Examice

Periodically, a sample of 20 items are randomly selected from a population that is normally distributed, the average is computed, and plotted. Which of the following statements is correct?

I. The average of the logarithms of the values is lognormally distributed.

II. The standard deviation of the averages is equal to the standard deviation of the individuals divided by the square root of 20.

III. The variance of the averages is equal to the variance of the individuals divided by the square root of 20.

Correct Answer:

When dealing with a population that is normally distributed and selecting samples of size 20, the averages of the samples follow the central limit theorem, which states that the distribution of the sample means will be approximately normal. The standard deviation of the sample means (also known as the standard error) is equal to the standard deviation of the individuals divided by the square root of the sample size, which is 20 in this case. This makes statement II correct. Statement I is incorrect because while logarithms of a normal distribution result in a lognormal distribution, this doesn't imply that the average of the logarithms of the values will be lognormally distributed. Statement III is incorrect because the variance of the averages is equal to the variance of the individuals divided by the sample size (not the square root of the sample size). Thus, among the given statements, only statement II is correct.