The manufacturer of the ColorSmart-5000 television set claims 95 percent of its
ID: 3382371 • 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 403 consumers who have owned a ColorSmart-5000 television set for five years. Of these 403 consumers, 330 say their ColorSmart-5000 television sets did not need a repair, whereas 73 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. (Round your answers to 3 decimal places.)
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 403 consumers who have owned a ColorSmart-5000 television set for five years. Of these 403 consumers, 330 say their ColorSmart-5000 television sets did not need a repair, whereas 73 say their ColorSmart-5000 television sets did need at least one repair.
Explanation / Answer
a)
Note that
p^ = point estimate of the population proportion = x / n = 0.818858561
Also, we get the standard error of p, sp:
sp = sqrt[p^ (1 - p^) / n] = 0.019184957
Now, for the critical z,
alpha/2 = 0.005
Thus, z(alpha/2) = 2.575829304
Thus,
Margin of error = z(alpha/2)*sp = 0.049417175
lower bound = p^ - z(alpha/2) * sp = 0.769441386
upper bound = p^ + z(alpha/2) * sp = 0.868275735
Thus, the confidence interval is
( 0.769441386 , 0.868275735 ) [ANSWER]
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