Questions: Construct a relative frequency marginal distribution for the given contingency table. Round values to the nearest thousandth. x1 x2 x3 -------------------- y1 20 10 25 y2 60 35 45

Solution Steps

To construct a relative frequency marginal distribution from the given contingency table, follow these steps:

1. Calculate the total sum of all the values in the table.
2. For each cell, divide the cell value by the total sum to get the relative frequency.
3. Sum the relative frequencies for each row and each column to get the marginal distributions.
4. Round the values to the nearest thousandth.

The total sum of all the values in the contingency table is calculated as follows: \[ \text{Total Sum} = 20 + 10 + 25 + 60 + 35 + 45 = 195 \]

Each cell value is divided by the total sum to obtain the relative frequency: \[ \begin{array}{l|lll} & x_{1} & x_{2} & x_{3} \\ \hline y_{1} & \frac{20}{195} & \frac{10}{195} & \frac{25}{195} \\ y_{2} & \frac{60}{195} & \frac{35}{195} & \frac{45}{195} \\ \end{array} \] This results in the relative frequency table: \[ \begin{array}{l|lll} & x_{1} & x_{2} & x_{3} \\ \hline y_{1} & 0.103 & 0.051 & 0.128 \\ y_{2} & 0.308 & 0.179 & 0.231 \\ \end{array} \]

The sum of the relative frequencies for each row is calculated: \[ \begin{array}{l|l} y_{1} & 0.103 + 0.051 + 0.128 = 0.282 \\ y_{2} & 0.308 + 0.179 + 0.231 = 0.718 \\ \end{array} \]

The sum of the relative frequencies for each column is calculated: \[ \begin{array}{l|lll} x_{1} & 0.103 + 0.308 = 0.410 \\ x_{2} & 0.051 + 0.179 = 0.231 \\ x_{3} & 0.128 + 0.231 = 0.359 \\ \end{array} \]

Final Answer