Question

# Here are the row and column totals for a two-way table with two rows and two columns: \begin{array}{ll|r} a & b & 50 \ c & d & 50 \ \hline 60 & 40 & 1

Two-way tables

Here are the row and column totals for a two-way table with two rows and two columns:$$\begin{array}{ll|r} a & b & 50 \\ c & d & 50 \\ \hline 60 & 40 & 100 \end{array}$$ Find two different sets of counts a, b, c, and d for the body of the table that give these same totals. This shows that the relationship between two variables cannot be obtained from the two individual distributions of the variables.

2021-05-14

Given:
$$\begin{array}{ll|r} a & b & 50 \\ c & d & 50 \\ \hline 60 & 40 & 100 \end{array}$$
We then need to find a set of values a, 6,c and d such that row totals are 50 and the column totals are 60 and 40 respectively. This then implies that the following equations need to be satisfied:
a+b=50
c+d=50
a+c=60
b+d=40
One possible set of values for a, b, c and d is then a = 30, b = 30, c = 20 and d= 20.
$$\begin{array}{ll|r} 30 & 30 & 50 \\ 20 & 20 & 50 \\ \hline 60 & 40 & 100 \end{array}$$
Another possible set of values fora, 8, c and dis then a = 40, b = 10, c = 30 and d = 50.
$$\begin{array}{ll|r} 40 & 10 & 50 \\ 20 & 30 & 50 \\ \hline 60 & 40 & 100 \end{array}$$