Question

Occasionally, warning flares of the type contained in most automobile emergency kits fail to ignite. A consumer advocacy group wants to investigate a claim against a manufacturer of flares brought by a person who claims that the proportion of defective flares is much higher than the value of .1 claimed by the manufacturer. A large number of flares will be tested, and the results will be used to decide between \(H_{0}: p=.1\) and \(H_{a}: p>.1,\) where \(p\) represents the proportion of defective flares made by this manufacturer. If \(H_{0}\) is rejected, charges of false advertising will be filed against the manufacturer. a. Explain why the alternative hypothesis was chosen to be \(H_{a}: p>.1 .\) b. In this context, describe Type I and Type II errors, and discuss the consequences of each.

Short answer

The alternative hypothesis is chosen to represent that the failure rate of flares is higher than the manufacturer's claim. Type I error in this case could harm the reputation of the company and possibly lead to unwarranted legal action, while Type II error could allow the company to continue false advertising. Both errors have significant consequences in real-world situations.

Step-by-step solution

Understanding the Null and Alternative Hypotheses

The null hypothesis, denoted by \(H_{0}\), is a statement that the value of a population parameter (such as proportion in this case) is equal to some claimed value. Here, \(H_{0}: p=.1\) represents the manufacturer's claim that the failure rate of their flares is 10%. The alternative hypothesis, denoted by \(H_{a}\) is a statement that the population parameter has a value that somehow differs from the null hypothesis. The alternative hypothesis here is \(H_{a}: p>.1\), which is the claim by the consumer advocacy group that the failure rate of the flares is higher than the manufacturer's claim.

Explain Why the Alternative Hypothesis Was Chosen to be \(H_{a}: p>.1\)

The alternative hypothesis \(H_{a}: p>.1\) was chosen because the consumer advocacy group suspects that the failure rate of the flares is higher than the claim of the manufacturer. If the group can gather enough evidence to support this claim, it will suggest that the manufacturer's claim is false and charges of false advertising may be filed.

Understanding Type I and Type II Errors

In hypothesis testing, a Type I error occurs when the null hypothesis is true, but it is rejected. A Type II error occurs when the null hypothesis is false, but it is not rejected.

Describe Type I and Type II Errors in This Context

In this context, a Type I error would occur if the manufacturer's claim that only 10% of its flares are defective is true, but the group rejects this hypothesis and accuses the company of false advertising. This would have negative implications for the company's reputation and may lead to unnecessary legal action. A Type II error would occur if the manufacturer's claim is false (the failure rate is more than 10%), but the advocacy group fails to reject the null hypothesis, thus letting the company continue its false advertising.

Key concepts

Null Hypothesis

When approaching hypothesis testing, the null hypothesis serves as a default statement that there is no effect or no difference; it represents the status quo. It is symbolized as H_0 in statistical notation. In our example of the automobile emergency flares, the null hypothesis H_0: p = .1 is the manufacturer's claim that only 10% of the flares are defective. It is not an assertion of absolute truth but rather a baseline assumption to be tested against the evidence collected in the study.

The significance of the null hypothesis lies in its role as a starting point. By attempting to disprove H_0, we apply a rigorous standard for finding evidence against it. Only if there is sufficient evidence to suggest that the null hypothesis is highly unlikely are we justified in favoring the alternative hypothesis.

Alternative Hypothesis

The alternative hypothesis, noted as H_a or H_1, posits a different state of affairs and is what researchers are trying to provide evidence for. Contrary to the null hypothesis, it is a claim that there is a significant effect or difference. In the case of the emergency flares, the alternative hypothesis H_a: p > .1 suggests that the proportion of defective flares is greater than 10%, which contradicts the manufacturer's claim.

Choosing the alternative hypothesis as p > .1 allows the consumer advocacy group to focus on detecting whether the defect rate is unacceptably high. Should the group find enough statistical evidence, it can challenge the manufacturer's assertions and potentially bring charges of false advertising. The alternative hypothesis is essentially the suspicion that drives the need for testing, and it presents a specific claim to be measured against the null.

Type I and Type II Errors

In hypothesis testing, two types of errors can occur: Type I and Type II errors. A Type I error is essentially a false alarm; it happens when we incorrectly reject the null hypothesis even though it is true. In our flare example, this would mean erroneously concluding that more than 10% of flares are defective when, in reality, the defect rate is as the manufacturer claims. This error would lead to unjustly accusing the manufacturer of false advertising, with potential legal and reputational damage.

Type II errors, on the other hand, are misses. This error occurs when the null hypothesis is false - meaning the product defect rate actually exceeds 10% - but we fail to reject the null hypothesis. This would result in the consumer group missing the opportunity to hold the manufacturer accountable for a potentially greater proportion of defective flares. Type II errors permit ongoing false advertising if not identified.

Minimizing these errors is crucial. However, trying to reduce one type of error can often increase the risk of the other, thus it is essential to find a balance that minimizes the impact of both errors in the context of the research.

Statistical Significance

The concept of statistical significance is key to interpreting the results of hypothesis testing. It quantifies the probability that the observed difference or effect could have occurred by random chance. Significance is often reflected by a p-value, and a common threshold for declaring statistical significance is a p-value of less than 0.05. If the p-value falls below this level, the null hypothesis is considered unlikely enough to reject in favor of the alternative hypothesis.

In the instance of the automotive flare analysis, if the evidence collected produces a p-value less than 0.05 for the observed proportion of defective flares, it implies that the deviation from the claimed defect rate is significant enough to not just be due to random chance. This would justify taking further action against the manufacturer. However, it's important to keep in mind that statistical significance does not necessarily imply practical importance, and it must be interpreted in the context of the real-world consequences of Type I and Type II errors.