A population has a mean of 80 and a standard deviation of 7. A sample of 49 observations will be taken. The probability that the mean from that sample will be larger than 82 is
A)0.5228
B)0.9772
C)0.4772
D)0.0228

Question

Quizplus · Accepted Answer

To solve this problem, we can use the central limit theorem which states that as the sample size increases, the sampling distribution of the mean 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 sample size is 49, so the standard deviation of the sampling distribution of the mean is 7/sqrt(49) = 1. Therefore, we can standardize the sample mean as:

z = (x - mu) / (sigma / sqrt(n))
z = (82 - 80) / (7 / sqrt(49))
z = 2 / 1
z = 2

Next, we need to find the probability that a standard normal variable is greater than 2. We can use a standard normal table or calculator to find that the probability is approximately 0.0228.

Therefore, the answer is (D) 0.0228.

A Population Has a Mean of 80 and a Standard