To determine an estimate for the mean height of the plants, we can follow the steps below:
1. Identify the midpoints of each height interval: The midpoint of an interval gives a representative value for that interval, which helps in estimating the mean height. The midpoint can be calculated as the average of the lower and upper bounds of the interval.
2. Multiply each midpoint by the frequency of the interval: This step helps to weight the midpoints by how many plants fall in each interval.
3. Sum these weighted midpoints: This gives us a cumulative sum that considers the frequency of plants at each height.
4. Sum the frequencies: This gives us the total number of plants.
5. Divide the cumulative sum of midpoints by the total number of plants: This will give us the estimated mean height of the plants.
Let's go through these steps in detail:
Step 1: Calculate the midpoints.
- For the interval [tex]\(0 < h < 10\)[/tex], the midpoint is [tex]\(\frac{0 + 10}{2} = 5.0\)[/tex].
- For the interval [tex]\(10 < h < 20\)[/tex], the midpoint is [tex]\(\frac{10 + 20}{2} = 15.0\)[/tex].
- For the interval [tex]\(20 < h < 30\)[/tex], the midpoint is [tex]\(\frac{20 + 30}{2} = 25.0\)[/tex].
- For the interval [tex]\(30 < h < 40\)[/tex], the midpoint is [tex]\(\frac{30 + 40}{2} = 35.0\)[/tex].
- For the interval [tex]\(40 < h < 50\)[/tex], the midpoint is [tex]\(\frac{40 + 50}{2} = 45.0\)[/tex].
- For the interval [tex]\(50 < h < 60\)[/tex], the midpoint is [tex]\(\frac{50 + 60}{2} = 55.0\)[/tex].
So, the midpoints are [tex]\(5.0, 15.0, 25.0, 35.0, 45.0, 55.0\)[/tex].
Step 2: Multiply each midpoint by the frequency.
- [tex]\(5.0 \times 1 = 5.0\)[/tex]
- [tex]\(15.0 \times 4 = 60.0\)[/tex]
- [tex]\(25.0 \times 6 = 150.0\)[/tex]
- [tex]\(35.0 \times 3 = 105.0\)[/tex]
- [tex]\(45.0 \times 3 = 135.0\)[/tex]
- [tex]\(55.0 \times 3 = 165.0\)[/tex]
Step 3: Sum these products.
Summing these gives [tex]\(5.0 + 60.0 + 150.0 + 105.0 + 135.0 + 165.0 = 620.0\)[/tex].
Step 4: Sum the frequencies.
The sum of the frequencies is [tex]\(1 + 4 + 6 + 3 + 3 + 3 = 20\)[/tex].
Step 5: Divide the cumulative sum by the total number of plants to find the mean height.
The mean height is [tex]\(\frac{620.0}{20} = 31.0 \, \text{cm}\)[/tex].
Therefore, the estimated mean height of the plants is [tex]\(31.0 \, \text{cm}\)[/tex].
