Consumer Reports (January 2005) indicates that profit margins on extended warran
ID: 3132295 • Letter: C
Question
Consumer Reports (January 2005) indicates that profit margins on extended warranties are much greater than on the purchase of most products. In this exercise we consider a major electronics retailer that wishes to increase the proportion of customers who buy extended warranties on digital cameras. Historically, 20 percent of digital camera customers have purchased the retailer’s extended warranty. To increase this percentage, the retailer has decided to offer a new warranty that is less expensive and more comprehensive. Suppose that three months after starting to offer the new warranty, a random sample of 527 customer sales invoices shows that 132 out of 527 digital camera customers purchased the new warranty. Find a 95 percent confidence interval for the proportion of all digital camera customers who have purchased the new warranty. Are we 95 percent confident that this proportion exceeds .20? (Round your answers to 3 decimal places.)
The 95 percent confidence interval is [, ]. (Click to select)NoYes , the entire interval (Click to select)is notis above .20.Explanation / Answer
Note that
p^ = point estimate of the population proportion = x / n = 0.250474383
Also, we get the standard error of p, sp:
sp = sqrt[p^ (1 - p^) / n] = 0.018874245
Now, for the critical z,
alpha/2 = 0.025
Thus, z(alpha/2) = 1.959963985
Thus,
Margin of error = z(alpha/2)*sp = 0.036992841
lower bound = p^ - z(alpha/2) * sp = 0.213481543
upper bound = p^ + z(alpha/2) * sp = 0.287467224
Thus, the confidence interval is
( 0.213481543 , 0.287467224 ) [ANSWER]
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As this interval is entirely above 0.20, then
YES, the entire interval IS above 20. [ANSWER]
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