Question

The manufacturer of the ColorSmart-5000 television set claims that 95 percent of its television sets last at least five years without needing a single repair. In order to test this claim, a consumer group randomly selects 400 consumers who have owned a ColourSmart-5000 television set for five years. Of these 400 consumers, 316 say that their ColourSmart-5000 television sets did not need repair, while 84 say that their ColourSmart-5000 television sets did need at least one repair. Conduct an appropriate test of hypothesis at a 5% level of confidence to test the validity of the manufacturer's claim.

Answer 1

Answer

Given claim is that 95 percent of its television sets last at least five years without needing a single repair.

we have to check whether the percent is actually 95% or not. So, it is a two tailed hypothesis test.

$H_o: p = 0.95 \\ H_1: p \ne 0.95$

we have population proportion po = 95/100 = 0.95

sample proportion p(hat) = x/n = 316/400 = 0.79

sample size n = 400

test statistic z =

$(\hat{p}-p_o)/\sqrt{((1-p_o)*p_o)/n} \\ = (0.79-0.95)/\sqrt{((1-0.95)*0.95)/400} \\ = -0.16/0.10897 \\ = -14.68$

using z table with -14.68, we get the following p value

p value = 0.0000

it is clear that the p value is less than significance level of 0.05, so we can reject the null hypothesis.

Therefore, there is insufficient evidence to support the validity of manufacturer's claim.