Testing the Difference Between Two Means (a) identify the claim and state Ho and Ha
[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 claim and hypotheses. The claim is that injections of onabotulinumtoxinA reduce the number of days per month that chronic migraine sufferers have headaches. The null hypothesis (H₀) states that there is no difference in the mean number of migraine days before and after treatment (μ₁ = μ₂). The alternative hypothesis (Hₐ) states that the mean number of migraine days after treatment is less than before treatment (μ₁ > μ₂).
2. Step 2: Calculate the differences for each patient. For each patient, subtract the number of migraine days after treatment from the number of migraine days before treatment. This will give the difference (d) for each patient.
3. Step 3: Compute the mean and standard deviation of the differences. Use the formula for the mean: $d=\frac{\sum d}{n}$, where d is the difference for each patient and n is the total number of patients. For the standard deviation, use the formula: $s=\sqrt{\frac{\sum (d-d)2^{)}}{n-1}}$.
4. Step 4: Perform a t-test for paired samples. Use the formula for the t-statistic: $t=\frac{d-\mu }{\frac{s}{\sqrt{n}}}$, where μ is the hypothesized mean difference (0 in this case), s is the standard deviation of the differences, and n is the number of patients.
5. Step 5: Compare the calculated t-statistic to the critical t-value at α = 0.01. Determine the degrees of freedom (df = n - 1) and find the critical t-value from the t-distribution table. If the calculated t-statistic is greater than the critical t-value, reject the null hypothesis and conclude that there is enough evidence to support the claim. Otherwise, fail to reject the null hypothesis.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating two competing hypotheses: the null hypothesis (H0), which states there is no effect or difference, and the alternative hypothesis (Ha), which suggests there is an effect or difference. In this context, the null hypothesis would assert that onabotulinumtoxinA has no effect on the number of migraine days, while the alternative would claim it does reduce the number of days.

Paired Sample t-Test

A paired sample t-test is used to compare the means of two related groups to determine if there is a statistically significant difference between them. In this scenario, the test will analyze the number of migraine days before and after treatment for the same patients. This method accounts for the fact that the samples are not independent, as they come from the same subjects, thus providing a more accurate assessment of the treatment's effect.

Significance Level (α)

The significance level, denoted as α, is the threshold for determining whether the results of a statistical test are significant. In this case, α is set at 0.01, meaning there is a 1% risk of concluding that a difference exists when there is none (Type I error). If the p-value obtained from the t-test is less than 0.01, the null hypothesis will be rejected, providing strong evidence to support the researcher's claim that the treatment reduces migraine days.