To estimate the mean height of a plant given the table of heights and their frequencies, we can follow these steps:
1. Find the midpoints for each height range:
- For the range [tex]\(0 \leqslant h < 10\)[/tex], the midpoint is [tex]\(\frac{0 + 10}{2} = 5.0\)[/tex].
- For the range [tex]\(10 \leqslant h < 20\)[/tex], the midpoint is [tex]\(\frac{10 + 20}{2} = 15.0\)[/tex].
- For the range [tex]\(20 \leqslant h < 30\)[/tex], the midpoint is [tex]\(\frac{20 + 30}{2} = 25.0\)[/tex].
- For the range [tex]\(30 \leqslant h < 40\)[/tex], the midpoint is [tex]\(\frac{30 + 40}{2} = 35.0\)[/tex].
- For the range [tex]\(40 \leqslant h < 50\)[/tex], the midpoint is [tex]\(\frac{40 + 50}{2} = 45.0\)[/tex].
- For the range [tex]\(50 \leqslant h < 60\)[/tex], the midpoint is [tex]\(\frac{50 + 60}{2} = 55.0\)[/tex].
Thus, the list of midpoints is [tex]\([5.0, 15.0, 25.0, 35.0, 45.0, 55.0]\)[/tex].
2. Calculate the total number of plants:
- Sum of frequencies = [tex]\(1 + 4 + 7 + 2 + 3 + 3 = 20\)[/tex].
3. Calculate the weighted sum of the midpoints:
- For each midpoint, multiply it by the corresponding frequency and then sum these products:
- [tex]\(5.0 \times 1 = 5.0\)[/tex]
- [tex]\(15.0 \times 4 = 60.0\)[/tex]
- [tex]\(25.0 \times 7 = 175.0\)[/tex]
- [tex]\(35.0 \times 2 = 70.0\)[/tex]
- [tex]\(45.0 \times 3 = 135.0\)[/tex]
- [tex]\(55.0 \times 3 = 165.0\)[/tex]
- Sum of these products = [tex]\(5.0 + 60.0 + 175.0 + 70.0 + 135.0 + 165.0 = 610.0\)[/tex].
4. Calculate the mean height:
- Mean height = [tex]\(\frac{\text{Weighted sum of midpoints}}{\text{Total number of plants}} = \frac{610.0}{20} = 30.5\)[/tex].
Thus, the estimated mean height of a plant is [tex]\(30.5\)[/tex] cm.
