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Probability: Insurance, Population, Medicare and TV Viewing

1. National Association of Insurance Commissioners
The average annual cost of automobile insurance is $687 (National Association of Insurance Commissioners, January 2003). Use this value as the population mean and assume that the population standard deviation is σ = $230. Consider a sample of 45 automobile insurance policies.

a. Show the sampling distribution of x where x is the sample mean annual cost of automobile insurance.
b. Determine the probability that the sample mean is within $100 of the population mean.
c. Determine the probability that the sample mean is within $25 of the population mean.
d. What would you recommend if an insurance agency wanted the sample mean to estimate the population mean within ± $25?

2. The Democrat and Chronicle
The Democrat and Chronicle reported that 25% of the flights arriving at the San Diego airport during the first five months of 2001 were late (Democrat and Chronicle, July 23, 2001). Assume the population proportion is p = .25

a. Show the sampling distribution of p, the proportion of late flights in a sample of 1000 flights.
b. Determine the probability that the sample proportion will be within ± .03 of the population proportion if a sample of size 1000 is selected.
c. Determine the probability that the sample proportion will be within ± .03 of the population proportion if a sample of size 500 is selected.

3. Medicare
Americans have become increasingly concerned about the rising cost of Medicare. In 1990, the average annual Medicare spending per enrollee was $3267; in 2003, the average annual Medicare spending per enrollee was $6883 (Money, Fall 2003). Suppose you hired a consulting firm to take a sample of fifty 2003 Medicare enrollees to further investigate the nature of expenditure. Assume the population standard deviation for 2003 was $2000.

a. Show the sampling distribution of the mean amount of Medicare spending for a sample of fifty 2003 enrollees.
b. Determine the probability the sample mean will be within ±$300 of the population mean.
c. Determine the probability the sample mean will be greater than $7500. If the consulting firm tells you the sample mean for the Medicare enrollees they interviewed was $7500, would you question whether they followed correct simple random sampling procedures? Why or why not?

4. Television Viewing
The mean television viewing time for Americans is 15 hours per week (Money, November 2003). Suppose a sample of 60 Americans is taken to further investigate viewing habits. Assume the population standard deviation for weekly viewing time is Σ= 4 hours.

a. Determine the probability the sample mean will be within 1 hour of the population mean.
b. Determine the probability the sample mean will be within 45 minutes of the population mean.

Solution Summary

This solution looks at different probability scenarios and provides guidelines on calculating the sampling distributions.

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