Question

discuss whether you think the central limit theorem applies to each of the following situations. if it does apply, specify the population of interest and the measurement of interest. if it does not apply, explain why not.
(a) a university wants to know the average income of its alumni. staff members select a random sample of 200 alumni and mail them a questionnaire. they follow - up with a phone call to those who do not respond within 30 days.

• the theorem applies. the population is staff members and the measurement is income.
• the theorem applies. the population is alumni and the measurement is income.
• the theorem does not apply because the measurement may not be bell - shaped and the sample is not large enough.
• the theorem does not apply because the sample is not random.

(b) an automobile dealership wants to know the average price for which used cars of a particular model and year are selling in a certain state. they are able to obtain a list of buyers from the state motor vehicle division, from which they select a random sample of 20 buyers. they make every effort to find out what these people paid for the cars and use this information for analysis.

• the theorem applies. the population is used cars for sale and the measurement is number of miles on the car.
• the theorem applies. the population is used cars of a particular model and year and the measurement is price.
• the theorem does not apply because the measurement may not be bell - shaped and the sample is not large enough.
• the theorem does not apply because the sample is not random.

Explanation:

1. (a) The Central Limit Theorem applies. The population of interest is all alumni of the university. The measurement of interest is income. The sample is a random sample of 200 alumni and the sample size is reasonably large. Non - response bias from follow - up calls doesn't violate the core assumptions of the theorem regarding sample size and random sampling for the initial selection.
2. (b) The Central Limit Theorem does not apply. The sample is not random as it is a sample of buyers of a particular model and year of used cars. The theorem requires a random sample from the relevant population for its assumptions to hold.

Answer:

1. The theorem applies. The population is alumni and the measurement is income.
2. The theorem does not apply because the sample is not random.