Question

5 mark for review
let s represent the number of randomly selected adults in a community surveyed to find someone with a certain genetic trait.

the random variable s follows a geometric distribution with mean 4.66. which of the following is a correct interpretation of the mean?

a a value randomly selected from the distribution of s is expected to be 4.66.

b in repeated sampling from the distribution of s, the average of the values will approach 4.66.

c for a sample of values randomly selected from the distribution of s, the average of the sample will be 4.66.

d the probability is 0.66 that a value randomly selected from the distribution of s will be close to the mean.

e for a sample of values randomly selected from the distribution of s, the average of the sample will vary from the population mean by no more than 4.66.

Explanation:

To determine the correct interpretation of the mean of a geometric distribution:

• Option A: A single randomly selected value isn't "expected to be" the mean; the mean is about long - run averages, not individual values. So A is incorrect.
• Option B: By the law of large numbers, in repeated sampling (many trials), the average of the values of the random variable (here, \(S\)) will approach the population mean (4.66 in this case). This is a correct interpretation of the mean in the context of repeated sampling.
• Option C: A single sample's average is not guaranteed to be equal to the population mean. It is only in the long - run (with many samples) that the average approaches the mean. So C is incorrect.
• Option D: The mean has nothing to do with the probability that a value is close to the mean. This misinterprets the concept of the mean. So D is incorrect.
• Option E: There is no guarantee that the sample average will vary from the population mean by no more than the mean value. This is not a correct interpretation of the mean.

Answer:

B. In repeated sampling from the distribution of \(S\), the average of the values will approach 4.66.