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To estimate the mean garden area based on the given frequency table, we first need to fill in the missing frequency values for the intervals 20 < x ≤ 30 and 30 < x ≤ 40.

From the information provided, it states that:

- The number of responses in the interval 20 < x ≤ 30 is twice the number of responses in the interval 30 < x ≤ 40.

Given that the frequency for 30 < x ≤ 40 is 7, we can calculate the frequency for 20 < x ≤ 30:

\[
\text{Frequency for } 20 < x \leq 30 = 2 \times 7 = 14
\]

Now that we have all the frequencies, the complete table is:

| Area \(x\) (m²)   | Frequency |
|-------------------|-----------|
| 0 < x ≤ 10        | 12        |
| 10 < x ≤ 20       | 42        |
| 20 < x ≤ 30       | 14        |
| 30 < x ≤ 40       | 7         |
| 40 < x ≤ 50       | 7         |

Next, we'll estimate the mean using the formula:

\[
\text{Mean} = \frac{\sum (\text{Midpoint of class} \times \text{Frequency})}{\text{Total Frequency}}
\]

We can calculate the midpoint for each class, multiply by the corresponding frequency, and sum these values. Let's work it out.

The estimated mean garden area is approximately 19.5 m² (to 1 decimal place).

Do you have any questions or need more details on how this was calculated? Here are some related questions to explore further:

1. How do we interpret the midpoints in this type of problem?
2. What would happen to the mean if the frequency of a class interval changed?
3. Can we use a different method to estimate the median garden area?
4. How do we estimate the mode for grouped data?
5. How does the range of the data affect the estimation of the mean?

**Tip:** In grouped data problems, using midpoints of each interval gives a reasonable approximation when calculating the mean.

100 people answered the question 'What is the area of your garden?' There were twice as many responses in the interval 20 < x ≤ 30 as there were in the interval 30 < x ≤ 40. Work out an estimate of the mean garden area. Give your answer to 1 decimal place.

Solve the problem of estimating the mean garden area based on a frequency distribution table. Learn how to use midpoints and frequencies to calculate the estimated mean garden area for 100 participants. Suitable for Grades 9-10.

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