8.3.2 - Hypothesis Testing

Below are the procedures for conducting a hypothesis test for two paired means. This is often referred to as a "paired means \(t\) test," "dependent means \(t\) test," or "matched pairs \(t\) test." 

1. Check any necessary assumptions and write null and alternative hypotheses.

Data must be paired. The difference between the two groups must be normally distributed in the population or the sample size must be at least 30.

The possible combinations of null and alternative hypotheses are:

Research Question Is the mean difference different from 0? Is the mean difference greater than 0? Is the mean difference less than 0?
Null Hypothesis, \(H_{0}\) \(\mu_d = 0 \) \(\mu_d = 0 \) \(\mu_d = 0 \)
Alternative Hypothesis, \(H_{a}\) \(\mu_d \neq 0 \) \(\mu_d > 0 \) \(\mu_d < 0 \)
Type of Hypothesis Test Two-tailed, non-directional Right-tailed, directional Left-tailed, directional

Where \( \mu_d \) is the hypothesized difference in the population.

2. Calculate an appropriate test statistic.

The calculation of the test statistic for dependent samples is similar to the calculation you performed earlier in this lesson for a single sample mean. In this formula, \(\overline{x}_d\) is used in place of \(\overline{x}\) and \(s_d\) is used in place of \(s\):

Test Statistic for Dependent Means

\(t=\frac{\bar{x}_d-\mu_0}{\frac{s_d}{\sqrt{n}}}\)

\(\overline{x}_d\) = observed sample mean difference
\(\mu_0\) = mean difference specified in the null hypothesis
\(s_d\) = standard deviation of the differences
\(n\) = sample size (i.e., number of unique individuals)

Observed Sample Mean Difference
\(\overline{x}_d=\frac{\Sigma{x}_d}{n}\)
\(x_d\) = observed difference
Standard Deviation of the Differences
\(s_d=\sqrt{\frac{\sum (x_d-\overline{x}_d)^{2}}{n-1}}\)
3. Determine the p value associated with the test statistic.

When testing hypotheses about a mean difference, a \(t\) distribution is used to find the \(p\) value. The degrees of freedom are equal to \(n-1\) where \(n\) is the number of pairs. 

4. Decide between the null and alternative hypotheses.

 If \(p \leq \alpha\) reject the null hypothesis. If \(p>\alpha\) fail to reject the null hypothesis.

5. State a "real world" conclusion.

Based on your decision in Step 4, write a conclusion in terms of the original research question.