To determine whether there is an association between shirt size and hat size, we will analyze the conditional relative frequencies given in the table:
[tex]\[
\begin{array}{|c|c|c|c|}
\hline
& \text{Child-sized Hat} & \text{Adult-sized Hat} & \text{Total} \\
\hline
\text{Medium Shirt} & \approx 0.67 & \approx 0.33 & 1.0 \\
\hline
\text{Large Shirt} & 0.2 & 0.8 & 1.0 \\
\hline
\text{Total} & 0.48 & 0.52 & 1.0 \\
\hline
\end{array}
\][/tex]
1. Analysis of Conditional Relative Frequencies:
For children with a Medium Shirt size:
- About 67% ([tex]$\approx 0.67$[/tex]) have a Child-sized Hat.
- About 33% ([tex]$\approx 0.33$[/tex]) have an Adult-sized Hat.
For children with a Large Shirt size:
- 20% (0.2) have a Child-sized Hat.
- 80% (0.8) have an Adult-sized Hat.
2. Comparison for Association Determination:
To assess whether there is an association, we compare the percentages for each hat size across different shirt sizes:
- For the Child-sized Hats:
- Medium Shirt: [tex]$\approx 0.67$[/tex]
- Large Shirt: [tex]$0.2$[/tex]
Here we see a noticeable difference: [tex]$0.67$[/tex] is not similar to [tex]$0.2$[/tex].
- For the Adult-sized Hats:
- Medium Shirt: [tex]$\approx 0.33$[/tex]
- Large Shirt: [tex]$0.8$[/tex]
Here we also see a significant difference: [tex]$0.33$[/tex] is not similar to [tex]$0.8$[/tex].
3. Conclusion:
The values of [tex]$0.8$[/tex] and [tex]$0.33$[/tex] show a significant difference for the Adult-sized Hat category when comparing Medium and Large shirts. Similarly, the values of [tex]$0.67$[/tex] and [tex]$0.2$[/tex] show a significant difference for the Child-sized Hat category when comparing Medium and Large shirts. These differences indicate a likely association between shirt size and hat size.
Given these observations, the most appropriate conclusion is:
There is likely an association because 0.8 is not similar to [tex]$\approx 0.33$[/tex].
Thus, the correct answer is:
There is likely an association because 0.8 is not similar to [tex]$\approx 0.33$[/tex].
