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The conditional relative frequency table was generated by row using frequency table data comparing the hat size and shirt size of children on a baseball team.

Hat Size and Shirt Size

\begin{tabular}{|c|c|c|c|}
\cline { 2 - 4 }
\multicolumn{1}{c|}{} & Child-sized Hat & Adult-sized Hat & Total \\
\hline Medium Shirt & [tex]$\approx 0.67$[/tex] & [tex]$\approx 0.33$[/tex] & 1.0 \\
\hline Large Shirt & 0.2 & 0.8 & 1.0 \\
\hline Total & 0.48 & 0.52 & 1.0 \\
\hline
\end{tabular}

The coach attempts to determine an association between shirt size and hat size. Which is most likely true?

A. An association cannot be determined because 0.48 is similar to 0.52.

B. An association cannot be determined because the sum of each column is not 1.0.

C. There is likely an association because 0.8 is not similar to 0.2.

D. There is likely an association because 0.8 is not similar to [tex]$\approx 0.33$[/tex].



Answer :

GinnyAnswer
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].

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