The manufacturer of the ColorSmart-5000 television set claims 95 percent of its
ID: 397138 • 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. (a) 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 99 percent confidence interval is [ , ].
Explanation / Answer
Solution:
Given in the question
Population proportion = 0.95
Standard error = sqrt(0.95*0.05/403) = (0.0475/403) = sqrt(0.000117866) = 0.0108566
Here alpha = 0.01
so alpha/2 =0.005
and from z table we found Z0.005 = 2.575
So 99% confidence interval can be calculated as
Population proportion+/- Zalpha/2 * Standard error
0.95+/- 2.575*0.01085
0.95+0.02793 to 0.95-0.02793
so 99% confidence interval is [0.922,0.978]
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