The manufacturer of the ColorSmart5000 television set claims

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.

Solution

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.