Solve the problem.
-A final exam in Math 160 has a mean of 73 with standard deviation 7.8. If 24 students are randomly selected, find the probability that the mean of their test scores is greater than 71.
A) 0.8962
B) 0.5036
C) 0.9012
D) 0.0008

Question

Quizplus · Accepted Answer

Since the sample size is greater than or equal to 30, we can use the Central Limit Theorem and assume that the sampling distribution of sample means will be approximately normal. The mean of the sampling distribution of sample means will be equal to the population mean (73) and the standard deviation of the sampling distribution of sample means (also called the standard error) will be equal to the population standard deviation divided by the square root of the sample size, which, in this case, is 7.8 / sqrt(24) = 1.6.

To find the probability that the mean of 24 students' test scores is greater than 71, we need to standardize the sample mean using the standard error and then find the area to the right of the resulting z-score.

z = (71 - 73) / 1.6 = -1.25

Using a standard normal distribution table or calculator, we find that the area to the right of z = -1.25 is 0.8962.

Therefore, the probability that the mean of 24 students' test scores is greater than 71 is 0.8962, or approximately 89.62%. The best choice is A.

Solve the Problem. -A Final Exam in Math 160 Has a Mean of a Mean