Testing the Difference Between Two Means (f) interpret the decision in the context of the original claim. 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)

Verified step by step guidance

1. Step 1: Identify the type of test to use. Since the data involves paired samples (before and after treatment for the same patients), this is a dependent samples t-test (also known as a paired t-test). The goal is to test if the mean difference in migraine days before and after treatment is significantly less than zero, indicating a reduction in migraine days.
2. Step 2: State the null and alternative hypotheses. The null hypothesis (H₀) is that the mean difference in migraine days (before - after) is zero: H₀: μ_d = 0. The alternative hypothesis (Hₐ) is that the mean difference is less than zero: Hₐ: μ_d < 0.
3. Step 3: Calculate the differences for each patient. Subtract the 'Days after' values from the 'Days before' values to get the differences. For example, for Patient 1, the difference is 20 - 0 = 20. Repeat this for all patients to create a list of differences.
4. Step 4: Compute the test statistic. First, calculate the mean of the differences (d̄), the standard deviation of the differences (s_d), and the number of paired observations (n). Then, use the formula for the t-statistic: t = (d̄ - 0) / (s_d / √n).
5. Step 5: Compare the test statistic to the critical value. Using a t-distribution table and α = 0.01 with degrees of freedom df = n - 1, find the critical value for a one-tailed test. If the calculated t-statistic is less than the critical value, reject the null hypothesis. Otherwise, fail to reject the null hypothesis. Interpret the result in the context of the claim.

Key Concepts

Dependent Samples

Dependent samples, also known as paired samples, occur when the same subjects are measured before and after a treatment. This design is crucial for analyzing the effect of an intervention, as it controls for individual variability. In this case, the number of migraine days before and after treatment for the same patients allows for a direct comparison of the treatment's effectiveness.

Hypothesis Testing

Hypothesis testing is a statistical method used to determine if there is enough evidence to support a specific claim about a population parameter. In this scenario, the null hypothesis would state that there is no difference in the number of migraine days before and after treatment, while the alternative hypothesis posits that the treatment does reduce the number of days. The significance level (α = 0.01) indicates the threshold for rejecting the null hypothesis.

Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, indicating that data near the mean are more frequent in occurrence than data far from the mean. Many statistical tests, including those for comparing means, assume that the populations being studied are normally distributed. This assumption is important for the validity of the results when analyzing the differences in migraine days before and after treatment.