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Sagot :
To solve this problem, we need to construct a relative frequency distribution table for the given data on per capita disposable income in 25 cities.
### Step-by-Step Solution:
#### Step 1: Identify the Classes
The first class limit is given as [tex]$30,000 and the class width is $[/tex]6,000. So, we break down the income ranges as follows:
- Class 1: [tex]$30,000 - $[/tex]35,999
- Class 2: [tex]$36,000 - $[/tex]41,999
- Class 3: [tex]$42,000 - $[/tex]47,999
- Class 4: [tex]$48,000 - $[/tex]53,999
#### Step 2: Count the Frequencies
We count how many data points fall into each class:
- Class 1 ([tex]$30,000 - $[/tex]35,999): 14 data points
- Class 2 ([tex]$36,000 - $[/tex]41,999): 10 data points
- Class 3 ([tex]$42,000 - $[/tex]47,999): 0 data points
- Class 4 ([tex]$48,000 - $[/tex]53,999): 1 data point
This directly matches the information provided in the question:
[tex]\[ \begin{tabular}{|c|c|} \hline Class & Frequency \\ \hline $30,000-35,999$ & 14 \\ $36,000-41,999$ & 10 \\ $42,000-47,999$ & 0 \\ $48,000-53,999$ & 1 \\ \hline \end{tabular} \][/tex]
#### Step 3: Calculate Relative Frequencies
The relative frequency for each class is calculated by dividing the frequency of each class by the total number of data points, which is 25.
[tex]\[ \text{Relative Frequency} = \frac{\text{Frequency}}{\text{Total Data Points}} \][/tex]
So, for each class:
- Class 1: [tex]\( \frac{14}{25} = 0.56 \)[/tex]
- Class 2: [tex]\( \frac{10}{25} = 0.40 \)[/tex]
- Class 3: [tex]\( \frac{0}{25} = 0.00 \)[/tex]
- Class 4: [tex]\( \frac{1}{25} = 0.04 \)[/tex]
#### Step 4: Construct the Relative Frequency Distribution Table
The relative frequency distribution table will be:
[tex]\[ \begin{tabular}{|c|c|} \hline Class & Relative Frequency \\ \hline $30,000-35,999$ & 0.56 \\ $36,000-41,999$ & 0.40 \\ $42,000-47,999$ & 0.00 \\ $48,000-53,999$ & 0.04 \\ \hline \end{tabular} \][/tex]
### Conclusion:
The final relative frequency distribution table is:
[tex]\[ \begin{tabular}{|c|c|} \hline Class & Relative Frequency \\ \hline $30,000-35,999$ & 0.56 \\ $36,000-41,999$ & 0.40 \\ $42,000-47,999$ & 0.00 \\ $48,000-53,999$ & 0.04 \\ \hline \end{tabular} \][/tex]
### Step-by-Step Solution:
#### Step 1: Identify the Classes
The first class limit is given as [tex]$30,000 and the class width is $[/tex]6,000. So, we break down the income ranges as follows:
- Class 1: [tex]$30,000 - $[/tex]35,999
- Class 2: [tex]$36,000 - $[/tex]41,999
- Class 3: [tex]$42,000 - $[/tex]47,999
- Class 4: [tex]$48,000 - $[/tex]53,999
#### Step 2: Count the Frequencies
We count how many data points fall into each class:
- Class 1 ([tex]$30,000 - $[/tex]35,999): 14 data points
- Class 2 ([tex]$36,000 - $[/tex]41,999): 10 data points
- Class 3 ([tex]$42,000 - $[/tex]47,999): 0 data points
- Class 4 ([tex]$48,000 - $[/tex]53,999): 1 data point
This directly matches the information provided in the question:
[tex]\[ \begin{tabular}{|c|c|} \hline Class & Frequency \\ \hline $30,000-35,999$ & 14 \\ $36,000-41,999$ & 10 \\ $42,000-47,999$ & 0 \\ $48,000-53,999$ & 1 \\ \hline \end{tabular} \][/tex]
#### Step 3: Calculate Relative Frequencies
The relative frequency for each class is calculated by dividing the frequency of each class by the total number of data points, which is 25.
[tex]\[ \text{Relative Frequency} = \frac{\text{Frequency}}{\text{Total Data Points}} \][/tex]
So, for each class:
- Class 1: [tex]\( \frac{14}{25} = 0.56 \)[/tex]
- Class 2: [tex]\( \frac{10}{25} = 0.40 \)[/tex]
- Class 3: [tex]\( \frac{0}{25} = 0.00 \)[/tex]
- Class 4: [tex]\( \frac{1}{25} = 0.04 \)[/tex]
#### Step 4: Construct the Relative Frequency Distribution Table
The relative frequency distribution table will be:
[tex]\[ \begin{tabular}{|c|c|} \hline Class & Relative Frequency \\ \hline $30,000-35,999$ & 0.56 \\ $36,000-41,999$ & 0.40 \\ $42,000-47,999$ & 0.00 \\ $48,000-53,999$ & 0.04 \\ \hline \end{tabular} \][/tex]
### Conclusion:
The final relative frequency distribution table is:
[tex]\[ \begin{tabular}{|c|c|} \hline Class & Relative Frequency \\ \hline $30,000-35,999$ & 0.56 \\ $36,000-41,999$ & 0.40 \\ $42,000-47,999$ & 0.00 \\ $48,000-53,999$ & 0.04 \\ \hline \end{tabular} \][/tex]
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