91Ó°ÊÓ

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 \(\hat{p}\) 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 \(p<.9\). The appropriate hypotheses are then \(H_{0}: p=.9\) versus \(H_{a}: p<.9\). 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.

Step by step solution

01

Identify Type I and Type II errors and their consequences

A Type I error occurs when a true null hypothesis is rejected. In this case, it means the consumer agency would falsely accuse the manufacturer of lying when in fact \(90\%\) of TV sets would not require service in the first three years. Implication of a Type I error here would lead to potential damage to manufacturer's reputation and could possibly result in legal action against the consumer agency. On the other hand, a Type II error is made when a false null hypothesis is not rejected - the consumer agency fails to identify that the manufacturer's claim is false when it actually is. The consequence of a Type II error could lead to consumers buying TV sets believing they will probably not need repair in the first three years when in fact they do.

02

Deciding between \(\alpha=.10\) and \(\alpha=.01\)

The \(\alpha\) level, or significance level, determines the threshold for rejecting the null hypothesis. A smaller \(\alpha\) (such as \(\alpha=.01\)) means the test is more stringent. In the context of this question, using \(\alpha=.01\) would mean stronger evidence (a smaller p-value) is required to claim that the manufacturer's claim is false. Given the potential consequences of a Type I error (falsely accusing the manufacturer), it is suggested to use a more stringent test, i.e., \(\alpha=.01\). This reduces the likelihood of wrongly accusing the manufacturer.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Type I Error

In hypothesis testing, a Type I error occurs when the null hypothesis is true, but it is mistakenly rejected. This means you conclude there is an effect or a difference, when in fact there isn't one. In the TV manufacturer example, a Type I error would occur if the consumer agency accuses the company of false advertising, even though 90% of the TV sets truly do not require service in the first three years.
Potential consequences of a Type I error in this scenario can include:

• Damage to the manufacturer's reputation: Consumers may lose trust in the brand, avoiding future purchases.
• Legal implications: False accusations can lead to lawsuits or financial penalties against the consumer agency.
• Unnecessary business implications: The manufacturer may have to alter its practices or create unnecessary quality checks.

The goal is to minimize Type I errors, as they can have significant and often costly consequences for businesses.

Type II Error

A Type II error happens when the null hypothesis is false, yet it is wrongly accepted. Essentially, this error leads to missing an actual effect or difference. In our example, a Type II error would occur if the consumer agency fails to identify the manufacturer's false claim, believing incorrectly that 90% of their TV sets really do not need repairs within three years.
The implications of a Type II error include:

• Consumers may be misled into buying products under false pretenses, leading to dissatisfaction and increased repair costs.
• The manufacturer might continue lowering quality control because they believe their claims are confirmed.
• Overall market trust issues, as similar situations repeated could degrade consumer trust across the industry.

Type II errors can affect consumer confidence and product quality over time, so agencies must try to balance avoiding both types of errors in decision-making.

Significance Level

The significance level, denoted as \( \alpha \), is a critical threshold in hypothesis testing that determines when we should reject the null hypothesis. A smaller \( \alpha \), like 0.01, means we require stronger evidence to reject the null hypothesis compared to a larger \( \alpha \), such as 0.10.
Choosing a significance level requires weighing the risks of Type I and Type II errors. In the TV set claim scenario, using an \( \alpha = 0.01 \) means being very cautious before accusing the manufacturer of false advertising. This low significance level minimizes the risk of a Type I error, known as a false positive.
Factors influencing the choice of \( \alpha \) include:

• The severity of potential Type I errors: If accusing falsely has severe consequences, a lower \( \alpha \) is preferred.
• Sensitivity to Type II errors: If missing an actual effect is significant, a higher \( \alpha \) might be more appropriate to catch true differences.
• Practical or financial considerations: Deciding the repercussions of both error types in a real-world context.

Appropriately selecting \( \alpha \) helps balance these errors to protect both consumers and manufacturers effectively.