Question

If random samples of size n = 36 are drawn from a nonnormal population with finite mean = 75 and standard deviation = 15, then the sampling distribution of the sample mean is approximately normally distributed with mean = 75 and standard deviation = 2.5.

a. False
b. True

Answer 1

This statement is false.

The Central Limit Theorem (CLT) states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the shape of the population distribution, provided the sample size is sufficiently large.

For a non-normal population, the distribution of the sample mean will tend to be approximately normally distributed when the sample size is large enough (typically n > 30) due to the CLT. However, the standard deviation of the sampling distribution of the sample mean is given by the population standard deviation divided by the square root of the sample size (σ / √n).

For this case with a population standard deviation of 15 and a sample size of 36 (n = 36), the standard deviation of the sampling distribution of the sample mean would be σ / √n = 15 / √36 = 15 / 6 = 2.5, not 2.5 as stated in the statement.

Therefore, the correct standard deviation for the sampling distribution of the sample mean from a population with a standard deviation of 15 and a sample size of 36 would be 2.5, but the mean of this sampling distribution would remain the same as the population mean, which is 75.