The formula for the test statistic follows the same general format as the others that we have seen this week:
- Test Statistic
- \(test\; statistic = \dfrac{sample \; statistic - null\;parameter}{standard \;error}\)
Minitab will compute the test statistic for you! You will just need to determine if equal variances should be assumed or not. There is one example below walking through these procedures by hand, but you are strongly encouraged to use Minitab whenever possible.
There are two assumptions: (1) the two samples are independent and (2) both populations are normally distributed or \(n_1 \geq 30\) and \(n_2 \geq 30\). If the second assumption is not met then you can conduct a randomization test.
Below are the possible null and alternative hypothesis pairs:
| Research Question | Are the means of group 1 and group 2 different? | Is the mean of group 1 greater than the mean of group 2? | Is the mean of group 1 less than the mean of group 2? |
|---|---|---|---|
| Null Hypothesis, \(H_{0}\) | \(\mu_1 = \mu_2\) | \(\mu_1 = \mu_2\) | \(\mu_1 = \mu_2\) |
| Alternative Hypothesis, \(H_{a}\) | \(\mu_1 \neq \mu_2\) | \(\mu_1 > \mu_2\) | \(\mu_1 < \mu_2\) |
| Type of Hypothesis Test | Two-tailed, non-directional | Right-tailed, directional | Left-tailed, directional |
Standard Error
\(\sqrt{\dfrac{s_1^2}{n_1}+\dfrac{s_2^2}{n_2}}\)
Test Statistic for Independent Means
\(t=\dfrac{\bar{x}_1-\bar{x}_2}{ \sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}\)
Estimated Degrees of Freedom
\(df=smallest\;n - 1\)
The \(t\) test statistic found in Step 2 is used to determine the p-value.
If \(p \leq \alpha\) reject the null hypothesis. If \(p>\alpha\) fail to reject the null hypothesis.
Based on your decision in Step 4, write a conclusion in terms of the original research question.