Looking at the problem, we need to construct a frequency distribution and draw a frequency polygon based on the provided data.

Part (a): Complete the grouped frequency distribution

The data provided (at the top of the image) consists of the following values: $21, 3, 20, 16, 10, 6, 11, 9, 12, 13, 8, 22, 11, 11, 12, 21$

The classes are already defined in the table:

• 2.5 to 6.5
• 6.5 to 10.5
• 10.5 to 14.5
• 14.5 to 18.5
• 18.5 to 22.5

To complete the frequency distribution, we must count how many data points fall into each class.

Counting Data Points for Each Class:

1. 2.5 to 6.5: Data points in this range: 3, 6 → Frequency = 2
2. 6.5 to 10.5: Data points in this range: 9, 10, 8 → Frequency = 3
3. 10.5 to 14.5: Data points in this range: 12, 13, 11, 11, 11, 12 → Frequency = 6
4. 14.5 to 18.5: Data points in this range: 16 → Frequency = 1
5. 18.5 to 22.5: Data points in this range: 20, 21, 22, 21 → Frequency = 4

Thus, the frequency distribution is:

Part (b): Frequency Polygon

In a frequency polygon, we plot the midpoints of each class on the x-axis and their corresponding frequencies on the y-axis. The midpoint of each class is calculated as the average of the lower and upper limits:

• Midpoint of 2.5 to 6.5: $\frac{2.5 + 6.5}{2} = 4.5$
• Midpoint of 6.5 to 10.5: $\frac{6.5 + 10.5}{2} = 8.5$
• Midpoint of 10.5 to 14.5: $\frac{10.5 + 14.5}{2} = 12.5$
• Midpoint of 14.5 to 18.5: $\frac{14.5 + 18.5}{2} = 16.5$
• Midpoint of 18.5 to 22.5: $\frac{18.5 + 22.5}{2} = 20.5$

So, the midpoints and frequencies to plot are:

• (4.5, 2)
• (8.5, 3)
• (12.5, 6)
• (16.5, 1)
• (20.5, 4)

Now, you can use this information to plot the points and connect them in a frequency polygon.

Would you like a more detailed explanation of how to plot the frequency polygon? Here are 5 related questions:

1. How do we decide the class width for a frequency distribution?
2. What are the key differences between a frequency polygon and a histogram?
3. Can a frequency polygon be used for continuous data, and why?
4. How does the choice of class interval affect the shape of the frequency polygon?
5. How can cumulative frequency be used to plot an ogive curve?

Tip: In a frequency polygon, always start and end the graph at the x-axis, which means you should add an extra class before the first and after the last class with zero frequency.