To find the estimated mean height of a plant using the given frequency table, we follow these steps:
1. Identify the Midpoints of Each Class Interval:
- For the interval [tex]\(0 \leqslant h < 10\)[/tex], the midpoint is [tex]\(\frac{0 + 10}{2} = 5 \, cm\)[/tex].
- For the interval [tex]\(10 \leqslant h < 20\)[/tex], the midpoint is [tex]\(\frac{10 + 20}{2} = 15 \, cm\)[/tex].
- For the interval [tex]\(20 \leqslant h < 30\)[/tex], the midpoint is [tex]\(\frac{20 + 30}{2} = 25 \, cm\)[/tex].
- For the interval [tex]\(30 \leqslant h < 40\)[/tex], the midpoint is [tex]\(\frac{30 + 40}{2} = 35 \, cm\)[/tex].
- For the interval [tex]\(40 \leqslant h < 50\)[/tex], the midpoint is [tex]\(\frac{40 + 50}{2} = 45 \, cm\)[/tex].
- For the interval [tex]\(50 \leqslant h < 60\)[/tex], the midpoint is [tex]\(\frac{50 + 60}{2} = 55 \, cm\)[/tex].
2. Calculate the Weighted Sum of the Midpoints: This is done by multiplying each midpoint by the frequency of its class interval and then summing these products.
- [tex]\(5 \, cm \times 1 = 5\)[/tex]
- [tex]\(15 \, cm \times 4 = 60\)[/tex]
- [tex]\(25 \, cm \times 7 = 175\)[/tex]
- [tex]\(35 \, cm \times 2 = 70\)[/tex]
- [tex]\(45 \, cm \times 3 = 135\)[/tex]
- [tex]\(55 \, cm \times 3 = 165\)[/tex]
When we sum these values, we get:
[tex]\[
5 + 60 + 175 + 70 + 135 + 165 = 610
\][/tex]
3. Calculate the Total Frequency: Add up the frequencies for all class intervals.
[tex]\[
1 + 4 + 7 + 2 + 3 + 3 = 20
\][/tex]
4. Compute the Mean Height: Divide the weighted sum by the total frequency.
[tex]\[
\text{Mean height} = \frac{610}{20} = 30.5 \, cm
\][/tex]
Thus, the estimated mean height of the plants is [tex]\(30.5 \, cm\)[/tex].
