Question

2. 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 400 consumers who have owned a ColorSmart-5000 television set for five years. Of these 400 consumers, 316 say that their ColorSmart-5000 television sets did not need repair, while 84 say that their ColorSmart-5000 television sets did need at least one repair. a. Formulate the null and alterative hypotheses that the consumer group should use to attempt to show that the manufacturer’s claim is false. b. Use the critical value approach to test the hypotheses by setting α equal to 0.10 and 0.001.

Answer 1

(a) Null Hypothesis: p = 0.95

Alternate Hypothesis: p ≠ 0.95

(b) We will run a two-tailed z test for a population proportion:

SE =

p = 316/400 = 0.79

z = (p' - p)/SE = (0.79 - 0.95)/0.0109 = -14.679

Critical values for α = 0.10 and 0.001 are respectively ±1.645 and ±3.291

As the z-statistic < z-critical for both of the confidence levels, we will reject the null hypothesis and conclude that the proportion is not equal to 0.95.

Hence, the manufacturer's claim about the 95 percent of its sets lasting more than 5 years without a repair is false.