Chi-Square Homogeneity or Independence (Setup) - AP Statistics

1. Determine the test: Evaluating if gender is related to voting preference.

   Chi-Square Test of Independence. Tests if gender and voting preference are associated variables.

2. What is the alternative hypothesis for a Chi-Square Test of Homogeneity?

   The distributions are different. Homogeneity tests check if categorical distributions vary across groups.

3. What assumption does a Chi-Square test make about samples?

   Samples are randomly selected. Random sampling ensures independence and representative results.

4. Calculate degrees of freedom for a 4x3 contingency table.

   $(4-1)(3-1) = 6$. Use the formula $(r-1)(c-1)$ where $r=4$ rows and $c=3$ columns.

5. Which test uses the contingency table to organize data?

   Both Chi-Square Tests. Both tests use contingency tables to organize observed frequencies.

6. Which test checks if proportions differ across multiple populations?

   Chi-Square Test of Homogeneity. Homogeneity tests compare categorical proportions across multiple populations.

7. What does a significant Chi-Square statistic indicate?

   Reject the null hypothesis. Significant results lead to rejecting the null hypothesis.

8. What is the formula for the Chi-Square statistic?

   $\text{Chi-Square} = \frac{(O-E)^2}{E}$ summed over all cells. Sum the squared standardized deviations across all contingency table cells.

9. What is the null hypothesis for a Chi-Square Test of Independence?

   The variables are independent. For independence tests, we assume no relationship exists between variables.

10. What is the degree of freedom formula for a Chi-Square Test of Independence?

    $(r-1)(c-1)$, where $r$ is rows, $c$ is columns. Degrees of freedom account for constraints in the contingency table.

11. What is the null hypothesis for a Chi-Square Test of Homogeneity?

    The distributions are the same. Assumes categorical distributions are identical across all groups.

12. What type of test is the Chi-Square Test of Homogeneity?

    Non-parametric test. Non-parametric tests don't assume specific population distributions.

13. Which test assesses if sample proportions are equal across groups?

    Chi-Square Test of Homogeneity. Homogeneity tests compare proportions of categorical outcomes across groups.

14. What describes a contingency table with 3 rows and 2 columns?

    3x2 table. Describes the dimensions of a two-way frequency table.

15. Calculate $\text{Chi-Square}$: $O=10, E=8$ in one cell.

    $\frac{(10-8)^2}{8} = 0.5$. Apply the Chi-Square formula: $(O-E)^2/E$ for this cell.

16. What is the critical value for a Chi-Square test with 2 df at 0.05 significance?

    5.991. Standard critical value for 2 degrees of freedom at 5% significance level.

17. Determine the test: Evaluating if study fields affect job types.

    Chi-Square Test of Independence. Tests association between study field and job type variables.

18. Identify the test: Comparing categorical distributions across multiple groups.

    Chi-Square Test of Homogeneity. This test compares how categorical variables are distributed across groups.

19. State the alternative hypothesis for a Chi-Square Test of Independence.

    The variables are dependent. Independence alternative hypothesis states variables are related or associated.

20. Identify the test: Comparing response frequency across different age groups.

    Chi-Square Test of Homogeneity. Compares response distributions across different age group populations.

21. Find the expected frequency: $60$ total, $15$ in row, $4$ in column.

    $E = \frac{(15 \times 4)}{60} = 1$. Apply the expected frequency formula: row total × column total ÷ grand total.

22. How are the observed frequencies denoted in Chi-Square tests?

    $O$. O represents the actual counts observed in each cell.

23. What is a common use of the Chi-Square Test of Independence?

    To assess relationships between categorical variables. Independence tests determine if categorical variables are associated.

24. Which test applies when examining if two variables interact?

    Chi-Square Test of Independence. Independence tests examine relationships between two categorical variables.

25. State the formula for the expected frequency in a contingency table cell.

    $E = \frac{(row\text{ }total \times column\text{ }total)}{grand\text{ }total}$. Expected frequency equals the product of marginal totals divided by sample size.

26. Identify the type of variable suitable for Chi-Square tests.

    Categorical variables. Chi-Square tests require qualitative data with distinct categories.

27. Find expected frequency: $100$ total, $25$ in row, $10$ in column.

    $E = \frac{(25 \times 10)}{100} = 2.5$. Calculate using the standard expected frequency formula.

28. What assumption must be met regarding expected frequencies for Chi-Square tests?

    Expected frequencies should be at least 5. This ensures the Chi-Square distribution approximation is valid.

29. When is a Chi-Square test considered invalid?

    When expected frequencies are less than 5. Low expected frequencies violate the Chi-Square distribution assumption.

30. What is the minimum sample size recommended for Chi-Square tests?

    No specific size, but expected counts > 5. Focus is on expected counts rather than total sample size.

31. What is the general purpose of a Chi-Square test?

    To test the association between categorical variables. Chi-Square tests evaluate relationships and patterns in categorical data.

32. What type of distribution is used in Chi-Square tests?

    Chi-Square distribution. Chi-Square distribution is the theoretical basis for these hypothesis tests.

33. Identify the symbol commonly used for the Chi-Square statistic.

    $\text{Chi-Square}$ or $\text{X}^2$. Both symbols represent the calculated test statistic in Chi-Square tests.

34. Identify the test: Assessing relationship between two categorical variables.

    Chi-Square Test of Independence. This test determines if two categorical variables are associated.

35. What is required for valid Chi-Square results concerning data collection?

    Random sampling. Random sampling ensures valid statistical inferences from Chi-Square tests.

36. What condition must be met for Chi-Square tests regarding sample size?

    Expected frequencies must be at least 5. All expected frequencies must meet the minimum threshold of 5.

37. What is the range of values for a Chi-Square statistic?

    0 to $\text{infinity}$. Chi-Square statistics are always non-negative with no upper bound.

38. What do you compare the Chi-Square statistic to for hypothesis testing?

    Chi-Square critical value. Compare calculated test statistic to tabled critical value for decision.

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