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 \(H_{0}=\pi=.1\) and \(H_{a}: \bar{\pi}>.1\), where \(\pi\) represents the true 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 \(H_{a}: \pi>.1\). b. In this context, describe Type I and Type II errors, and discuss the consequences of each.

Short answer

The alternative hypothesis \(H_{a}: \pi>.1\) is chosen as the advocate suspects the defect rate to be greater than 0.1. A Type I error would falsely accuse the manufacturer of false advertising, leading to potential loss of reputation and legal sanctions for the manufacturer. A Type II error would mean failing to detect a true high defective rate, potentially jeopardizing consumer's safety.

Step-by-step solution

Identify why the alternative hypothesis was chosen.

The alternative hypothesis \(H_{a}: \pi>.1\) was chosen as the advocacy group or person suspect that the true proportion of defective flares is greater than 0.1. This suspicion is opposing the claim made by the manufacturer, hence it forms our alternative hypothesis. The idea here is to provide enough statistical evidence to support this hypothesis instead of the null hypothesis \(H_{0}: \pi = .1\) made by the manufacturer, which states that the proportion of defective flares is exactly 0.1.

Define Type I Error

A Type I error in this context would happen if the advocacy group incorrectly rejects the null hypothesis that the defect rate is 0.1 when in fact, it truly is 0.1. Essentially, a Type I error would mean that the group accuses the manufacturer of false advertising unjustly.

Define Type II Error

A Type II error in this context is failing to reject the null hypothesis when the alternative hypothesis is true. This means that if the true defect rate of flares by this manufacturer is higher than 0.1, but the statistics do not provide robust evidence to reject the null hypothesis, a Type II error would have occurred.

Discuss the Consequences of Type I and Type II Errors

For a Type I error, the consequence could be legal sanctions, loss of reputation, and financial losses for the manufacturer if the false claim is publicized. A Type II error would result in consumers being given faulty flares, potentially leading to safety risks if they are relying on the flares in an emergency situation.

Key concepts

Null Hypothesis

In hypothesis testing in statistics, the null hypothesis, denoted as \( H_{0} \) serves as a baseline assumption for the statistical test. It usually states that there is 'no effect' or 'no difference' or, as in the context of our exercise, that a certain parameter such as the proportion of defective flares is equal to a specified value. In this case, \( H_{0}: \(\pi = .1\) \) suggests that the alleged proportion of defective flares made by the manufacturer is exactly 10%.
Typically, the goal of statistical testing is to gather enough evidence to reject the null hypothesis in favor of an alternative. The null hypothesis is presumed true unless statistical evidence suggests otherwise. Rejecting \( H_{0} \) is a statement that there is a statistically significant effect or difference that needs attention. If it's rejected incorrectly, however, there can be significant consequences, which we'll explore under the Type I error section.

Alternative Hypothesis

The alternative hypothesis, denoted as \( H_{a} \) or \( H_{1} \) is a statement that contradicts the null hypothesis and represents what the researcher is trying to demonstrate statistically. Alternative hypotheses are formulated based on suspicion, prior evidence, or research questions. In our exercise, the alternative hypothesis is \( H_{a}: \(\pi > .1\) \), proposing that the true proportion of defective flares is greater than the stated 10%.
A test's ability to find support for the alternative hypothesis relies on several factors, including the sample size and the level of statistical significance chosen. Establishing a significant difference that supports the alternative hypothesis could lead to changes in perception, policy, or practice regarding the topic under study.

Type I and Type II Errors

Hypothesis testing isn't infallible; it can result in Type I and Type II errors, which are essentially false positives and false negatives, respectively.

Type I Errors

A Type I error occurs when the null hypothesis is true, but we wrongly reject it. In the flare scenario, it corresponds to incorrect charges against the manufacturer for false advertising, when the flares' defect rate is genuinely at 10%. This error affects the manufacturer's credibility and might have legal and financial repercussions.

Type II Errors

Conversely, a Type II error happens when the null hypothesis is false, yet the test fails to reject it. That is, the defect rate is indeed higher than 10%, but our test doesn't provide sufficient evidence to support the claim. For consumers, such an error implies a risk of using defective flares, which is particularly dangerous in emergency situations. Both types of errors are related to the test's sensitivity and potential impacts on decision-making, indicating the careful balance needed between avoiding false positives and false negatives.

Statistical Evidence

In the context of hypothesis testing, statistical evidence refers to the data-based substantiation used to decide whether to reject the null hypothesis in favor of the alternative hypothesis. It's typically based on a test statistic that measures the degree of agreement between the sample data and the null hypothesis.
Evidence is evaluated against a predetermined significance level (often \( \alpha = 0.05 \)), which acts as a threshold for determining whether the observed results are likely due to random chance or to a true effect. If the test statistic falls into the critical region beyond this threshold, the evidence is considered strong enough to reject the null hypothesis. In our flare defect rate example, sufficient statistical evidence could lead to a conclusion that more than 10% of the flares are, in fact, defective — potentially bringing significant changes in manufacturing practices or regulatory actions.