Question

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Explanation:

To determine the best interpretation of the mean of the binomial random variable \( B \) (representing the number of people without the characteristic in a sample of 45), we analyze each option:

• Option A: The mean of a binomial distribution represents the expected value (average) over many trials (groups of 45 people). So, over all groups of 45, the average number without the characteristic is 2.7. This matches the definition of the mean.
• Option B: The mean refers to people without the characteristic, and it is an average, not a guarantee for every group. Also, it incorrectly refers to people with the characteristic.
• Option C: The mean is an average, not a fixed number for every group. It is impossible to have 2.7 people (a fraction) in a single group.
• Option D: The problem involves a binomial distribution (fixed number of trials, \( n = 45 \)), not a geometric distribution (waiting for a success). This option describes a geometric context.
• Option E: Similarly, this describes a geometric distribution (waiting for a "success" of finding someone without the characteristic), but the problem uses a binomial distribution (counting successes in 45 trials).

Answer:

A. For all groups of 45 people, the average number of people without the characteristic is 2.7.