Summarizing Categorical Data: Two-Way Tables - Statistics and Probability

1. State the formula for a marginal relative frequency for row $i$.

   $\frac{\text{row total } i}{\text{grand total}}$. Shows what proportion of all data is in that row.

2. What is the joint relative frequency for a cell with count $n_{ij}$ and grand total $N$?

   $\frac{n_{ij}}{N}$. Divide the cell count by grand total for proportion in both categories.

3. State the formula for a conditional relative frequency of column $j$ given row $i$.

   $\frac{\text{cell}(i,j)}{\text{row total } i}$. Shows distribution of columns within a specific row.

4. What is the correct comparison to check association using conditional relative frequencies?

   Compare $P(A\mid B_1)$ to $P(A\mid B_2)$ (and other $B$ categories). Different conditional probabilities indicate association between variables.

5. What is the marginal relative frequency for a row total $n_{i\cdot}$ with grand total $N$?

   $\frac{n_{i\cdot}}{N}$. Divide row total by grand total for proportion in that row category.

6. Identify the meaning of $P(A)$ in a two-way table context.

   The marginal proportion for category $A$ (row or column total over grand total). Represents the overall probability of event $A$ occurring.

7. What is the sum of all joint relative frequencies in a complete two-way table?

   $1$. All proportions must sum to the whole.

8. Find the joint relative frequency: cell count $12$, grand total $80$.

   $\frac{12}{80}=0.15$. Apply the joint relative frequency formula.

9. What is the conditional relative frequency of column category $C_j$ given row category $R_i$?

   $\frac{n_{ij}}{n_{i\cdot}}$. Divide cell count by row total for proportion within that row.

10. What is a marginal frequency in a two-way table?

    A row total or column total (a total along the margin). Found at the edges where row/column totals are displayed.

11. What is a correct interpretation of a conditional relative frequency such as $\frac{n_{ij}}{n_{i\cdot}}$?

    Among those in $R_i$, the proportion who are also in $C_j$. Conditional restricts to a subset before calculating proportion.

12. Identify the denominator for $P(A\mid B)$ when computed from a two-way table.

    The total count (or total proportion) for category $B$. Condition on $B$ means we only consider those in category $B$.

13. What is a correct interpretation of a marginal relative frequency such as $\frac{n_{i\cdot}}{N}$?

    The proportion of all individuals in row category $R_i$. Marginal ignores the other variable and gives overall proportion.

14. What pattern in conditional relative frequencies suggests an association?

    Conditional relative frequencies differ noticeably between groups. Different distributions indicate dependence between variables.

15. Find the marginal relative frequency for a row total $18$ when the grand total is $72$.

    $\frac{18}{72}=0.25$. Marginal relative frequency equals margin total divided by grand total.

16. What is a two-way frequency table used to display for two categorical variables?

    Counts for each category combination (rows and columns) and totals. Two-way tables organize data by two categorical variables simultaneously.

17. What is a correct interpretation of a joint relative frequency such as $\frac{n_{ij}}{N}$?

    The proportion of all individuals in both categories $R_i$ and $C_j$. Joint means the proportion satisfying both conditions simultaneously.

18. What is a two-way frequency table used to summarize?

    Counts for two categorical variables, organized by rows and columns. Two-way tables display relationships between two categorical variables.

19. Find the joint relative frequency for the cell count $12$ when the grand total is $60$.

    $\frac{12}{60}=0.20$. Joint relative frequency equals cell count divided by grand total.

20. Identify the meaning of $P(A\cap B)$ in a two-way table context.

    The joint proportion in the cell where $A$ and $B$ occur together. Represents the probability of both events occurring together.

21. What condition suggests no association between two categorical variables in a two-way table?

    Conditional relative frequencies are (approximately) equal across groups. Independence means conditioning doesn't change the proportions.

22. Find the marginal relative frequency: row total $30$, grand total $120$.

    $\frac{30}{120}=0.25$. Divide the row total by the grand total.

23. What is the conditional relative frequency of row category $R_i$ given column category $C_j$?

    $\frac{n_{ij}}{n_{\cdot j}}$. Divide cell count by column total for proportion within that column.

24. In a two-way table, what is the sum of the relative frequencies within a fixed row when row-conditional frequencies are used?

    $1$. Row-conditional frequencies distribute the row total completely.

25. Identify whether variables are associated if $P(A\mid B_1)=0.40$ and $P(A\mid B_2)=0.10$.

    Associated (conditional relative frequencies differ substantially). Large difference (0.40 vs 0.10) indicates strong association.

26. What is the marginal frequency for a row or column in a two-way table?

    The row total or column total (a total along the margin). Marginal frequencies appear at the edges (margins) of the table.

27. What is a relative frequency?

    A proportion: $ frac{\text{count}}{\text{total}}$. Converts counts to proportions for comparison.

28. State the formula for a joint relative frequency for cell $(i,j)$.

    $\frac{\text{cell}(i,j)}{\text{grand total}}$. Divides the cell count by the grand total.

29. Which phrase best matches $P(A\mid B)$ in a two-way table context?

    The proportion in category $A$ among those in category $B$. Conditional probability restricts to those already in category $B$.

30. State the formula for a conditional relative frequency of row $i$ given column $j$.

    $\frac{\text{cell}(i,j)}{\text{column total } j}$. Shows distribution of rows within a specific column.

31. What condition indicates no association between two categorical variables using conditional relative frequencies?

    Conditional relative frequencies are (approximately) the same across groups. Independence means the condition doesn't affect the distribution.

32. What is the grand total in a two-way table?

    The sum of all cell frequencies in the table. Add all interior cells to get the overall total.

33. Identify the association: $P(\text{Yes}\mid A)=0.70$ and $P(\text{Yes}\mid B)=0.20$.

    Association is suggested because the conditional proportions differ. Large difference (0.70 vs 0.20) indicates variables are related.

34. Find and correct the formula error: joint relative frequency written as $\frac{n_{ij}}{n_{i\cdot}}$.

    Correct joint relative frequency: $\frac{n_{ij}}{N}$. Joint uses grand total $N$, not row total $n_{i\cdot}$, as denominator.

35. Which denominator is used for $P(A\mid B)$ in a two-way table?

    The total for category $B$ (the given condition). Conditional probability uses the condition's total as denominator.

36. Identify whether variables show little association if $P(A\mid B_1)=0.33$ and $P(A\mid B_2)=0.34$.

    Little to no association (conditional relative frequencies are similar). Nearly equal values (0.33 vs 0.34) suggest independence.

37. Find $P(\text{Yes}\mid\text{Group A})$: Yes in Group A $18$, Group A total $60$.

    $\frac{18}{60}=0.30$. Divide Yes count by Group A total for conditional probability.

38. Find the conditional relative frequency $P(C_1\mid R_2)$ if $n_{21}=9$ and $n_{2\cdot}=30$.

    $\frac{9}{30}=0.30$. Conditional divides cell count by the conditioning category's total.

Study these flashcards