To work out an estimate of the mean garden area, let's go through the problem step-by-step.
1. Interpreting the information:
- We know the frequencies for the ranges [tex]\(0 < x \leq 10\)[/tex], [tex]\(10 < x \leq 20\)[/tex], and [tex]\(40 < x \leq 50\)[/tex].
- The frequencies are:
- [tex]\(0 < x \leq 10\)[/tex]: 16
- [tex]\(10 < x \leq 20\)[/tex]: 44
- [tex]\(40 < x \leq 50\)[/tex]: 4
2. Finding the frequencies for the unknown intervals:
- We are told that the number of responses in the interval [tex]\(20 < x \leq 30\)[/tex] is twice that in the interval [tex]\(30 < x \leq 40\)[/tex].
- Let the frequency for the interval [tex]\(30 < x \leq 40\)[/tex] be [tex]\(x\)[/tex].
- Then, the frequency for the interval [tex]\(20 < x \leq 30\)[/tex] will be [tex]\(2x\)[/tex].
3. Creating an equation for the total number of responses:
- The total number of responses is 100.
- Therefore, we can set up the equation:
[tex]\[
16 + 44 + 2x + x + 4 = 100
\][/tex]
Simplify:
[tex]\[
64 + 3x = 100
\][/tex]
Solve for [tex]\(x\)[/tex]:
[tex]\[
3x = 36 \implies x = 12
\][/tex]
- Thus, the frequency for the interval [tex]\(30 < x \leq 40\)[/tex] is 12, and for the interval [tex]\(20 < x \leq 30\)[/tex] is [tex]\(2 \times 12 = 24\)[/tex].
4. Calculating midpoints for each interval:
- For the interval [tex]\(0 < x \leq 10\)[/tex], midpoint = [tex]\(\frac{0 + 10}{2} = 5\)[/tex]
- For the interval [tex]\(10 < x \leq 20\)[/tex], midpoint = [tex]\(\frac{10 + 20}{2} = 15\)[/tex]
- For the interval [tex]\(20 < x \leq 30\)[/tex], midpoint = [tex]\(\frac{20 + 30}{2} = 25\)[/tex]
- For the interval [tex]\(30 < x \leq 40\)[/tex], midpoint = [tex]\(\frac{30 + 40}{2} = 35\)[/tex]
- For the interval [tex]\(40 < x \leq 50\)[/tex], midpoint = [tex]\(\frac{40 + 50}{2} = 45\)[/tex]
5. Calculating the total frequency-weighted area:
- Sum of all frequencies [tex]\(= 16 + 44 + 24 + 12 + 4 = 100\)[/tex]
- Total area calculated using midpoints and frequencies:
[tex]\[
(5 \times 16) + (15 \times 44) + (25 \times 24) + (35 \times 12) + (45 \times 4) = 80 + 660 + 600 + 420 + 180 = 1940
\][/tex]
6. Calculating the mean garden area:
- Mean [tex]\(= \frac{\text{total area}}{\text{total frequency}} = \frac{1940}{100} = 19.4\)[/tex]
Hence, the estimated mean garden area is [tex]\(\boxed{19.4 \ \text{m}^2}\)[/tex].
