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 \(\hat{p}\) 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.

Short answer

Type I Error: Falsely claiming false advertising, consequences might involve damaging the manufacturer's reputation and possible legal issues. Type II Error: Not catching false advertising when it occurs, consequences involve consumers buying a product under false assumptions. An alpha level of 0.10 might be more appropriate given the context of the problem.

Step-by-step solution

Understand the Hypotheses

First, define the null hypothesis \(H_0\): this manufacturer's TV sets have exactly a 0.9 probability of needing no service in the first three years. The alternative hypothesis \(H_a\): the proportion of TV sets that need no service in the first 3 years is less than 0.9.

Conceptualize Type I and II Errors

A Type I error occurs when you reject the null hypothesis, but it is actually true. In this context, a Type I error would be claiming the manufacturer is falsely advertising, even though there is a 90% probability that the TVs will need no service in the first three years. A Type II error, on the other hand, happens when you do not reject the null hypothesis when it is in fact false. In this scenario, a Type II error would occur if the consumer agency did not claim the manufacturer was falsely advertising and the probability of the TVs needing no service in the first three years was less than 90%.

Assess the Consequences of Errors

The consequence of a Type I error would be wrongfully accusing the manufacturer of false advertising, which could damage the manufacturer's reputation and possibly lead to legal implications. The consequence of a Type II error would be failing to identify false advertising, leading to consumers buying the product under a wrong assumption.

Choose the Appropriate Significance Level

The appropriate significance level depends on weighing the consequences of Type I and II errors. An alpha level of 0.10 increases the probability of making a Type I error (claiming false advertising when the claim is true). An alpha level of 0.01, while reducing the chance of a Type I error, increases the likelihood of a Type II error (failing to identify false advertising when it exists). Given the context, it would be less harmful to the consumers to falsely accusing the manufacturer (Type I error), than failing to catch false advertising (Type II error). Therefore, an alpha level of 0.10 might be recommended.

Key concepts

Type I and Type II Errors

When conducting hypothesis testing, two types of errors can occur: Type I and Type II errors. These are crucial concepts in the statistical field, as they help to understand the potential mistakes that can be made when making decisions based on sample data. Let's explore each one.

Type I Error

Type I error, also known as a false positive, happens when the null hypothesis is true, but we mistakenly reject it. For the TV manufacturer scenario, committing a Type I error would lead the consumer agency to conclude that less than 90% of TV sets will need no service, while the claim is accurate. This could undesirably harm the manufacturer's reputation and might involve legal consequences.

Type II Error

Conversely, a Type II error, or false negative, occurs when the null hypothesis is false, but we fail to reject it. In our television example, this would mean the agency does not flag the manufacturer for false advertising when, in fact, over 10% of TV sets do require service within the first three years. This might leave consumers misled, purchasing products based on incorrect information.

Understanding these errors is critical because it impacts how we set the threshold for making decisions, balancing the risk of both types of errors. The significance level we choose influences the likelihood of each error type.

Sample Proportion

The concept of sample proportion, denoted as \(\hat{p}\), is an estimate that refers to the fraction of items in a sample that have a particular attribute. In our exercise, the sample proportion represents the percentage of TV sets from the sample that did not require repair in the first three years. It serves as a practical estimate of the population proportion (\(p\)), the true proportion of all TV sets made by the manufacturer that will not need service during that time.

Sample proportions are used in hypothesis testing to compare against a claimed population proportion (the null hypothesis). As such, they are pivotal in determining whether there is sufficient evidence to support or refute a claim, in this case, the manufacturer’s assertion about their TV sets’ durability.

Null and Alternative Hypotheses

Hypothesis testing is a systematic way to evaluate claims about a population based on sample data. Each test involves two competing statements: the null hypothesis (\(H_0\)) and the alternative hypothesis (\(H_a\)).

The null hypothesis is essentially the statement being tested and is assumed true until evidence suggests otherwise. It often represents a status quo or a claim that there is no effect or difference. For our scenario, the null hypothesis is that the true proportion of TV sets not requiring service is 90% (\(p = 0.9\)).

The alternative hypothesis contradicts the null hypothesis and is what you suspect might be true instead. It represents a new claim aiming to provide evidence against the null. In the context of the television sets, the alternative hypothesis is that the true proportion needing no service is less than 90% (\(H_a: p < 0.9\)). The type of alternative hypothesis (one-tailed or two-tailed) guides the direction in which we look for evidence against the null hypothesis in our statistical test.

Significance Level

The significance level, denoted by \(\alpha\), represents the threshold for deciding whether observed data is statistically significant. It is the probability of rejecting the null hypothesis when it is actually true, thus the probability of making a Type I error. In simpler terms, it's the level of risk you are willing to accept to erroneously conclude an effect or difference exists.

Commonly used significance levels are 0.05, 0.01, and 0.10. A lower \(\alpha\) value indicates a more strict criterion for claiming significance, reducing the risk of Type I errors. The trade-off, however, is generally an increased risk of Type II errors. Selecting the appropriate significance level depends on the relative cost of the errors in the context. For instance, a 0.10 significance level might be chosen if the consumer agency prioritized minimizing false negatives (Type II error) over false positives (Type I error), accepting a slightly higher likelihood of accusing the manufacturer without sufficient evidence.