Question

Interpreting a P-value In each case, indicate whether the statement is a proper interpretation of what a p-value measures. (a) The probability the null hypothesis \(H_{0}\) is true. (b) The probability that the alternative hypothesis \(H_{a}\) is true. (c) The probability of seeing data as extreme as the sample, when the null hypothesis \(H_{0}\) is true. (d) The probability of making a Type I error if the null hypothesis \(H_{0}\) is true. (e) The probability of making a Type II error if the alternative hypothesis \(H_{a}\) is true.

Short answer

The only correct interpretation of a P-value among the given options is (c). The P-value is the probability of seeing data as extreme as the sample, when the null hypothesis \(H_{0}\) is true.

Step-by-step solution

Evaluate Statement (a)

Statement (a) indicates that the P-value is the probability the null hypothesis \(H_{0}\) is true. However, the P-value is not the probability of the null hypothesis being true but the measure of the strength of evidence against the null hypothesis.

Evaluate Statement (b)

Statement (b) suggests the P-value is the probability that the alternative hypothesis \(H_{a}\) is true. This interpretation is not correct because the P-value is not a measure of the probability that the alternative hypothesis is true.

Evaluate Statement (c)

Statement (c) states that the P-value is the probability of seeing data as extreme as the sample when the null hypothesis \(H_{0}\) is true. This interpretation is correct. The P-value measures the probability of observing a sample as extreme (or more extreme)than our current one, given that the null hypothesis is true.

Evaluate Statement (d)

Statement (d) suggests that the P-value is the probability of making a Type I error if the null hypothesis \(H_{0}\) is true. A Type I error occurs when the null hypothesis is true but is rejected. However, the P-value is not a measure of the probability of a Type I error. Therefore, this statement is incorrect.

Evaluate Statement (e)

Statement (e) states that the P-value is the probability of making a Type II error if the alternative hypothesis \(H_{a}\) is true. A Type II error happens when the null hypothesis is false, yet is not rejected. But, the P-value is not a measure of a Type II error. Thus, this interpretation is incorrect.

Key concepts

Null Hypothesis

The null hypothesis, denoted as \(H_0\), is a staple concept in statistical testing, where it represents the default or baseline assumption that there is no effect or no difference in the context of the study. Specifically, it stands for the statement that any observed data patterns are due to chance and not due to a systematic effect. For example, if we are testing a new drug, the null hypothesis might assert that the drug has no effect on patients compared to a placebo.

When interpreting p-values, it's essential to understand that they pertain to the null hypothesis. A p-value is not the probability that the null hypothesis is true but rather the likelihood of obtaining test results at least as extreme as the ones observed during the study, assuming the null hypothesis is true. A low p-value indicates that, under the assumption of the null hypothesis, the observed results would be quite unlikely, thus leading to consideration of rejecting the null hypothesis in favor of the alternative.

Alternative Hypothesis

In contrast to the null hypothesis, the alternative hypothesis, denoted by \(H_a\), proposes that there is a statistically significant effect or difference. Going back to the drug example, the alternative hypothesis might claim that the new drug does have a positive effect on patient outcomes.

A widespread misconception is thinking that a p-value can confirm the alternative hypothesis; however, statistically, it only quantifies evidence against the null hypothesis. The p-value itself does not validate the presence of an effect or difference (i.e., the alternative hypothesis), nor does it convey the probability of the alternative hypothesis being true. Instead, if the p-value is small enough, researchers may reject the null hypothesis and consider the alternative hypothesis more seriously.

Type I Error

A Type I error, often denoted as \(\alpha\), is the mistake of rejecting the null hypothesis when it is actually true. This error is analogous to a 'false positive' in diagnostic testing, where one incorrectly concludes that there is an effect or difference when none exists.

A crucial subtlety here is that the p-value is not the probability of committing a Type I error. The significance level, pre-selected by the researcher, indicates the threshold at which they risk a Type I error. Common levels are 5% or 1%. If the p-value is below this chosen significance level, the likelihood of the observed data given the null hypothesis is sufficiently small, and the null hypothesis is rejected, accepting the risk of Type I error inherent in that significance level.

Type II Error

Conversely, a Type II error, represented as \(\beta\), occurs if the null hypothesis is not rejected when it is, in fact, false. This kind of error is known as a 'false negative' where one fails to detect a true effect or difference. Unlike Type I error, it isn’t directly related to the p-value but more to the power of the test, which is the probability of correctly rejecting the null hypothesis when the alternative hypothesis is true.

The probability of a Type II error depends on several factors, including the sample size, the effect size, and the chosen significance level. A larger sample size or a greater effect size will generally reduce the likelihood of a Type II error. Researchers can use power analysis to estimate and reduce the potential for this type of error. Reducing the incidence of Type II errors enhances the sensitivity of an experiment or study to detect actual effects when they exist.