Step 1

It is given that the population of women is approximately normal with mean 65 inches and standard deviation of 3.5 inches. In the scenario, it is assumed that two different random samples were taken from the population, one of size 5 and one of size 85.

When the sample size is large, the sampling distribution of the sample mean is close to the normal distribution. The standard deviation will decrease with an increase in sample size. However, the mean will remain the same as the population means.

Step 2

The random sample of size 5 is small and may have a mean 65 inches with different standard deviation. In the same way, the random sample of size 85 is large and may have the mean 65 inches with a standard deviation 3.5

Therefore, the true information about the sampling distributions of the sample means for the two sample sizes is, “Both distributions are approximately normal with the same mean. The standard deviation for size 5 is greater than that for size 85.”

Therefore, the correct answer is “Option C”.

It is given that the population of women is approximately normal with mean 65 inches and standard deviation of 3.5 inches. In the scenario, it is assumed that two different random samples were taken from the population, one of size 5 and one of size 85.

When the sample size is large, the sampling distribution of the sample mean is close to the normal distribution. The standard deviation will decrease with an increase in sample size. However, the mean will remain the same as the population means.

Step 2

The random sample of size 5 is small and may have a mean 65 inches with different standard deviation. In the same way, the random sample of size 85 is large and may have the mean 65 inches with a standard deviation 3.5

Therefore, the true information about the sampling distributions of the sample means for the two sample sizes is, “Both distributions are approximately normal with the same mean. The standard deviation for size 5 is greater than that for size 85.”

Therefore, the correct answer is “Option C”.