Hypothesis Testing
Learning Module 8: Hypothesis Testing
Exhibit 2: Test Statistics and Their Distribution
| What We Want to Test | Test Statistic | Probability Distribution of the Statistic | Degrees of Freedom |
|---|---|---|---|
| Test of a single mean | \(t = \frac{\bar{X} - \mu_0}{s / \sqrt{n}}\) | \(t\)-distributed | \(n - 1\) |
| Test of the difference in means | \(t = \frac{(\bar{X}_{d1} - \bar{X}_{d2}) - (\mu_{d1} - \mu_{d2})}{\sqrt{\frac{s_p^2}{n_{d1}} + \frac{s_p^2}{n_{d2}}}}\) | \(t\)-distributed | \(n_1 + n_2 - 2\) |
| Test of the mean of differences | \(t = \frac{\bar{d} - \mu_{d0}}{s_{\bar{d}}}\) | \(t\)-distributed | \(n - 1\) |
| Test of a single variance | \(\chi^2 = \frac{(n - 1)s^2}{\sigma_0^2}\) | Chi-square distributed | \(n - 1\) |
| Test of the difference in variances | \(F = \frac{s^2_{\text{before}}}{s^2_{\text{after}}}\) | \(F\)-distributed | \(n_1 - 1\), \(n_2 - 1\) |
| Test of a correlation | \(t = \frac{r \sqrt{n - 2}}{\sqrt{1 - r^2}}\) | \(t\)-Distributed | \(n - 2\) |
| Test of independence (categorical data) | \(\chi^2 = \sum_{i=1}^{m} \frac{(O_{ij} - E_{ij})^2}{E_{ij}}\) | Chi-square distributed | \((r - 1)(c - 1)\) |
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## Exhibit 2: Test Statistics and Their Distribution
| What We Want to Test | Test Statistic | Probability Distribution of the Statistic | Degrees of Freedom |
|--------------------------------------------|--------------------------------------------------------------------------------|-------------------------------------------|----------------------------|
| Test of a single mean | $t = \frac{\bar{X} - \mu_0}{s / \sqrt{n}}$ | $t$-distributed | $n - 1$ |
| Test of the difference in means | $t = \frac{(\bar{X}_{d1} - \bar{X}_{d2}) - (\mu_{d1} - \mu_{d2})}{\sqrt{\frac{s_p^2}{n_{d1}} + \frac{s_p^2}{n_{d2}}}}$ | $t$-distributed | $n_1 + n_2 - 2$ |
| Test of the mean of differences | $t = \frac{\bar{d} - \mu_{d0}}{s_{\bar{d}}}$ | $t$-distributed | $n - 1$ |
| Test of a single variance | $\chi^2 = \frac{(n - 1)s^2}{\sigma_0^2}$ | Chi-square distributed | $n - 1$ |
| Test of the difference in variances | $F = \frac{s^2_{\text{before}}}{s^2_{\text{after}}}$ | $F$-distributed | $n_1 - 1$, $n_2 - 1$ |
| Test of a correlation | $t = \frac{r \sqrt{n - 2}}{\sqrt{1 - r^2}}$ | $t$-Distributed | $n - 2$ |
| Test of independence (categorical data) | $\chi^2 = \sum_{i=1}^{m} \frac{(O_{ij} - E_{ij})^2}{E_{ij}}$ | Chi-square distributed | $(r - 1)(c - 1)$ |