Let's solve the step by step: --- ## Step 1: Tally the Counts for Each (x, y) Pair We'll go through all 30 observations and count the occurrences for each combination of x (A, B, C) and y (1, 2). ### Tally Table | x | y=1 | y=2 | |---|-----|-----| | A | | | | B | | | | C | | | Let's fill this in: #### For A: - y=1: Obs 2, 3, 11, 26, 30 → 5 times - y=2: None #### For B: - y=1: Obs 1, 6, 9, 10, 17, 20, 23, 24, 27, 28 → 10 times - y=2: Obs 7, 22 → 2 times #### For C: - y=1: Obs 8 → 1 time - y=2: Obs 4, 5, 12, 13, 14, 15, 16, 18, 19, 21, 25, 29 → 12 times Now sum the rows and columns. --- ## Step 2: Crosstabulation Table ### (a) Crosstabulation | | y=1 | y=2 | Total | |-----|-----|-----|-------| | A | 5 | | 5 | | B | 10 | 2 | 12 | | C | 1 | 12 | 13 | |**Total**| 16 | 14 | 30 | --- ## Step 3: Row Percentages Each cell is: (cell count / row total) × 100 ### (b) Row Percentages | | y=1 | y=2 | Total | |-----|---------|---------|-------| | A | 100.% | .% | 5 | | B | 83.3% | 16.7% | 12 | | C | 7.7% | 92.3% | 13 | _Calculations:_ - A-y=1: 5/5 = 1 → 100% - A-y=2: /5 = % - B-y=1: 10/12 ≈ .833 → 83.3% - B-y=2: 2/12 ≈ .167 → 16.7% - C-y=1: 1/13 ≈ .077 → 7.7% - C-y=2: 12/13 ≈ .923 → 92.3% --- ## Step 4: Column Percentages Each cell is: (cell count / column total) × 100 ### (c) Column Percentages | | y=1 | y=2 | Total | |-----|---------|---------|-------| | A | 31.3% | .% | 5 | | B | 62.5% | 14.3% | 12 | | C | 6.3% | 85.7% | 13 | |**Total**| 100% | 100% | 30 | _Calculations:_ - y=1 (total=16): A: 5/16=.313 → 31.3%; B: 10/16=.625 → 62.5%; C: 1/16=.063 → 6.3% - y=2 (total=14): A: /14=.%; B: 2/14≈.143 → 14.3%; C: 12/14≈.857 → 85.7% --- ## Step 5: Relationship Between x and y ### (d) Relationship - **Category A values for x are**: highest for y=1 - **Category B values for x are**: highest for y=1 - **Category C values for x are**: highest for y=2 #### Conclusion: - **A and B** are more likely associated with **y=1** - **C** is strongly associated with **y=2** --- ## **Final Answers** ### (a) Crosstabulation Table | | y=1 | y=2 | Total | |-----|-----|-----|-------| | A | 5 | | 5 | | B | 10 | 2 | 12 | | C | 1 | 12 | 13 | |**Total**| 16 | 14 | 30 | --- ### (b) Row Percentages | | y=1 | y=2 | Total | |-----|---------|---------|-------| | A | 100.% | .% | 5 | | B | 83.3% | 16.7% | 12 | | C | 7.7% | 92.3% | 13 | --- ### (c) Column Percentages | | y=1 | y=2 | Total | |-----|---------|---------|-------| | A | 31.3% | .% | 5 | | B | 62.5% | 14.3% | 12 | | C | 6.3% | 85.7% | 13 | --- ### (d) Relationship - **Category A values for x are**: highest for y=1 - **Category B values for x are**: highest for y=1 - **Category C values for x are**: highest for y=2 --- **Summary:** There is a clear relationship: x = C is most likely to occur when y = 2, while x = A or B are more likely when y = 1.

# Step-by-Step Solution ## Step 1: Tally the Counts for Each (x, y) Pair We begin by identifying the counts of observations for each combination of the categories \( x \) (A, B, C) and \( y \) (1, 2). ### Tally Table | x | y=1 | y=2 | |---|-----|-----| | A | 5 | 0 | | B | 10 | 2 | | C | 1 | 12 | ### Explanation - For each category \( x \): - **A** had 5 occurrences with \( y=1 \) and none with \( y=2 \). - **B** had 10 occurrences with \( y=1 \) and 2 with \( y=2 \). - **C** had 1 occurrence with \( y=1 \) and 12 with \( y=2 \). ## Step 2: Crosstabulation Table Now, we will organize these counts into a crosstabulation table, including the totals for each row and column. ### (a) Crosstabulation | | y=1 | y=2 | Total | |-----|-----|-----|-------| | A | 5 | 0 | 5 | | B | 10 | 2 | 12 | | C | 1 | 12 | 13 | |**Total**| 16 | 14 | 30 | ### Explanation - The total number of observations for \( y=1 \) is 16, and for \( y=2 \) is 14, which sums to 30, confirming the table is complete. ## Step 3: Row Percentages Next, we compute the percentages of each row, which reflect the distribution of \( y \) values within each category \( x \). ### (b) Row Percentages | | y=1 | y=2 | Total | |-----|---------|---------|-------| | A | 100.0% | 0.0% | 5 | | B | 83.3% | 16.7% | 12 | | C | 7.7% | 92.3% | 13 | ### Explanation - **A**: 5 out of 5 → \( 100\% \) for \( y=1 \). - **B**: \( \frac{10}{12} \times 100 \approx 83.3\% \) for \( y=1 \) and \( \frac{2}{12} \times 100 \approx 16.7\% \) for \( y=2 \). - **C**: \( \frac{1}{13} \times 100 \approx 7.7\% \) for \( y=1 \) and \( \frac{12}{13} \times 100 \approx 92.3\% \) for \( y=2 \). ## Step 4: Column Percentages Now, we calculate the percentages for each column, which illustrate the distribution of categories \( x \) within each \( y \) value. ### (c) Column Percentages | | y=1 | y=2 | Total | |-----|---------|---------|-------| | A | 31.3% | 0.0% | 5 | | B | 62.5% | 14.3% | 12 | | C | 6.3% | 85.7% | 13 | |**Total**| 100% | 100% | 30 | ### Explanation - **y=1**: - \( A: \frac{5}{16} \times 100 \approx 31.3\% \) - \( B: \frac{10}{16} \times 100 \approx 62.5\% \) - \( C: \frac{1}{16} \times 100 \approx 6.3\% \) - **y=2**: - \( A: \frac{0}{14} = 0.0\% \) - \( B: \frac{2}{14} \times 100 \approx 14.3\% \) - \( C: \frac{12}{14} \times 100 \approx 85.7\% \) ## Step 5: Relationship Between x and y We now analyze the relationships between the categories \( x \) and \( y \). ### (d) Relationship - **Category A values for x are**: highest for \( y=1 \). - **Category B values for x are**: highest for \( y=1 \). - **Category C values for x are**: highest for \( y=2 \). ### Conclusion - Categories \( A \) and \( B \) are more likely associated with \( y=1 \). - Category \( C \) shows a strong association with \( y=2 \). ## Final Summary - **Crosstabulation** shows how the counts are distributed across different categories. - **Row Percentages** indicate the proportion of each \( y \) within each \( x \). - **Column Percentages** reflect the proportion of each \( x \) within each \( y \). This analysis helps to understand the relationship between categorical variables \( x \) and \( y \), highlighting the associations clearly.