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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.

Step by step solution

01

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.

02

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.

03

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.

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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 incorrectly rejected. In simpler terms, it's like sounding a false alarm.

Imagine a company being falsely accused of a wrong it didn't commit—this is precisely what a Type I error would mean in our context. For our television manufacturer, this would mean the agency wrongly claims they were misleading consumers without factual basis.

The consequences of a Type I error can be serious.

• The manufacturer could suffer reputational damage.
• There could be legal repercussions.
• It creates an unfair business environment.

It's crucial to understand why avoiding false positives (Type I errors) is vital, especially when accusations can damage businesses or lives.

Type II Error

A Type II error is the counterpart to a Type I error and occurs when the null hypothesis fails to be rejected when it is indeed false. Think of it as missing the alarm altogether.

In this situation, the consumer agency might miss out on detecting false advertising by the television manufacturer.

The consequences might not appear immediate, but can be severe over time:

• Consumers may suffer by purchasing subpar products under false pretenses.
• There can be financial losses due to constant repairs or faulty products.
• The credibility of the testing agency could be questioned.

Recognizing the impacts of Type II errors is equally as important as recognizing Type I errors.

Significance Level

The significance level, often denoted by \( \alpha \), plays a critical role in hypothesis testing. It represents the probability of committing a Type I error.

Choosing a significance level depends on the balance one wants between Type I and Type II errors:

• A lower \( \alpha \), like 0.01, reduces the likelihood of a Type I error. This is suitable when the cost of falsely rejecting the null hypothesis is high.
• A higher \( \alpha \), like 0.10, allows for a greater chance of detecting real effects, but at the risk of more Type I errors.

For our case, if we are concerned about falsely accusing the TV manufacturer, opting for \( \alpha = 0.01 \) minimizes this risk. But remember, a lower significance level increases the probability of committing a Type II error, meaning we might miss a genuine case of false advertising. Balancing the significance level is key to making informed, accurate statistical decisions.