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According 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. (a) Determine the null and alternative hypotheses that would be used to test the effectiveness of the massage therapist's techniques. (b) A sample of 500 American adults who participated in the massage therapist's program results in data that indicate that the null hypothesis should be rejected. Provide a statement that supports the massage therapist's program. (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?

Step by step solution

01

Identify Null Hypothesis (H0)

The null hypothesis is a statement that there is no effect or no difference, and it is the statement that will be tested. In this case, the null hypothesis (H0) would be that the percentage of American adults who experience migraine headaches remains the same, which is 15.2%. Formally, we can write this as: \( H_0: p = 0.152 \)

02

Identify Alternative Hypothesis (H1)

The alternative hypothesis is a statement that indicates the presence of an effect or difference. Here, the alternative hypothesis (H1) would be that the massage therapist's technique reduces the frequency and intensity of migraine headaches, so the percentage of American adults who experience migraines is less than 15.2%. Formally, we can write this as: \( H_1: p < 0.152 \)

03

Rejecting the Null Hypothesis

Given that the sample of 500 American adults who participated in the massage therapist's program leads to rejection of the null hypothesis, we can support the claim that the massage therapist's technique is effective. Therefore, a supporting statement would be: 'There is significant evidence to suggest that the massage therapist's technique reduces the frequency and intensity of migraine headaches in American adults.'

04

Analyze Error Type

In the context of hypothesis testing, a Type I error occurs when the null hypothesis is incorrectly rejected (false positive), while a Type II error occurs when the null hypothesis is not rejected when it is false (false negative). Since the true percentage of patients in the program who experience migraines is 15.3%, which is very close to the initial 15.2%, and we rejected the null hypothesis, a Type I error was committed. This is because we rejected the null hypothesis even though it was actually true.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Null Hypothesis

The null hypothesis (H0) is a fundamental aspect of hypothesis testing. It is a statement that assumes no effect or no difference exists in the population. For instance, in the original exercise, the null hypothesis assumes that the massage therapist's technique does not reduce migraine frequency and intensity. Therefore, if the known migraine rate is 15.2%, the null hypothesis would state: \( H_0: p = 0.152 \). The null hypothesis is what we aim to test against with our collected data. It serves as the default or starting assumption.

Alternative Hypothesis

The alternative hypothesis (H1) contrasts the null hypothesis by suggesting that there is an effect or a difference. In the exercise, the alternative hypothesis proposes that the massage therapist's technique reduces the frequency and intensity of migraines. Consequently, it can be written as: \( H_1: p < 0.152 \). This hypothesis represents the therapist's belief and what they aim to support with their evidence. The key distinction is that while the null hypothesis assumes no change, the alternative hypothesis indicates the anticipated difference due to the intervention.

Type I Error

A Type I error occurs when we wrongly reject a true null hypothesis. It is also known as a false positive. In the context of the exercise, if the null hypothesis stated that the migraine rate remains 15.2%, and the alternative hypothesis suggested it was lower due to the therapist's technique, a Type I error would occur if we wrongly concluded that the technique was effective at reducing migraines. Consequently, although our hypothesis testing indicated a reduction, the actual migraine rate was still around 15.3%, leading to a Type I error.

Type II Error

Conversely, a Type II error is the mistake of not rejecting a false null hypothesis, also known as a false negative. This error arises when the data fails to show sufficient evidence against the null hypothesis when, in reality, the null hypothesis is false. If, in the exercise, the massage technique had indeed reduced the migraine rate, but the hypothesis test results did not reflect this, we would have failed to support the alternative hypothesis. This would constitute a Type II error, missing the effect of the treatment.

Significance Level

The significance level, denoted as \( \alpha \), is a threshold set by the researcher before conducting a hypothesis test. It defines the probability of committing a Type I error. Commonly, a significance level of 0.05 is selected, meaning there's a 5% risk of rejecting the null hypothesis when it is true. Lowering this level decreases the risk of Type I error but increases the risk of Type II error. In the exercise, if a result falls below this significance level, it suggests strong enough evidence to reject the null hypothesis, supporting the effectiveness of the massage technique.