Question

A television manufacturer claims that (at least) \(90 \%\) of its TV sets will need no 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 \(p\) be the sample proportion of responses indicating no repair (so that no repair is identified with a success). Let \(\pi\) denote the true 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 \(\pi<.9 .\) The appropriate hypotheses are then \(H_{0}: \pi=.9\) versus \(H_{a}: \pi<.9\). a. In the context of this problem, describe Type \(I\) and Type II errors, and discuss the possible consequences of each. b. Would you recommend a test procedure that uses \(\alpha=.10\) or one that uses \(\alpha=.01 ?\) Explain.

Short answer

Type I error would lead to wrongly accusing the manufacturer of false advertising. Its consequences could be legal or reputational harm to the manufacturer. Type II error would result in failure to identify false advertising leading to poor consumer choices and potential damage to the agency's credibility. The choice between using \(\alpha = .10\) or \(\alpha = .01\) depends on the balance between the risks of Type I and Type II errors one is willing to take. If the cost associated with Type I error is high, one might use a lower \(\alpha = .01\).

Step-by-step solution

Define Type I and Type II Errors

In the context of this problem, a Type I error occurs when the consumer agency incorrectly rejects the null hypothesis \(H_{0}: \pi=.9\) when in fact it is true, i.e., they mistakenly accuse the manufacturer of false advertising when their claims are accurate. On the other hand, a Type II error is made when the agency fails to reject the null hypothesis when it is false i.e., they fail to detect the manufacturer's false advertising.

Discuss the Consequences of Each Error

The consequences of a Type I error in this scenario could lead to potential legal issues or harm to the manufacturer’s reputation, as they would be wrongly accused of false advertising. A Type II error, on the other hand, involves the failure to identify false advertising. This could lead to consumers purchasing lower-quality TV sets under false premises, resulting in financial losses and potential damage to the agency's credibility.

Recommending a test procedure

Whether one should use a test procedure that uses \(\alpha=.10\) or \( \alpha=.01\) depends heavily on how much risk of committing Type I and Type II errors one is willing to accept. The smaller the value of alpha, the smaller the chance of committing a Type I error but the greater the chance of committing a Type II error. If the potential costs or consequences of a Type I error (falsely accusing the TV manufacturer) are considered to be quite high, then a smaller alpha, such as \(\alpha = 0.01\), would be advisable. However, this comes at the expense of a higher risk for a Type II error.