The manufacturer of the ColorSmart-5000 television set claims 95 percent of its
ID: 3382980 • Letter: T
Question
The manufacturer of the ColorSmart-5000 television set claims 95 percent of its sets last at least five years without needing a single repair. In order to test this claim, a consumer group randomly selects 400 consumers who have owned a ColorSmart-5000 television set for five years. Of these 400 consumers, 316 say their ColorSmart-5000 television sets did not need a repair, whereas 84 say their ColorSmart-5000 television sets did need at least one repair. Find a 99 percent confidence interval for the proportion of all ColorSmart-5000 television sets that have lasted at least five years without needing a single repair, Does this confidence interval provide strong evidence that the percentage of all ColorSmart-5000 television sets that last at least five years without a single repair is less than the 95 percent claimed by the manufacturer? Explain.Explanation / Answer
a)
proportion = p' = 308/403 = 0.764
z for 99% CI = 2.575
so,
lower limit of CI = p' - z*sqrt(p'*(1-p')/n) = 0.764-2.575*sqrt(0.764*(1-0.764)/403) = 0.710
Upper limit of CI = p' + z*sqrt(p'*(1-p')/n) = 0.764+2.575*sqrt(0.764*(1-0.764)/403) = 0.818
99% CI = (0.710,0.818)
b)
Yes.
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