A television manufacturer claims that (at least) 90% of its TV sets will not need service during the first 3 years of operation. A consumer agency wishes to check this claim, so it obtains a random sample of
n = 100 purchasers and asks each whether the set purchased needed repair during the first 3 years after purchase. Let $\hat{p}$ be the sample proportion of responses indicating no repair (so that no repair is
identified with a success). Let p denote the actual proportion of successes for all sets made by this manufacturer.
The agency does not want to claim false advertising unless sample evidence strongly suggests that p < 0.9. The appropriate hypotheses are then $H_0: p = 0.9$ versus $H_a: p < 0.9$.
(a) In the context of this problem, describe Type I and Type II errors. (Select all that apply.)
A Type II error would be obtaining convincing evidence that less than 90% of the TV sets need no repair when in fact (at least) 90% need no repair.
A Type I error would be not obtaining convincing evidence that less than 90% of the TV sets need no repair when in fact less than 90% need no repair.
A'Type II error would be not obtaining convincing evidence that less than 90% of the TV sets need no repair when in fact less than 90% need no repair.
A Type I error would be obtaining convincing evidence that less than 90% of the TV sets need no repair when in fact (at least) 90% need no repair.
Discuss the possible consequences of each. (Select all that apply.)
The consumer agency would not take action against the manufacturer when in fact the manufacturer is making true claims about the reliability of the TV sets.
The consumer agency might take action against the manufacturer when in fact the manufacturer is at fault.
The consumer agency would not take action against the manufacturer when in fact the manufacturer is making untrue claims about the reliability of the TV sets.
The consumer agency might take action against the manufacturer when in fact the manufacturer is not at fault.
(b) Would you recommend a test procedure that uses $\alpha = 0.10$ or one that uses $\alpha = 0.01$? Explain.
Use $\alpha = 0.10$, as making a Type II error involves not catching the manufacturer when they are at fault.
Use $\alpha = 0.10$, as making a Type I error involves not catching the manufacturer when they are at fault.
Use $\alpha = 0.01$, as making a Type II error involves taking action against the manufacturer when in fact the manufacturer is not at fault.
Use $\alpha = 0.01$, as making a Type I error involves taking action against the manufacturer when in fact the manufacturer is not at fault.
