The following data are for 30 observations involving two categorical variables, x and y. The categories for x are A, B, and C; the categories for y are 1 and 2.
Observation	x	y
1	A	1
2	B	1
3	B	1
4	C	2
5	B	1
6	C	2
7	B	1
8	C	2
9	A	1
10	B	1
11	A	1
12	B	1
13	C	2
14	C	2
15	C	2
 
Observation	x	y
16	B	2
17	C	1
18	B	1
19	C	1
20	B	1
21	C	2
22	B	1
23	C	2
24	A	1
25	B	1
26	C	2
27	C	2
28	A	1
29	B	1
30	B	2
(a)
Develop a crosstabulation for the data, with x as the row variable and y as the column variable.
y	Total
1	2
x	A	
5
 
Correct: Your answer is correct.
0
 
Correct: Your answer is correct.
5
 
Correct: Your answer is correct.
B	
11
 
Correct: Your answer is correct.
2
 
Correct: Your answer is correct.
13
 
Correct: Your answer is correct.
C	
2
 
Correct: Your answer is correct.
10
 
Correct: Your answer is correct.
12
 
Correct: Your answer is correct.
Total	
18
 
Correct: Your answer is correct.
12
 
Correct: Your answer is correct.
30
 
Correct: Your answer is correct.
(b)
Compute the row percentages. (Round your answers to one decimal place.)
y	Total
1	2
x	A	
100.0
 
Correct: Your answer is correct.
0.0
 
Correct: Your answer is correct.
100.0
 
Correct: Your answer is correct.
B	
84.6
 
Correct: Your answer is correct.
15.4
 
Correct: Your answer is correct.
100.0
 
Correct: Your answer is correct.
C	
16.7
 
Correct: Your answer is correct.
83.3
 
Correct: Your answer is correct.
100.0
 
Correct: Your answer is correct.
(c)
Compute the column percentages. (Round your answers to one decimal place.)
y
1	2
x	A	
 
Incorrect: Your answer is incorrect.
0.0
 
Correct: Your answer is correct.
B	
 
Incorrect: Your answer is incorrect.
16.7
 
Correct: Your answer is correct.
C	
 
Incorrect: Your answer is incorrect.
83.3
 
Correct: Your answer is correct.
Total	
100
 
Correct: Your answer is correct.
100
 
Correct: Your answer is correct.


Let's walk through the steps for part (c) to compute the **column percentages**, rounding to one decimal place as required. The idea is to express each cell count as a percentage of the total for each column (y = 1 and y = 2).

### Cross-tabulation dataLet's walk through the steps for part (c) to compute the **column percentages**, rounding to one decimal place as required. The idea is to express each cell count as a percentage of the total for each column (y = 1 and y = 2).

### Cross-tabulation 
\[
\begin{array}{c|c|c|c}
 & y = 1 & y = 2 & \text{Total} \\
\hline
x = A & 5 & 0 & 5 \\
x = B & 11 & 2 & 13 \\
x = C & 2 & 10 & 12 \\
\hline
\text{Total} & 18 & 12 & 30 \\
\end{array}
\]

### Column Percentages:
To calculate the column percentages, we compute:

\[
\text{Column Percentage} = \left( \frac{\text{cell count}}{\text{column total}} \right) \times 100
\]

#### For Column \( y = 1 \):
- For \( x = A \): 
\[
\frac{5}{18} \times 100 = 27.8\%
\]
- For \( x = B \): 
\[
\frac{11}{18} \times 100 = 61.1\%
\]
- For \( x = C \): 
\[
\frac{2}{18} \times 100 = 11.1\%
\]

#### For Column \( y = 2 \):
- For \( x = A \): 
\[
\frac{0}{12} \times 100 = 0.0\%
\]
- For \( x = B \): 
\[
\frac{2}{12} \times 100 = 16.7\%
\]
- For \( x = C \): 
\[
\frac{10}{12} \times 100 = 83.3\%
\]

### Final Column Percentage Table:
\[
\begin{array}{c|c|c}
 & y = 1 & y = 2 \\
\hline
x = A & 27.8\% & 0.0\% \\
x = B & 61.1\% & 16.7\% \\
x = C & 11.1\% & 83.3\% \\
\hline
\text{Total} & 100\% & 100\% \\
\end{array}
\]

This table shows the percentage contribution of each category \(x\) to each category \(y\).

Would you like more details or clarification?

Here are 5 related questions to deepen your understanding:
1. How would you compute the row percentages if the total for each row varied?
2. Can you calculate joint probabilities from this data?
3. What insights can be drawn about the relationship between \(x\) and \(y\) based on the percentages?
4. How would the percentages change if the total observations were different?
5. Can we visualize this data using a bar chart or other graphs?

**Tip:** When calculating percentages for categorical data, always ensure you are dividing by the appropriate total (row or column) based on the desired perspective.

The following data are for 30 observations involving two categorical variables, x and y. The categories for x are A, B, and C; the categories for y are 1 and 2.
Observation	 x	 y
1	 A	 1
2	 B	 1
3	 B	 1
4	 C	 2
5	 B	 1
6	 C	 2
7	 B	 1
8	 C	 2
9	 A	 1
10	 B	 1
11	 A	 1
12	 B	 1
13	 C	 2
14	 C	 2
15	 C	 2

Observation	 x	 y
16	 B	 2
17	 C	 1
18	 B	 1
19	 C	 1
20	 B	 1
21	 C	 2
22	 B	 1
23	 C	 2
24	 A	 1
25	 B	 1
26	 C	 2
27	 C	 2
28	 A	 1
29	 B	 1
30	 B	 2
(a)
Develop a crosstabulation for the data, with x as the row variable and y as the column variable.
y	 Total
1	 2
x	 A	 5	 0	 5
x	 B	 11	 2	 13
x	 C	 2	 10	 12
Total	 18	 12	 30
(b)
Compute the row percentages.
y	 Total
1	 2
x	 A	 100.0	 0.0	 100.0
x	 B	 84.6	 15.4	 100.0
x	 C	 16.7	 83.3	 100.0
(c)
Compute the column percentages.
y	 1	 2
x	 A	 27.8%	 0.0%
x	 B	 61.1%	 16.7%
x	 C	 11.1%	 83.3%
Total	 100%	 100%

Learn how to create a crosstabulation and compute row and column percentages for 30 observations involving two categorical variables. This problem covers essential concepts in statistics, including crosstabulation and percentage calculations. Suitable for Grades 9-12 students.

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