gen-define-a-chi-squared-test-and-give-one-short-worked-example-2259

Define a chi-squared test and give one short worked example.

# Chi-Squared Test ## Definition A **chi-squared test** is a statistical hypothesis test that determines whether there is a significant difference between observed frequencies and expected frequencies in one or more categories. It measures how well observed data fit an expected distribution. The test statistic is: **χ² = Σ(O - E)² / E** where O = observed frequency and E = expected frequency ## Worked Example **Question:** A die is rolled 60 times. Are the results consistent with a fair die? | Number | 1 | 2 | 3 | 4 | 5 | 6 | |--------|---|---|---|---|---|---| | Observed | 8 | 12 | 9 | 11 | 14 | 6 | **Solution:** **Expected frequency** for a fair die: 60 ÷ 6 = 10 for each number **Calculate χ²:** χ² = (8-10)²/10 + (12-10)²/10 + (9-10)²/10 + (11-10)²/10 + (14-10)²/10 + (6-10)²/10 χ² = 4/10 + 4/10 + 1/10 + 1/10 + 16/10 + 16/10 χ² = **4.2** **Degrees of freedom:** 6 - 1 = 5 **Critical value** at α = 0.05: 11.07 **Conclusion:** Since 4.2 < 11.07, we fail to reject the null hypothesis. The die appears to be fair.