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 401 consumers who have owned a ColorSmart-5000 television set for five years. Of these 401 consumers, 316 say that their ColorSmart-5000 television sets did not need repair, while 85 say that their ColorSmart-5000 television sets did need at least one repair. (a) 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. Ho :p [(Click to select)y 95 versus Ho: P[(Click to select)y95 (b) Use critical values and the previously given sample information to test the hypotheses you set up in part a by setting a equal to 10, .05.01, and .001. How much evidence is there that the manufacturer's claim is false? (Click to select) Ho at each value of a. (Click to select evidence.

Answer 1

a) As we are trying to test here whether 95% of its sets last at least five years without needing a single repair, therefore the null and the alternative hypothesis here are given as:

$H_a: p \neq 0.95$

b) The test statistic here is computed as:

$z^* = \frac{\frac{x}{n} - P}{\sqrt{\frac{P(1-P)}{n}}} = \frac{\frac{316}{401} - 0.95}{\sqrt{\frac{0.95*0.05}{401}}} = -14.88$

From standard normal tables, we have here:
P(Z < 3.291) = 0.9995
Therefore, due to symmetry, we have here:
P( -3.291 < Z < 3.291) = 0.999

Therefore the critical value at 0.001 level of significance here is 3.291

As the test statistic value here is -14.88 which is even lower than -3.291, therefore we can reject the null hypothesis at all the level of significance here.

Therefore we Reject H0 at all $\alpha$ here. We have sufficient evidence here that not 95% of its sets last at least five years without needing a single repair