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A department store stocks pants and shorts in the colors black and blue. The relative frequency of their orders for blue pants is half the relative frequency of their orders for black shorts. Which two-way frequency table could represent the data from the store's orders?

A.
\begin{tabular}{|l|c|c|}
\cline { 2 - 3 } \multicolumn{1}{c|}{} & Pants & Shorts \\
\hline Black & 91 & 150 \\
\hline Blue & 44 & 75 \\
\hline
\end{tabular}

B.
\begin{tabular}{|l|c|c|}
\cline { 2 - 3 } \multicolumn{1}{c|}{} & Pants & Shorts \\
\hline Black & 92 & 114 \\
\hline Blue & 57 & 27 \\
\hline
\end{tabular}

C.
\begin{tabular}{|l|c|c|}
\cline { 2 - 3 } \multicolumn{1}{c|}{} & Pants & Shorts \\
\hline Black & 148 & 41 \\
\hline Blue & 82 & 74 \\
\hline
\end{tabular}

D.
\begin{tabular}{|l|c|c|}
\cline { 2 - 3 } \multicolumn{1}{c|}{} & Pants & Shorts \\
\hline Black & 55 & 110 \\
\hline Blue & 78 & 39 \\
\hline
\end{tabular}



Answer :

GinnyAnswer
To determine which two-way frequency table could represent the data from the store's orders given that the relative frequency of orders for blue pants is half the relative frequency of orders for black shorts, we need to check each table. Let's analyze each option one by one.

### Option A:
[tex]\[ \begin{tabular}{|l|c|c|} \cline { 2 - 3 } \multicolumn{1}{c|}{} & Pants & Shorts \\ \hline Black & 91 & 150 \\ \hline Blue & 44 & 75 \\ \hline \end{tabular} \][/tex]
Here, the frequencies are:
- Blue Pants: 44
- Black Shorts: 150

The relative frequency condition is:
[tex]\[ \frac{\text{Blue Pants}}{\text{Black Shorts}} = \frac{44}{150} \][/tex]
[tex]\[ \frac{44}{150} = \frac{22}{75} \neq 0.5 \][/tex]
So, Option A does not meet the condition.

### Option B:
[tex]\[ \begin{tabular}{|l|c|c|} \cline { 2 - 3 } \multicolumn{1}{c|}{} & Pants & Shorts \\ \hline Black & 92 & 114 \\ \hline Blue & 57 & 27 \\ \hline \end{tabular} \][/tex]
Here, the frequencies are:
- Blue Pants: 57
- Black Shorts: 114

The relative frequency condition is:
[tex]\[ \frac{\text{Blue Pants}}{\text{Black Shorts}} = \frac{57}{114} = 0.5 \][/tex]

Since [tex]\( 57 \div 114 = 0.5 \)[/tex], Option B meets the condition.

### Option C:
[tex]\[ \begin{tabular}{|l|c|c|} \cline { 2 - 3 } \multicolumn{1}{c|}{} & Pants & Shorts \\ \hline Black & 148 & 41 \\ \hline Blue & 82 & 74 \\ \hline \end{tabular} \][/tex]
Here, the frequencies are:
- Blue Pants: 82
- Black Shorts: 41

The relative frequency condition is:
[tex]\[ \frac{\text{Blue Pants}}{\text{Black Shorts}} = \frac{82}{41} = 2 \][/tex]

Since [tex]\( \frac{82}{41} = 2 \)[/tex], Option C does not meet the condition.

### Option D:
[tex]\[ \begin{tabular}{|l|c|c|} \cline { 2 - 3 } \multicolumn{1}{c|}{} & Pants & Shorts \\ \hline Black & 55 & 110 \\ \hline Blue & 78 & 39 \\ \hline \end{tabular} \][/tex]
Here, the frequencies are:
- Blue Pants: 78
- Black Shorts: 110

The relative frequency condition is:
[tex]\[ \frac{\text{Blue Pants}}{\text{Black Shorts}} = \frac{78}{110} \approx 0.709 \][/tex]

Since [tex]\( \frac{78}{110} \approx 0.709 \)[/tex], Option D does not meet the condition.

Hence, after evaluating each option, the correct two-way frequency table is Option B.

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