Example 1
\(\text{Test if die is fair: } O = [30,20,25,25,22,28], E = [25,25,25,25,25,25]\)
Chi-Square Test
Perform chi-square goodness of fit and independence tests
Results
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Chi-Square Test
The chi-square test is used to test if observed frequencies differ significantly from expected frequencies. It is commonly used for:
- Goodness-of-Fit Test: Tests if observed data fits an expected distribution
- Test of Independence: Tests if two categorical variables are independent
- Test of Homogeneity: Tests if different populations have the same distribution
Key Formulas:
- χ² = Σ[(O - E)² / E], where O = observed, E = expected
- Degrees of Freedom: df = number of categories - 1
- If p-value < α: Reject null hypothesis (significant difference)
- If p-value ≥ α: Fail to reject null hypothesis (no significant difference)
The test assumes that expected frequencies are at least 5 in each category for reliable results. For smaller expected frequencies, consider combining categories or using Fisher's exact test.
\(\chi^2 = \sum \frac{(O - E)^2}{E}, df = k - 1\)
Worked Examples
Example 2
\(\text{Test product preferences: } O = [45,30,25], E = [33.3,33.3,33.3]\)