A population has a standard deviation of 16. If a sample of size 64 is selected from this population, what is the probability that the sample mean will be within ±2 of the population mean?
A) 0.6826
B) 0.3413
C) -0.6826
D) Since the mean is not given, there is no answer to this question.

Question

Quizplus · Accepted Answer

This problem requires the use of the central limit theorem, which states that as sample size increases, the distribution of sample means approaches a normal distribution with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size. In this case, the standard deviation of the population is 16, so the standard deviation of the sample mean is 16/sqrt(64) = 2. Therefore, we want to find the probability that a random variable with a normal distribution with mean equal to the population mean and standard deviation equal to 2 is within ±2 of the population mean. Using a standard normal distribution table or calculator, we find that this probability is approximately 0.6826, which is choice A.

A Population Has a Standard Deviation of 16