A sample of size 85 will be drawn from a population with mean 20 and standard deviation 8. Find the probability that $\bar { x }$ will be between 19 and 22.
A) 0.8256
B) 0.8642
C) 0.0107
D) 0.1251

Question

Quizplus · Accepted Answer

We use the central limit theorem to approximate the sampling distribution of the sample mean as normal with mean = population mean = 20 and standard deviation = population standard deviation / (sample size)^(1/2) = 8 / (85)^(1/2) ≈ 0.867. Then, we standardize the values of the sample mean:
z = (x - mean) / standard deviation = (19 - 20) / 0.867 ≈ -1.15 and z = (22 - 20) / 0.867 ≈ 2.31. Finally, we use the standard normal distribution table or calculator to find the probabilities:
P(-1.15 < z < 2.31) ≈ 0.8642
Therefore, the answer is B.

A Sample of Size 85 Will Be Drawn from a Population