A population has a mean of 84 and a standard deviation of 12. A sample of 36 observations will be taken. The probability that the sample mean will be between 80.54 and 88.9 is
A)0.0347
B)0.7200
C)0.9511
D)None of the alternative answers is correct.

Question

Quizplus · Accepted Answer

We can use the central limit theorem to approximate the distribution of the sample mean as normal with mean = 84 and standard deviation = 12/√36 = 2. Thus, we can standardize the sample mean as follows:
Z = (X - μ) / σ = (80.54 - 84) / 2 + (88.9 - 84) / 2 = -1.73 + 2.45 = 0.72
Using a standard normal distribution table or a calculator, we can find the probability that Z is between -0.72 and 0.72. This probability is approximately 0.9511, so the answer is C.

A Population Has a Mean of 84 and a Standard