Question

# The distribution of height for a certain population of women is approximately normal with mean 65 inches and standard deviation 3.5 inches. Consider t

Sampling distributions
The distribution of height for a certain population of women is approximately normal with mean 65 inches and standard deviation 3.5 inches. Consider two different random samples taken from the population, one of size 5 and one of size 85.
Which of the following is true about the sampling distributions of the sample mean for the two sample sizes?
Both distributions are approximately normal with mean 65 and standard deviation 3.5.
A
Both distributions are approximately normal. The mean and standard deviation for size 5 are both less than the mean and standard deviation for size 85.
B
Both distributions are approximately normal with the same mean. The standard deviation for size 5 is greater than that for size 85.
C
Only the distribution for size 85 is approximately normal. Both distributions have mean 65 and standard deviation 3.5.
D
Only the distribution for size 85 is approximately normal. The mean and standard deviation for size 5 are both less than the mean and standard deviation for size 85.
E

2021-02-20
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”.