1. **State the problem:** We have a 2x3 data table with values: $$\begin{array}{c|ccc} & x_1 & x_2 & x_3 \\ \hline y_1 & 20 & 25 & 30 \\ y_2 & 30 & 15 & 50 \end{array}$$ We need to construct the frequency marginal distribution for this data. 2. **Understand marginal distribution:** A marginal distribution sums the frequencies across rows or columns to show totals for each category. 3. **Calculate row sums (marginal distribution for y):** $$\text{Sum for } y_1 = 20 + 25 + 30 = 75$$ $$\text{Sum for } y_2 = 30 + 15 + 50 = 95$$ 4. **Calculate column sums (marginal distribution for x):** $$\text{Sum for } x_1 = 20 + 30 = 50$$ $$\text{Sum for } x_2 = 25 + 15 = 40$$ $$\text{Sum for } x_3 = 30 + 50 = 80$$ 5. **Interpretation:** The marginal distribution for rows (y) is \(\{75, 95\}\) and for columns (x) is \(\{50, 40, 80\}\). This means, for example, that the total frequency for category \(y_1\) is 75, and for \(x_3\) is 80. **Final answer:** - Marginal distribution for rows (y): \(y_1=75, y_2=95\) - Marginal distribution for columns (x): \(x_1=50, x_2=40, x_3=80\)