The manufacturer of the ColorSmart-5000 television set claims 95 percent of its

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 366 consumers who have owned a ColorSmart-5000 television set for five years. Of these 366 consumers, 309 say their ColorSmart-5000 television sets did not need a repair, whereas 57 say their ColorSmart-5000 television sets did need at least one repair.

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 366 consumers who have owned a ColorSmart-5000 television set for five years. Of these 366 consumers, 309 say their ColorSmart-5000 television sets did not need a repair, whereas 57 say their ColorSmart-5000 television sets did need at least one repair.

Determine the sample size needed in order to be 99 percent confident that , the sample proportion of ColorSmart-5000 television sets that last at least five years without a single repair, is within a margin of error of .03 of p, the population proportion of sets that last at least five years without a single repair. (Round your p answer to 5 decimal places. Round your n answer to the next whole number.)

Explanation / Answer

Note that      
      
n = z(alpha/2)^2 p (1 - p) / E^2      
      
where      
      
alpha/2 =    0.005  
       
      
Using a table/technology,      
      
z(alpha/2) =    2.575  
      
Also,      
      
E =    0.03  
p = 309/366 =   0.844262295  
      
Thus,      
      
n =    968.6862197  
      
Rounding up,      
      
n =    969   [ANSWER]

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