Question

6 mark for review let w represent the number of attempted experiments to get one experiment that is not successful. the random variable w has a geometric distribution with mean 4 and standard deviation 3.5. which of the following is the best interpretation of the standard deviation? a a single value randomly selected from the distribution of w will vary from 4 by 3.5 attempted experiments. b a single value randomly selected from the distribution of w will vary from 3.5 by 4 attempted experiments. c values of w typically vary from 4 by about 3.5 attempted experiments, on average. d values of w typically vary from 3.5 by about 4 attempted experiments, on average. e the standard deviation of a random sample of values selected from the distribution of w will be 3.5 attempted experiments.

Explanation:

To determine the correct interpretation of the standard deviation for the geometric distribution of \( W \):

1. Recall the definition of standard deviation: It measures the typical (average) amount of variation or dispersion from the mean.
2. Analyze the mean and standard deviation of \( W \): The mean is 4, and the standard deviation is 3.5.
3. Evaluate the options:

• Option A: Incorrect. Standard deviation does not describe the variation of a single value from the mean (it describes typical variation across the distribution).
• Option B: Incorrect. The mean is 4, not 3.5, and the logic is flawed.
• Option C: Correct. Standard deviation describes how values of \( W \) typically vary from the mean (4) by about 3.5, on average.
• Option D: Incorrect. The mean is 4, not 3.5, and the variation direction is reversed.
• Option E: Incorrect. The standard deviation of a sample is not guaranteed to equal the population standard deviation (3.5); it describes the population’s variation, not a sample’s.

Answer:

C. Values of \( W \) typically vary from 4 by about 3.5 attempted experiments, on average.