Testing the Difference Between Two Means (d) find the standardized test statistic t, 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: Calculate the differences (d) 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.
2. Step 2: Compute the mean of the differences (d̄). Add all the differences calculated in Step 1 and divide by the total number of patients (n).
3. Step 3: Calculate the standard deviation of the differences (s_d). Use the formula: s_d = sqrt(Σ(d_i - d̄)^2 / (n - 1)), where d_i represents each individual difference, d̄ is the mean difference, and n is the number of patients.
4. Step 4: Compute the standardized test statistic t using the formula: t = d̄ / (s_d / sqrt(n)). Here, d̄ is the mean difference, s_d is the standard deviation of the differences, and n is the number of patients.
5. Step 5: Compare the calculated t-value to the critical t-value at α = 0.01 for a two-tailed test with degrees of freedom df = n - 1. If the calculated t-value exceeds the critical t-value, there is enough evidence to support the researcher's claim.

Key Concepts

Dependent Samples

Dependent samples, also known as paired samples, occur when the same subjects are measured under different conditions or at different times. In this context, the number of migraine days before and after treatment for the same patients is compared. This design helps control for individual variability, allowing for a more accurate assessment of the treatment's effect.

Standardized Test Statistic (t)

The standardized test statistic, denoted as t, is used to determine if there is a significant difference between the means of two groups. In this case, it assesses whether the mean number of migraine days before treatment significantly differs from the mean after treatment. The t-statistic is calculated using the difference in means, the standard deviation of the differences, and the sample size.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about population parameters based on sample data. In this scenario, the null hypothesis posits that there is no difference in the number of migraine days before and after treatment, while the alternative hypothesis suggests that the treatment does reduce migraine days. The significance level (α = 0.01) indicates the threshold for rejecting the null hypothesis.