Question

The manufacturer of the ColorSmart-5000 television set claims that 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 399 consumers who have owned a ColorSmart-5000 television set for five years. Of these 399 consumers, 310 say that their ColorSmart-5000 television sets did not need repair, while 89 say that their ColorSmart-5000 television sets did need at least one repair.

Letting p be the proportion of ColorSmart-5000 television sets that last five years without a single repair, set up the null and alternative hypotheses that the consumer group should use to attempt to show that the manufacturer’s claim is false.
H0 :p .95
Ha : p .95.

Answer 1

The null hypothesis for the consumer group is that the proportion of TV sets that last without repair is 0.95, while the alternative hypothesis is that this proportion is less than 0.95. A hypothesis test would involve calculating a p-value to determine whether to reject the null hypothesis.

Step-by-step explanation:

The student is asking about setting up a hypothesis test for a proportion. The consumer group is challenging the manufacturer's claim that 95 percent of ColorSmart-5000 television sets last at least five years without needing a single repair. To conduct the hypothesis test, the null hypothesis (H0) should state that the proportion of sets that last five years without repair is equal to 0.95 (p = 0.95).

The alternative hypothesis (Ha) is that the proportion is less than 0.95 (p < 0.95) because the consumer group is attempting to show that the manufacturer's claim is false. In a hypothesis test, the p-value helps determine whether to reject the null hypothesis. If the p-value is less than the chosen significance level, the null hypothesis is rejected, indicating that there is sufficient evidence to support the alternative hypothesis.