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 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.)

Solution

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]