12.6 Summary of formulas

The chi-square probability distribution

Goodness-of-fit hypothesis test

• Use goodness-of-fit to test whether a data set fits a particular probability distribution.
• The degrees of freedom are $\text{number of cells or categories - 1}$ .
• The test statistic is $\underset{k}{\Sigma }\frac{(O-E{)}^2}{E}$ , where $O$ = observed values (data), $E$ = expected values (from theory), and $k$ = the number of different data cells or categories.
• The test is right-tailed.

Test of independence

• Use the test of independence to test whether two factors are independent or not.
• The degrees of freedom are equal to $\text{(number of columns - 1)(number of rows - 1)}$ .
• The test statistic is $\underset{(i\cdot j)}{\Sigma }\frac{(O-E{)}^2}{E}$ where $O$ = observed values, $E$ = expected values, $i$ = the number of rows in the table, and $j$ = the number of columns in the table.
• The test is right-tailed.
• If the null hypothesis is true, the expected number $E=\frac{\text{(row total)(column total)}}{\text{total surveyed}}$ .

Test of homogeneity

• Use the test for homogeneity to decide if two populations with unknown distributions have the same distribution as each other.
• The degrees of freedom are equal to $\text{number of columns - 1}$ .
• The test statistic is $\underset{(i\cdot j)}{\Sigma }\frac{(O-E{)}^2}{E}$ where $O$ = observed values, $E$ = expected values, $i$ = the number of rows in the table, and $j$ = the number of columns in the table.
• The test is right-tailed.
• If the null hypothesis is true, the expected number $E=\frac{\text{(row total)(column total)}}{\text{total surveyed}}$ .

Test of a single variance

• Use the test to determine variation.
• The degrees of freedom are the number of samples - 1.
• The test statistic is $\frac{(n-1)\cdot s^2}{{\sigma }^2}$ , where $n$ = the total number of data, $s^2$ = sample variance, and ${\sigma }^2$ = population variance.
• The test may be left, right, or two-tailed.