To draw a frequency polygon for the given data, follow these steps:
1. Identify the income ranges and their corresponding frequencies:
[tex]\[
\begin{array}{|c|c|}
\hline \text{Income (thousands)} & \text{Frequency} \\
\hline 0 < I \leq 20 & 40 \\
\hline 20 < I \leq 40 & 75 \\
\hline 40 < I \leq 60 & 64 \\
\hline 60 < I \leq 80 & 20 \\
\hline 80 < I \leq 100 & 1 \\
\hline
\end{array}
\][/tex]
2. Determine the midpoints for each income range:
- For [tex]\(0 < I \leq 20\)[/tex]: Midpoint = [tex]\((0 + 20) / 2 = 10\)[/tex]
- For [tex]\(20 < I \leq 40\)[/tex]: Midpoint = [tex]\((20 + 40) / 2 = 30\)[/tex]
- For [tex]\(40 < I \leq 60\)[/tex]: Midpoint = [tex]\((40 + 60) / 2 = 50\)[/tex]
- For [tex]\(60 < I \leq 80\)[/tex]: Midpoint = [tex]\((60 + 80) / 2 = 70\)[/tex]
- For [tex]\(80 < I \leq 100\)[/tex]: Midpoint = [tex]\((80 + 100) / 2 = 90\)[/tex]
Thus, the midpoints are: 10, 30, 50, 70, and 90.
3. Plot the midpoints vs. frequencies:
The points to be plotted are:
[tex]\[
(10, 40),\, (30, 75),\, (50, 64),\, (70, 20),\, (90, 1)
\][/tex]
4. Draw the frequency polygon:
- Create a set of axes with the income midpoints on the horizontal axis (x-axis) and the frequencies on the vertical axis (y-axis).
- Plot the points [tex]\((10, 40)\)[/tex], [tex]\((30, 75)\)[/tex], [tex]\((50, 64)\)[/tex], [tex]\((70, 20)\)[/tex], and [tex]\((90, 1)\)[/tex].
- Connect the plotted points with straight lines to form the frequency polygon.
Below is a sketch representation of how the frequency polygon should look:
```
Frequency (y-axis)
|
|
|
| |
| |
| |
| | |
| | |
| | |
| |
| | | |
| | | |
| | | | *
| | | | |
| | | | |
| | | | |
|------------------------------------------- Income Midpoints (x-axis)
10 30 50 70 90
```
This is a simplified visual guide. Remember to label your axes and choose appropriate scales for accuracy when you plot the graph on paper or using graphing software.
