A particular TV manufacturer claims that its sets have such high reliability tha

Question

A particular TV manufacturer claims that its sets have such high reliability that 90% of these sets last at least five years without needing a single repair. In order to test this claim, a consumer group randomly selects 400 customers who have owned this particular set for five years. Of these 400 consumers, 320 say their TV did not need any repairs, whereas 80 said their TV needed at least one repair. (a) Find a 95% confidence interval for the proportion of this particular TV that last at least five years without needing a single repair. Interpret this interval. (b) Does the interval developed in part (a) above provide convincing evidence that the company's reliability claim is valid? Explain. (b) How many more samples should the consumer group take of these TV sets if they want to be 95% confident that their estimate of the true proportion of sets that last at least five years without a repair is within a margin of error of plusminus 2 percentage points the true proportion?

Explanation / Answer

p = 320/400 = 0.8

standard error = sqrt(0.8 * 0.2/400) = 0.02

a)

margin of error = 1.96 * 0.02 = 0.0392

lower bound = 0.8 - 0.0392 = 0.7608

upper bound = 0.8 + 0.0392 = 0.8392

proportion of tv that last without needing a single repair is between 0.7608 and 0.8392

b)

hence there is not enough evidence that company reliability claim is valid

c)

margin of error = 0.02

1.96 * sqrt(0.8 * 0.2/n) = 0.02

=>

n = 1536.64 = 1537

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