Statistics for Two Categorical Variables - AP Statistics

1. Identify the effect of sample size on chi-square statistic.

   Larger sample sizes generally increase the chi-square statistic. More data can detect smaller associations as statistically significant.

2. What does a contingency table with 2x2 dimensions represent?

   A table comparing two binary categorical variables. Each variable has exactly two possible categories or outcomes.

3. What is a p-value in the context of a chi-square test?

   The probability of observing a chi-square statistic as extreme as the one calculated. Used to determine statistical significance by comparing to alpha level.

4. What does a low p-value indicate in hypothesis testing?

   Strong evidence against the null hypothesis. Small p-values suggest the null hypothesis is likely false.

5. What is the general shape of a chi-square distribution?

   Right-skewed, becoming more symmetric with higher df. Always positive with mean equal to degrees of freedom.

6. What does a chi-square statistic measure?

   The difference between observed and expected frequencies. Larger values indicate greater deviation from what independence would predict.

7. What does it mean if two variables are independent?

   The occurrence of one does not affect the probability of the other. Knowledge of one variable provides no information about the other.

8. What is the null hypothesis in a chi-square test for independence?

   The two categorical variables are independent. Assumes no relationship exists between the categorical variables being tested.

9. Identify the test statistic used in a chi-square test for independence.

   Chi-square statistic. Measures how much observed frequencies deviate from expected frequencies.

10. Determine if this statement is true: High chi-square means high association.

    True, a high chi-square indicates strong association. Larger test statistics provide stronger evidence against independence.

11. What does a significant chi-square statistic indicate?

    There is evidence of an association between the variables. The variables are dependent rather than independent of each other.

12. What is a two-way table?

    A table that displays data on two categorical variables. Also called a contingency table, showing the relationship between variables.

13. What role do expected counts play in a chi-square test?

    Expected counts are under the assumption of independence. Represent what frequencies would occur if variables were truly independent.

14. Identify the expected count formula in a contingency table.

    Expected count = $\frac{(\text{row total}) (\text{column total})}{\text{grand total}}$. Assumes independence to calculate what count would be expected in each cell.

15. What is a contingency table used for?

    To display the frequency distribution of two categorical variables. Shows how categories of two variables relate to each other in a cross-tabulated format.

16. Calculate expected count for a cell with row total 20, column total 30, and grand total 200.

    Expected count = $\frac{(20)(30)}{200} = 3$. Multiplies marginal totals and divides by the overall sample size.

17. What is a significant result in the context of a chi-square test?

    A result that leads to the rejection of the null hypothesis. Indicates sufficient evidence to conclude variables are associated.

18. What is the purpose of a hypothesis test?

    To determine if there is enough evidence to reject a null hypothesis. Provides statistical framework for making decisions about population parameters.

19. State the relationship between p-value and significance level.

    If p-value < significance level, reject the null hypothesis. Compare p-value to chosen significance level to make decision.

20. What is the critical value in hypothesis testing?

    The value that separates the region where the null hypothesis is rejected. Determines the boundary for statistical significance in hypothesis testing.

21. Which condition must be met for chi-square test validity?

    All expected cell counts should be at least 5. Ensures the chi-square distribution is a good approximation for the test statistic.

22. Identify the role of the observed counts in a chi-square test.

    Observed counts are the actual data collected in the study. These are the frequencies actually recorded in the research study.

23. What does a joint distribution show?

    The frequency of each combination of categories of two variables. Cross-classifies observations by both variables simultaneously in each cell.

24. What is the significance level in hypothesis testing?

    The threshold p-value for rejecting the null hypothesis. Commonly set at 0.05 or 0.01 depending on desired confidence.

25. How do you interpret a p-value of 0.03 in a chi-square test?

    There is a 3% chance the observed association is due to random chance. Low probability suggests the association is unlikely due to chance alone.

26. What is a Type II error in hypothesis testing?

    Failing to reject the null hypothesis when it is false. False negative - missing a real association that actually exists.

27. Define conditional distribution.

    The distribution of one variable given a specific condition on another variable. Shows how one variable behaves when the other variable is fixed at specific values.

28. What is a cell in a contingency table?

    A cell is an intersection of a row and a column in the table. Contains the count for one specific combination of the two variable categories.

29. What is the formula for calculating chi-square statistic?

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

30. Find the chi-square statistic when $O = 15$ and $E = 10$.

    Chi-square = $\frac{(15 - 10)^2}{10} = 2.5$. Applies the formula for one cell contribution to the overall statistic.

31. What is a Type I error in hypothesis testing?

    Rejecting the null hypothesis when it is true. False positive - finding significance when none actually exists.

32. Define marginal distribution.

    The distribution of values of one variable among all individuals. Found by summing across rows or columns to get totals for each category.

33. Calculate degrees of freedom for a 3x4 table.

    $(3-1)(4-1) = 6$. Uses formula $df = (r-1)(c-1)$ where $r$ and $c$ are rows and columns.

34. State the purpose of a chi-square test for independence.

    To test if there is an association between two categorical variables. Determines whether observed frequencies differ significantly from expected under independence.

35. What assumption is made about data in a chi-square test?

    Data are a random sample from the population. Ensures results can be generalized to the broader population.

36. What is an association in statistics?

    A relationship between two categorical variables. Variables are dependent when one variable's distribution changes across levels of another.

37. What is the alternative hypothesis in a chi-square test for independence?

    The two categorical variables are not independent. Claims there is some relationship or association between the variables.

38. Identify the null hypothesis for a chi-square goodness-of-fit test.

    The observed distribution fits the expected distribution. Tests whether data follows a specified theoretical distribution.

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