9.2.2 - Hypothesis Testing

The formula for the test statistic follows the same general format as the others that we have seen this week:

Test Statistic
\(test\; statistic = \frac{sample \; statistic - null\;parameter}{standard \;error}\)

Minitab Express 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 Express whenever possible.

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

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
2. Calculate an appropriate test statistic.

Standard Error

\(\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}\)

Test Statistic for Independent Means

\(t=\frac{\bar{x}_1-\bar{x}_2}{ \sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}\)

Estimated Degrees of Freedom

\(df=smallest\;n - 1\)

3. Determine the p-value associated with the test statistic.

The \(t\) test statistic found in Step 2 is used to determine the p-value. 

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.