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)8.3600

Question

Quizplus · Accepted Answer

We can use the central limit theorem to approximate the sampling distribution of the mean as normal with a mean of 84 and a standard deviation of 12/sqrt(36) = 2.

To find the probability that the sample mean is between 80.54 and 88.9, we standardize the values:

Zlower = (80.54 - 84) / 2 = -1.73

Zupper = (88.9 - 84) / 2 = 2.45

Using a standard normal distribution table or calculator, we can find the probability that Z is between -1.73 and 2.45:

P(-1.73 < Z < 2.45) = 0.9511

Therefore, the probability that the sample mean will be between 80.54 and 88.9 is 0.9511 or approximately 95.11%.

The other answer choices are not correct because they either do not take into account the standard deviation of the sampling distribution (choices A and D), or they do not correctly calculate the probability using the standard normal distribution (choice B).

A Population Has a Mean of 84 and a Standard