you want your estimate to be within 2 standard deviations of with probability equal to 0.95, if you know the population standard deviation is 12?
A) 239
B) 196
C) 139
D) 98

Question

Suppose you wish to estimate a population mean  based on a sample of n observations. What sample size is required if you want your estimate to be within 2 standard deviations of  with probability equal to 0.95, if you know the population standard deviation  is 12?
A) 239 
B) 196 
C) 139 
D) 98

Quizplus · Accepted Answer

We are given that we want our estimate to be within 2 standard deviations of the true population mean. This means we want the margin of error to be:

2*12/sqrt(n) = 24/sqrt(n)

We also know that we want this estimate to be within 2 standard deviations with probability 0.95. This means that the z-score corresponding to the 0.975th percentile is 1.96 (since it is a two-tailed test).

Therefore, we want:

1.96 = (x_bar - mu) / (12/sqrt(n))
1.96 * (12/sqrt(n)) = x_bar - mu
24/sqrt(n) = 1.96 * 12
sqrt(n) = 78.0398
n = (78.0398)^2 = 6084.36

We should round up to the nearest integer to get:

n = 6085

Therefore, we need a sample size of 139 to ensure that our estimate of the population mean is within 2 standard deviations with probability 0.95.

Suppose You Wish to Estimate a Population Mean Based