Question

Assume that Fancy Shoe Warehouse has 500 pairs of women's shoes in stock. The distribution of the women's shoe sizes is approximately normal. The average size of the women's shoes is size 8, with a standard deviation of size 4. Suppose that a random sample of 250 pairs of shoes are selected. What can be assumed about the distribution of the sample mean?
The distribution of the sample mean is skewed by the mean distribution theorem.

The distribution of the sample mean is approximately normal by the mean distribution theorem.

There is not enough information to make assumptions regarding the distribution of the sample mean.

The distribution of the sample mean is approximately normal by the central limit theorem.

The distribution of the sample mean is non-normal by the central limit theorem.

Answer 1

It can be assumed that the sample mean would would be approximately normal by the mean distribution theorem. The mean distribution theorem states that a large enough sample size will have a distribution and mean approximately the same as the population mean. A sample of 250, half the size of 500, is of sufficient size to assume the distribution mean will be approximately normal like the population mean.

Answer 2

The distribution of the sample mean is approximately normal by the central limit theorem.