Question

which value for y in the table would be least likely to indicate an association between the variables?
0.06
0.24
0.69
1.0
a b total
c x 0.25 g
d y 0.68 h
e z 0.07 j
total 1.0 1.0 1.0

Explanation:

Step1: Recall the concept of independence

If two variables are independent, the joint - probabilities follow the rule that the probability of the intersection of two events is the product of their marginal probabilities. In a two - way table, when there is no association (i.e., independence), the values in the cells should be consistent with the marginal totals.

Step2: Analyze the marginal totals

The marginal total for column B is 1.0. The values in the rows for column B are 0.25, 0.68, and 0.07. The marginal total for row D is \(H = Y + 0.68\). The marginal total for column A is 1.0.
If there is no association, the distribution of values in column A should be similar to the distribution of values in column B across the rows. When \(Y = 0.24\), the distribution of values in row C (\(X\) and 0.25) and row D (\(Y\) and 0.68) is more in line with what we would expect if the variables were independent.
For example, if we consider the proportion of values in each row for the two columns, when \(Y = 0.24\), the ratio of values in row C and row D for column A and column B is more balanced compared to other values of \(Y\).
If \(Y = 0.06\), the difference in proportions between the rows for the two columns is large. If \(Y=0.69\) or \(Y = 1.0\), it will also create a large imbalance in the distribution of values between the two columns across the rows.

Answer:

0.24