Testing the Difference Between Two Means (e) decide whether to reject or fail to reject the null hypothesis, Assume the samples are random and dependent, and the populations are normally distributed.
[APPLET] Migraines
A researcher claims that injections of onabotulinumtoxinA reduce the number of days per month that chronic migraine sufferers have headaches. The table shows the number of days chronic migraine sufferers suffered migraines before and after using the treatment. At , α= 0.01 is there enough evidence to support the researcher’s claim? (Adapted from Journal of Headache and Pain)

검증된 단계별 안내

Step 1: Calculate the differences between the number of migraine days before and after treatment for each patient. For each patient, subtract the 'Days (after)' value from the 'Days (before)' value to find the difference.

Step 2: Compute the mean of the differences. Add up all the differences calculated in Step 1 and divide by the total number of patients to find the average difference.

Step 3: Compute the standard deviation of the differences. Use the formula for standard deviation: \( s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} \), where \( x_i \) represents each difference, \( \bar{x} \) is the mean difference, and \( n \) is the number of patients.

Step 4: Perform a paired t-test. Use the formula for the t-statistic: \( t = \frac{\bar{x}}{s / \sqrt{n}} \), where \( \bar{x} \) is the mean difference, \( s \) is the standard deviation of the differences, and \( n \) is the number of patients.

Step 5: Compare the calculated t-statistic to the critical t-value at \( \alpha = 0.01 \) for \( n-1 \) degrees of freedom. If the calculated t-statistic exceeds the critical t-value, reject the null hypothesis; otherwise, fail to reject the null hypothesis.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Null Hypothesis (H0)

The null hypothesis is a statement that there is no effect or no difference, serving as a default position in hypothesis testing. In this context, it posits that the mean number of migraine days before treatment is equal to the mean number after treatment. Testing this hypothesis helps determine if the treatment has a statistically significant impact on reducing migraine days.

Paired Sample t-Test

A paired sample t-test is a statistical method used to compare two related groups, such as measurements taken from the same subjects before and after a treatment. This test assesses whether the mean difference between the paired observations is significantly different from zero, which is crucial for evaluating the effectiveness of the treatment in this scenario.

Significance Level (α)

The significance level, denoted as α, is the threshold for determining whether to reject the null hypothesis. In this case, α is set at 0.01, indicating a 1% risk of concluding that a difference exists when there is none. This stringent level helps ensure that any observed effect is unlikely to be due to random chance, thus providing stronger evidence for the researcher's claim.