Question

Which of the following statements about the sampling distribution of the sample mean is incorrect?

a. The standard deviation of the sampling distribution will decrease as the sample size increases.

b. The standard deviation of the sampling distribution measures how far the sample mean typically varies from the population mean.

c. The sample mean is an unbiased estimator of the population mean.

d. The sampling distribution shows how the sample mean is distributed around the population mean.

e. The sampling distribution shows how the sample is distributed around the sample mean.

Short answer

(e). The sampling distribution is incorrect because it is the distribution of all possible sample means, which cannot be centred on the sample mean.

Step-by-step solution

Given Information

The choices are,

a. As the sample size grows, the sampling distribution's standard deviation decreases.

b. The sampling distribution's standard deviation indicates how far the sample mean normally differs from the population mean.

c. The sample mean is a reliable predictor of the population average.

d. The sampling distribution depicts the distribution of the sample mean around the population mean.

e. The sampling distribution depicts the distribution of the sample around the sample mean.

Explanation for correct option

a. Yes, because a bigger sample size provides more information about the population and so allows for more precise forecasts.

b. Correct, the variability of the sample mean over all potential samples is the standard deviation of the sampling distribution of the sample mean.

c. Yes, the sample mean is a reliable predictor of the population mean.

d. Correct, because the variation is determined by the sample distribution's standard deviation.

Because the sampling distribution represents the distribution of all potential sample means, it cannot be centred on the sample mean. e. Incorrect.

As a result, the best solution is (e)

Explanation for incorrect option

a. The standard deviation of the sampling distribution will decrease as the sample size increases is not the answer.

b. The standard deviation of the sampling distribution measures how far the sample mean typically varies from the population mean is not the answer.

c. The sample mean is an unbiased estimator of the population meanis not the answer.

d. The sampling distribution shows how the sample mean is distributed around the population mean is not the answer.