Answer :
To determine an estimate for the mean height of the plants, we follow these steps:
1. Identify the Midpoints of Each Height Interval:
To find the midpoint of each interval, we take the average of the lower and upper bounds of each height group. The intervals and their midpoints are:
- [tex]\( 0 \leq h < 10 \)[/tex]: Midpoint [tex]\( = \frac{0 + 10}{2} = 5 \)[/tex]
- [tex]\( 10 \leq h < 20 \)[/tex]: Midpoint [tex]\( = \frac{10 + 20}{2} = 15 \)[/tex]
- [tex]\( 20 \leq h < 30 \)[/tex]: Midpoint [tex]\( = \frac{20 + 30}{2} = 25 \)[/tex]
- [tex]\( 30 \leq h < 40 \)[/tex]: Midpoint [tex]\( = \frac{30 + 40}{2} = 35 \)[/tex]
- [tex]\( 40 \leq h < 50 \)[/tex]: Midpoint [tex]\( = \frac{40 + 50}{2} = 45 \)[/tex]
- [tex]\( 50 \leq h < 60 \)[/tex]: Midpoint [tex]\( = \frac{50 + 60}{2} = 55 \)[/tex]
2. List the Frequencies Corresponding to Each Interval:
The frequencies provided in the table are:
- [tex]\( 0 \leq h < 10 \)[/tex]: Frequency [tex]\( = 1 \)[/tex]
- [tex]\( 10 \leq h < 20 \)[/tex]: Frequency [tex]\( = 4 \)[/tex]
- [tex]\( 20 \leq h < 30 \)[/tex]: Frequency [tex]\( = 7 \)[/tex]
- [tex]\( 30 \leq h < 40 \)[/tex]: Frequency [tex]\( = 2 \)[/tex]
- [tex]\( 40 \leq h < 50 \)[/tex]: Frequency [tex]\( = 3 \)[/tex]
- [tex]\( 50 \leq h < 60 \)[/tex]: Frequency [tex]\( = 3 \)[/tex]
3. Calculate the Total Number of Plants:
The total number of plants is the sum of all frequencies:
[tex]\[ \text{Total Number of Plants} = 1 + 4 + 7 + 2 + 3 + 3 = 20 \][/tex]
4. Calculate the Weighted Sum of the Midpoints:
We multiply each midpoint by its corresponding frequency and then sum these products. This gives us the weighted sum:
[tex]\[ \text{Weighted Sum} = (5 \times 1) + (15 \times 4) + (25 \times 7) + (35 \times 2) + (45 \times 3) + (55 \times 3) \][/tex]
[tex]\[ = 5 + 60 + 175 + 70 + 135 + 165 = 610 \][/tex]
5. Calculate the Estimate for the Mean Height:
The mean height is obtained by dividing the weighted sum 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]
Thus, the estimated mean height of the plants is:
[tex]\[ \boxed{30.5 \text{ cm}} \][/tex]
1. Identify the Midpoints of Each Height Interval:
To find the midpoint of each interval, we take the average of the lower and upper bounds of each height group. The intervals and their midpoints are:
- [tex]\( 0 \leq h < 10 \)[/tex]: Midpoint [tex]\( = \frac{0 + 10}{2} = 5 \)[/tex]
- [tex]\( 10 \leq h < 20 \)[/tex]: Midpoint [tex]\( = \frac{10 + 20}{2} = 15 \)[/tex]
- [tex]\( 20 \leq h < 30 \)[/tex]: Midpoint [tex]\( = \frac{20 + 30}{2} = 25 \)[/tex]
- [tex]\( 30 \leq h < 40 \)[/tex]: Midpoint [tex]\( = \frac{30 + 40}{2} = 35 \)[/tex]
- [tex]\( 40 \leq h < 50 \)[/tex]: Midpoint [tex]\( = \frac{40 + 50}{2} = 45 \)[/tex]
- [tex]\( 50 \leq h < 60 \)[/tex]: Midpoint [tex]\( = \frac{50 + 60}{2} = 55 \)[/tex]
2. List the Frequencies Corresponding to Each Interval:
The frequencies provided in the table are:
- [tex]\( 0 \leq h < 10 \)[/tex]: Frequency [tex]\( = 1 \)[/tex]
- [tex]\( 10 \leq h < 20 \)[/tex]: Frequency [tex]\( = 4 \)[/tex]
- [tex]\( 20 \leq h < 30 \)[/tex]: Frequency [tex]\( = 7 \)[/tex]
- [tex]\( 30 \leq h < 40 \)[/tex]: Frequency [tex]\( = 2 \)[/tex]
- [tex]\( 40 \leq h < 50 \)[/tex]: Frequency [tex]\( = 3 \)[/tex]
- [tex]\( 50 \leq h < 60 \)[/tex]: Frequency [tex]\( = 3 \)[/tex]
3. Calculate the Total Number of Plants:
The total number of plants is the sum of all frequencies:
[tex]\[ \text{Total Number of Plants} = 1 + 4 + 7 + 2 + 3 + 3 = 20 \][/tex]
4. Calculate the Weighted Sum of the Midpoints:
We multiply each midpoint by its corresponding frequency and then sum these products. This gives us the weighted sum:
[tex]\[ \text{Weighted Sum} = (5 \times 1) + (15 \times 4) + (25 \times 7) + (35 \times 2) + (45 \times 3) + (55 \times 3) \][/tex]
[tex]\[ = 5 + 60 + 175 + 70 + 135 + 165 = 610 \][/tex]
5. Calculate the Estimate for the Mean Height:
The mean height is obtained by dividing the weighted sum 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]
Thus, the estimated mean height of the plants is:
[tex]\[ \boxed{30.5 \text{ cm}} \][/tex]
