Overview
This lecture covers the chi-square goodness of fit test, focusing on calculating expected counts, computing the test statistic, interpreting results, and reviewing key assumptions for validity.
Calculating Expected Counts
- Expected count = hypothesized proportion × total sample size.
- Proportions in the null hypothesis determine the expected counts for each group.
- Expected counts may not be equal if null hypothesis proportions are not equal.
Chi-Square Test Statistic
- Use the formula: sum of (observed - expected)² divided by expected for each group.
- Each group's result is called a "contribution to chi-squared."
- Add all contributions together to get the chi-square test statistic.
- Large chi-square values indicate a significant difference between observed and expected counts.
- Chi-square is not interpreted like a z-score or t-score and can be much larger.
Interpreting Results
- A larger chi-square statistic suggests the sample data disagrees more with the null hypothesis.
- Comparing chi-square values across different null hypotheses shows which is more inconsistent with the sample data.
Assumptions of the Chi-Square Goodness of Fit Test
- Samples must be random and representative of the population.
- Observations should be independent from each other.
- All expected counts must be at least 5 for the test to be valid.
- If any expected count is below 5, the test may not be appropriate.
Key Terms & Definitions
- Expected Count — the number predicted in each category if the null hypothesis is true.
- Observed Count — the actual number observed in each category.
- Chi-Square Test Statistic (χ²) — sum of (observed - expected)² / expected across all groups.
- Contribution to Chi-Squared — the value for each group before summing to get χ².
- Null Hypothesis — the initial assumption about population proportions for the groups.
Action Items / Next Steps
- Ensure all expected counts in future calculations are at least 5.
- Check for random and independent samples before applying the chi-square test.
- Practice calculating chi-square test statistics with given data and proportions.