Five years ago, an automobile manufacturer started offering an extended warranty to buyers of its sport-utility vehicle.  The extended warranty covered defects occurring after the initial three-year warranty expired.  Of the 10,000 people who bought the sport-utility vehicle in the first year of the program, 15% purchased the extended warranty.  In the Warranty Department, you have recently received data on a random sample of 200 of the cars sold in the first year that the extended warranty was available. For this sample, the average extended-warranty expenditure per car for the one-year period after the initial warranty elapsed was $350., with a standard deviation of $100.
a.  What is a 95 percent confidence interval for the mean one-year extended-warranty expenditure per automobile?
b.  At its introduction, the extended warranty was priced at $224 per year per automobile.  Compute a 95 percent confidence interval for the one-year profitability of the extended warranty.
c.  How large a sample would the Warranty Department require if it wanted its 95 percent confidence interval for the mean warranty expenditure to be no more than+_$5
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To answer these questions, we will use the formula for calculating confidence intervals:
Confidence Interval = Sample Mean ± (Critical Value * Standard Error)
First, we need to calculate the standard error:
Standard Error = Standard Deviation / sqrt(n)
Where n is the sample size.
a. Calculating the confidence interval for the mean one-year extended-warranty expenditure per automobile:
The sample mean is $350, and the standard deviation is $100. The sample size, from the given data, is 200.
Standard Error = 100 / sqrt(200) = 100 / 14.14 = 7.07 (approximately)
To find the critical value, we need to determine the z-score associated with a 95% confidence level. Looking up the z-score table, a 95% confidence level corresponds to a z-score of approximately 1.96.
Confidence Interval = 350 ± (1.96 * 7.07)
= 350 ± 13.85
= ($336.15, $363.85)
Therefore, the 95% confidence interval for the mean one-year extended-warranty expenditure per automobile is ($336.15, $363.85).
b. Calculating the confidence interval for the one-year profitability of the extended warranty:
The introduced price of the extended warranty is $224.
Standard Error remains the same as calculated before: 7.07 (approximately)
Confidence Interval = 224 ± (1.96 * 7.07)
= 224 ± 13.85
= ($210.15, $237.85)
Therefore, the 95% confidence interval for the one-year profitability of the extended warranty is ($210.15, $237.85).
c. Calculating the required sample size to achieve a 95% confidence interval no more than +_$5:
To calculate the required sample size, we need to use the formula:
Sample Size = (Z-score * Standard Deviation / Margin of Error)^2
Here, the Z-score is the critical value for a 95% confidence level which is approximately 1.96. The Margin of Error is $5, and the Standard Deviation is $100.
Sample Size = (1.96 * 100 / 5)^2
= (196 * 20)^2
= 3920^2
= 15,364,400
Therefore, the Warranty Department would require a sample size of approximately 15,364,400 to achieve a 95% confidence interval for the mean warranty expenditure to be no more than $5.