To estimate the mean height of the plants, we need to use the midpoint of each height interval and the frequencies provided in the table. Here's a detailed, step-by-step solution:
### Step 1: Determine the Midpoints of each Interval
The midpoint of an interval is calculated as the average of the lower and upper boundaries of that interval.
For the given intervals:
- [tex]\(0 < h < 10\)[/tex]: Midpoint = [tex]\(\frac{0 + 10}{2} = 5\)[/tex]
- [tex]\(10 < h < 20\)[/tex]: Midpoint = [tex]\(\frac{10 + 20}{2} = 15\)[/tex]
- [tex]\(20 < h < 30\)[/tex]: Midpoint = [tex]\(\frac{20 + 30}{2} = 25\)[/tex]
- [tex]\(30 < h < 40\)[/tex]: Midpoint = [tex]\(\frac{30 + 40}{2} = 35\)[/tex]
- [tex]\(40 < h < 50\)[/tex]: Midpoint = [tex]\(\frac{40 + 50}{2} = 45\)[/tex]
- [tex]\(50 < h < 60\)[/tex]: Midpoint = [tex]\(\frac{50 + 60}{2} = 55\)[/tex]
These midpoints represent the average height within each interval.
### Step 2: List the Frequencies for each Interval
From the given data, the frequencies for each interval are:
- [tex]\(0 < h < 10\)[/tex]: Frequency = 1
- [tex]\(10 < h < 20\)[/tex]: Frequency = 4
- [tex]\(20 < h < 30\)[/tex]: Frequency = 7
- [tex]\(30 < h < 40\)[/tex]: Frequency = 2
- [tex]\(40 < h < 50\)[/tex]: Frequency = 3
- [tex]\(50 < h < 60\)[/tex]: Frequency = 3
### Step 3: Calculate the Total Number of Plants
The total number of plants is the sum of the frequencies:
[tex]\[ 1 + 4 + 7 + 2 + 3 + 3 = 20 \][/tex]
### Step 4: Calculate the Weighted Sum of the Midpoints
To find the weighted sum, multiply each midpoint by its corresponding frequency and then sum these products:
[tex]\[
(5 \times 1) + (15 \times 4) + (25 \times 7) + (35 \times 2) + (45 \times 3) + (55 \times 3)
= 5 + 60 + 175 + 70 + 135 + 165 = 610
\][/tex]
### Step 5: Calculate the Estimated Mean Height
The estimated mean height is found by dividing the weighted sum of the midpoints by the total number of plants:
[tex]\[
\text{Mean Height} = \frac{\text{Weighted Sum}}{\text{Total Number of Plants}} = \frac{610}{20} = 30.5
\][/tex]
### Conclusion
The estimate for the mean height of a plant is:
[tex]\[
\boxed{30.5 \text{ cm}}
\][/tex]
