The probability that the sample mean lies between 498 and 502 is closest to
A).3174
B).9772
C).9544
D).3108
E).6826

Question

Quizplus · Accepted Answer

This problem involves the normal distribution and the central limit theorem. We are given the population mean and population standard deviation, but we do not know the sample size. However, we can still use the properties of the normal distribution for the sample mean.

To find the probability that the sample mean lies between 498 and 502, we need to standardize this range by subtracting the population mean and dividing by the standard deviation of the sample mean, which is given by the population standard deviation divided by the square root of the sample size. Since we do not know the sample size, we cannot directly calculate the standard deviation of the sample mean, but we can use a normal approximation.

Based on the central limit theorem, the distribution of the sample mean approaches a normal distribution with mean equal to the population mean and standard deviation equal to the population standard deviation divided by the square root of the sample size, as the sample size gets larger. Since the sample size is unknown, we can assume that it is large enough to use this approximation.

Thus, we can standardize the range of 498 to 502 by subtracting the population mean of 500 and dividing by the standard deviation of 5 (i.e., the population standard deviation of 10 divided by the square root of the sample size, which we assume is large enough to use the normal approximation). This gives us a z-score range of -0.4 to 0.4.

Using a standard normal table or calculator, we can find the probability that a standard normal random variable falls between -0.4 and 0.4. This probability is approximately 0.6826 or 68.26%.

Therefore, the best choice is C because it gives the closest probability of 68.26% to the correct answer.

The Probability That the Sample Mean Lies Between 498 and 502