Question

which value for r in the table would most likely indicate an association between the conditional variables?
0.09
0.10
0.13
0.79
a b c total
d 0.12 0.78 0.10 1.0
e r s t 1.0
total u x y 1.0

Explanation:

Step1: Recall independence condition

If two variables are independent, for a two - way table, the joint probabilities satisfy $P(A\cap D)=P(A)\times P(D)$. In terms of the table values, if the variables are independent, the values in the cells should follow a certain pattern. When there is an association, the values deviate from the pattern of independence.
For independence, we would expect the values in the cells to be distributed in a way that the row - column products match. If we assume independence, we can compare the values. Here, we note that the value in the first row and first column is $0.12$. If the variables were independent, we would expect $R$ to be different from what would be predicted by a simple product - rule if the variables were not related.
The closer $R$ is to a value that is far from what would be expected under independence, the more likely there is an association.
If we consider the first row probabilities, and assume a "random" or independent distribution, we note that a value of $R = 0.79$ would create a large deviation from what would be expected if the variables were independent.

Step2: Analyze options

• Option 0.09: This value is close to values that might be expected under a more "even" or independent distribution.
• Option 0.10: Similar to 0.09, it could be part of a distribution that might suggest independence.
• Option 0.13: Also could be in a range that might fit an independent - like distribution.
• Option 0.79: This value is very different from the values in the first row that are in the range of 0.12 and 0.10. A large value for $R$ like 0.79 would most likely indicate that the conditional variables are associated because it creates a significant deviation from what would be expected if the variables were independent.

Answer:

0.79