Sure, let's work through how the store owner could convert the frequency table to a conditional relative frequency table by row in detail:
### Step-by-Step Solution
Given the frequency table:
[tex]\[
\begin{array}{|c|c|c|c|}
\hline
& \text{Full Price} & \text{Discounted} & \text{Total} \\
\hline
\text{Month 1} & 49 & 1 & 50 \\
\hline
\text{Month 2} & 120 & 5 & 125 \\
\hline
\text{Month 3} & 101 & 24 & 125 \\
\hline
\text{Total} & 270 & 30 & 300 \\
\hline
\end{array}
\][/tex]
The goal is to convert this to a conditional relative frequency table by row, which involves calculating the relative frequency for each category within each month.
### 1. Month 1 Calculations
For Month 1:
- Total sales: \( 50 \)
- Full Price sales: \( 49 \)
- Discounted sales: \( 1 \)
To find the relative frequency:
[tex]\[ \text{Full Price Relative Frequency} = \frac{49}{50} = 0.98 \][/tex]
[tex]\[ \text{Discounted Relative Frequency} = \frac{1}{50} = 0.02 \][/tex]
Thus, for Month 1:
[tex]\[
\begin{array}{|c|c|c|c|}
\cline { 2 - 4 }
& \text{Full Price} & \text{Discounted} & \text{Total} \\
\hline
\text{Month 1} & 0.98 & 0.02 & 1.0 \\
\hline
\end{array}
\][/tex]
### 2. Month 2 Calculations
For Month 2:
- Total sales: \( 125 \)
- Full Price sales: \( 120 \)
- Discounted sales: \( 5 \)
To find the relative frequency:
[tex]\[ \text{Full Price Relative Frequency} = \frac{120}{125} = 0.96 \][/tex]
[tex]\[ \text{Discounted Relative Frequency} = \frac{5}{125} = 0.04 \][/tex]
Thus, for Month 2:
[tex]\[
\begin{array}{|c|c|c|c|}
\cline { 2 - 4 }
& \text{Full Price} & \text{Discounted} & \text{Total} \\
\hline
\text{Month 2} & 0.96 & 0.04 & 1.0 \\
\hline
\end{array}
\][/tex]
### 3. Month 3 Calculations
For Month 3:
- Total sales: \( 125 \)
- Full Price sales: \( 101 \)
- Discounted sales: \( 24 \)
To find the relative frequency:
[tex]\[ \text{Full Price Relative Frequency} = \frac{101}{125} = 0.808 \][/tex]
[tex]\[ \text{Discounted Relative Frequency} = \frac{24}{125} = 0.192 \][/tex]
Thus, for Month 3:
[tex]\[
\begin{array}{|c|c|c|c|}
\cline { 2 - 4 }
& \text{Full Price} & \text{Discounted} & \text{Total} \\
\hline
\text{Month 3} & 0.808 & 0.192 & 1.0 \\
\hline
\end{array}
\][/tex]
### Overall Calculations
To find overall relative frequencies, sum up for all months:
- Total sales: \( 300 \)
- Full Price sales: \( 270 \)
- Discounted sales: \( 30 \)
To find the relative frequency:
[tex]\[ \text{Full Price Relative Frequency} = \frac{270}{300} = 0.9 \][/tex]
[tex]\[ \text{Discounted Relative Frequency} = \frac{30}{300} = 0.1 \][/tex]
Thus, overall:
[tex]\[
\begin{array}{|c|c|c|c|}
\cline { 2 - 4 }
& \text{Full Price} & \text{Discounted} & \text{Total} \\
\hline
\text{Overall} & 0.9 & 0.1 & 1.0 \\
\hline
\end{array}
\][/tex]
### Final Conditional Relative Frequency Table
In conclusion, we transform the original frequency table into the following conditional relative frequency table:
[tex]\[
\begin{array}{|c|c|c|c|}
\cline { 2 - 4 }
& \text{Full Price} & \text{Discounted} & \text{Total} \\
\hline
\text{Month 1} & 0.98 & 0.02 & 1.0 \\
\hline
\text{Month 2} & 0.96 & 0.04 & 1.0 \\
\hline
\text{Month 3} & 0.808 & 0.192 & 1.0 \\
\hline
\text{Overall} & 0.9 & 0.1 & 1.0 \\
\hline
\end{array}
\][/tex]
This represents the relative frequency of sales for full price and discounted beach towels during each of the three months and overall.
