Testing the Difference Between Two Means, (b) find the critical value(s) and identify the rejection region(s), 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 between the 'Days (before)' and 'Days (after)' for each patient. This will give you the paired differences (d_i). For example, for Patient 1, the difference is 20 - 0 = 20.
2. Step 2: Compute the mean of the paired differences (d̄). Add all the paired differences together and divide by the total number of patients (n). Use the formula: $d=\frac{\sum d}{}$_in.
3. Step 3: Calculate the standard deviation of the paired differences (s_d). Use the formula: $s$_d=∑(d_i-d)2n-1.
4. Step 4: Compute the test statistic t using the formula: $t=\frac{d-0}{\frac{s}{}}$_dn, where d̄ is the mean of the paired differences, s_d is the standard deviation of the paired differences, and n is the number of patients.
5. Step 5: Determine the critical t-value for a two-tailed test at α = 0.01 with degrees of freedom df = n - 1. Compare the calculated t-value to the critical t-value to identify the rejection region. If the calculated t-value falls within the rejection region, reject the null hypothesis; otherwise, fail to reject the null hypothesis.

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, making it easier to detect the effect of the treatment.

Hypothesis Testing

Hypothesis testing is a statistical method used to determine whether there is enough evidence to reject a null hypothesis in favor of an alternative hypothesis. 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 migraine days. The significance level (α) helps determine the threshold for rejecting the null hypothesis.

Critical Value and Rejection Region

The critical value is a threshold that defines the boundary for rejecting the null hypothesis in hypothesis testing. It is determined based on the chosen significance level (α) and the distribution of the test statistic. The rejection region is the range of values for the test statistic that would lead to rejecting the null hypothesis. In this case, with α = 0.01, the critical value will help assess whether the observed difference in migraine days is statistically significant.