Question

The average annual cost of automobile insurance is $795 (National Association of Insurance Commissioners, 2007). Use this value as the population mean and assume that the population standard deviation is σ=$100. Consider a sampling distribution with sample size of 50 automobile insurance policies.

a. What is the standard error of the mean for the distribution of sample means of insurance costs?

b. What is the probability that the sample mean is within $50 (between $745 and $845) of the population mean?

c. Find the sample mean insurance cost in which 60% of the sample mean insurance costs are greater than it.

d. Consider that you selected a random sample of 40 auto insurance policies and find that the sample mean is $780. Assume the population standard deviation is the same as above. Calculate a 90% confidence interval for the true population mean cost of automobile insurance. Does the true population mean from above fall within this interval?

Answer 1

Solution:

Given:

Population mean of annual cost of automobile insurance = = $ 795
Population standard deviation = = $ 100
Sample Size = n = 50

Part a) What is the standard error of the mean for the distribution of sample means of insurance costs?

Part b) What is the probability that the sample mean is within $50 (between $745 and $845) of the population mean?

Look in z table for z = 3.5 and 0.04 as well as for z = -3.5 and 0.04 and find area.

P( Z< 3.54) = 0.99980

and

P( Z< -3.54) = 0.00020

Thus

Part c. Find the sample mean insurance cost in which 60% of the sample mean insurance costs are greater than it.

Thus find z such that:

P( Z> z) = 0.60

That is find:

P( Z< z) = 1 - P( Z > z )

P( Z< z) = 1 - 0.60

P( Z< z) = 0.40

Look in z table for Area = 0.4000 or its closest area and find z value.

Area 0.4013 is closest to 0.4000 and it corresponds to -0.2 and 0.05

Thus z = -0.25

Now use following formula to find sample mean:

Thus 60% of the sample mean insurance costs are greater than $791.46.

Part d) Calculate a 90% confidence interval for the true population mean cost of automobile insurance.

Sample size = n = 40

Population standard deviation = = $ 100

Formula:

where

Sample mean =

Z_c is z critical value for c = 90% confidence level.

Find Area = ( 1 + c ) / 2 = ( 1 + 0.90) / 2 = 1.90 / 2 = 0.9500

Look in z table for Area = 0.9500 or its closest area and find corresponding z value.

Area 0.9500 is in between 0.9495 and 0.9505 and both the area are at same distance from 0.9500

Thus we look for both area and find both z values

Thus Area 0.9495 corresponds to 1.64 and 0.9505 corresponds to 1.65

Thus average of both z values is : ( 1.64+1.65) / 2 = 1.645

Thus Z_c = 1.645

Thus

Thus

Does the true population mean from above fall within this interval?

Yes Population mean = = $ 795 fall within this interval.