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Daisy measured the heights of 20 plants. The table gives some information about the heights ([tex]\(h\)[/tex] in cm) of the plants.

[tex]\[
\begin{tabular}{|c|c|}
\hline
Height of plants (\(h\)) & Frequency \\
\hline
\(0 \ \textless \ h \ \textless \ 10\) & 1 \\
\hline
\(10 \ \textless \ h \ \textless \ 20\) & 4 \\
\hline
\(20 \ \textless \ h \ \textless \ 30\) & 7 \\
\hline
\(30 \ \textless \ h \ \textless \ 40\) & 2 \\
\hline
\(40 \ \textless \ h \ \textless \ 50\) & 3 \\
\hline
\(50 \ \textless \ h \ \textless \ 60\) & 3 \\
\hline
\end{tabular}
\][/tex]

Using the midpoints of each group, work out an estimate for the mean height of a plant.

Answer: ______ cm

Sagot :

GinnyAnswer
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]
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