To find an estimate for the mean height of the plants, we will follow these steps:
1. Identify the height intervals and their respective frequencies:
- [tex]\(0 \leqslant h < 10\)[/tex]: Frequency = 1
- [tex]\(10 \leqslant h < 20\)[/tex]: Frequency = 4
- [tex]\(20 \leqslant h < 30\)[/tex]: Frequency = 7
- [tex]\(30 \leqslant h < 40\)[/tex]: Frequency = 2
- [tex]\(40 \leqslant h < 50\)[/tex]: Frequency = 3
- [tex]\(50 \leqslant h < 60\)[/tex]: Frequency = 3
2. Calculate the midpoints of each height interval. The midpoint of an interval is found by averaging the lower and upper bounds of the interval.
- Midpoint of [tex]\(0 \leqslant h < 10\)[/tex]: [tex]\(\frac{0 + 10}{2} = 5\)[/tex]
- Midpoint of [tex]\(10 \leqslant h < 20\)[/tex]: [tex]\(\frac{10 + 20}{2} = 15\)[/tex]
- Midpoint of [tex]\(20 \leqslant h < 30\)[/tex]: [tex]\(\frac{20 + 30}{2} = 25\)[/tex]
- Midpoint of [tex]\(30 \leqslant h < 40\)[/tex]: [tex]\(\frac{30 + 40}{2} = 35\)[/tex]
- Midpoint of [tex]\(40 \leqslant h < 50\)[/tex]: [tex]\(\frac{40 + 50}{2} = 45\)[/tex]
- Midpoint of [tex]\(50 \leqslant h < 60\)[/tex]: [tex]\(\frac{50 + 60}{2} = 55\)[/tex]
So, the midpoints are: [tex]\(5, 15, 25, 35, 45, 55\)[/tex]
3. Multiply each midpoint by its corresponding frequency and find the total of these products.
- [tex]\(5 \times 1 = 5\)[/tex]
- [tex]\(15 \times 4 = 60\)[/tex]
- [tex]\(25 \times 7 = 175\)[/tex]
- [tex]\(35 \times 2 = 70\)[/tex]
- [tex]\(45 \times 3 = 135\)[/tex]
- [tex]\(55 \times 3 = 165\)[/tex]
The sum of these products is:
[tex]\[
5 + 60 + 175 + 70 + 135 + 165 = 610
\][/tex]
4. Find the total frequency:
[tex]\[
1 + 4 + 7 + 2 + 3 + 3 = 20
\][/tex]
5. Estimate the mean height: The mean is calculated by dividing the total of the products by the total frequency.
[tex]\[
\text{Mean height} = \frac{\text{Total of the products}}{\text{Total frequency}} = \frac{610}{20} = 30.5
\][/tex]
Therefore, the estimated mean height of the plants is 30.5 cm.
