To determine which value of [tex]\( Y \)[/tex] in the table would be least likely to indicate an association between the variables, we need to understand the concept of association in a contingency table.
An association between two categorical variables means that the distribution of one variable depends on the distribution of the other. Conversely, no association implies that the variables are independent, and the distribution of one variable is not influenced by the other variable.
Here's a step-by-step explanation of why the value [tex]\( 0.24 \)[/tex] is least likely to indicate an association:
1. Understanding the Marginal Totals and Existing Data:
The contingency table provides the joint distribution of two categorical variables [tex]\( A \)[/tex] and [tex]\( B \)[/tex] across different categories ([tex]\( C \)[/tex], [tex]\( D \)[/tex], and [tex]\( E \)[/tex]). The marginal totals (total sums for each row and column) help us understand expected frequencies if there were no association:
[tex]\[
\begin{array}{|c|c|c|c|}
\cline { 2 - 4 } \multicolumn{1}{c|}{} & A & B & \text{Total} \\
\hline C & X & 0.25 & G \\
\hline D & Y & 0.68 & H \\
\hline E & Z & 0.07 & J \\
\hline \text{Total} & 1.0 & 1.0 & 1.0 \\
\hline
\end{array}
\][/tex]
2. Expected Distributions Without Association:
If there is no association between the variables, the expected value for each cell should equal the product of the corresponding row and column totals. Given the marginal totals, let's calculate what we would expect for [tex]\( Y \)[/tex]:
The row totals for each category are [tex]\( G + H + J = 1.0 \)[/tex]. Similarly, the column totals are [tex]\( A + B = 1.0 \)[/tex] each.
3. Comparing Given Values to Independence:
We can compare the given values for [tex]\( Y \)[/tex] (0.06, 0.24, 0.69, 1.0) to the theoretical distribution. A value that falls closer to the expected value without strong deviation would indicate lesser association.
Among the provided options, [tex]\( 0.24 \)[/tex] falls closer towards the expected values in a scenario of no strong deviation or association. Thus:
Conclusion:
The value for [tex]\( Y \)[/tex] that would be least likely to indicate an association between the variables would be [tex]\( 0.24 \)[/tex]. This value suggests a distribution less deviated from the expected, indicating less likelihood of association.
