MigrainesAccording to the Centers for Disease Control, 15.2% of American adults experience migraine headaches. Stress is a major contributor to the frequency and intensity of headaches. A massage therapist feels that she has a technique that can reduce the frequency and intensity of migraine headaches.


c. Suppose, in fact, that the percentage of patients in the program who experience migraine headaches is 15.3%. Was a Type I or Type II error committed?

Verified step by step guidance

1. Step 1: Identify the null hypothesis (H0) and the alternative hypothesis (H1). Here, the null hypothesis typically states that the percentage of patients experiencing migraines is equal to the known population percentage, so H0: p = 0.152. The alternative hypothesis would be that the percentage is different (or less, depending on the claim), for example, H1: p < 0.152 if the therapist claims a reduction.
2. Step 2: Understand the definitions of Type I and Type II errors. A Type I error occurs when we reject the null hypothesis even though it is true. A Type II error occurs when we fail to reject the null hypothesis even though the alternative hypothesis is true.
3. Step 3: Given that the true percentage in the program is 15.3%, which is very close to the original 15.2%, consider what decision was made based on the data. If the test failed to detect a difference and did not reject H0, but the true percentage is actually different, then a Type II error was made.
4. Step 4: Conversely, if the test rejected H0 when the true percentage is actually 15.3% (close to 15.2%), then a Type I error would have been made because the null hypothesis is essentially true or very close to true.
5. Step 5: Summarize by comparing the true state of nature (p = 0.153) with the decision made in the hypothesis test to determine whether a Type I or Type II error occurred.

Key Concepts

Type I and Type II Errors

Type I error occurs when a true null hypothesis is incorrectly rejected, while Type II error happens when a false null hypothesis is not rejected. Understanding these errors helps determine the nature of mistakes in hypothesis testing, especially when evaluating claims about population parameters.

Null and Alternative Hypotheses

The null hypothesis typically represents the status quo or no effect, while the alternative hypothesis reflects the claim being tested. In this context, the null might state that the migraine rate is 15.2%, and the alternative that it differs, guiding the decision-making process in hypothesis testing.

Significance Level and Statistical Power

The significance level (alpha) is the threshold for rejecting the null hypothesis, controlling the probability of a Type I error. Statistical power is the probability of correctly rejecting a false null hypothesis, related to Type II error. These concepts influence error rates and interpretation of test results.