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

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

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



Answer :

To find the estimated mean height of a plant using the given frequency table, we follow these steps:

1. Identify the Midpoints of Each Class Interval:
- For the interval [tex]\(0 \leqslant h < 10\)[/tex], the midpoint is [tex]\(\frac{0 + 10}{2} = 5 \, cm\)[/tex].
- For the interval [tex]\(10 \leqslant h < 20\)[/tex], the midpoint is [tex]\(\frac{10 + 20}{2} = 15 \, cm\)[/tex].
- For the interval [tex]\(20 \leqslant h < 30\)[/tex], the midpoint is [tex]\(\frac{20 + 30}{2} = 25 \, cm\)[/tex].
- For the interval [tex]\(30 \leqslant h < 40\)[/tex], the midpoint is [tex]\(\frac{30 + 40}{2} = 35 \, cm\)[/tex].
- For the interval [tex]\(40 \leqslant h < 50\)[/tex], the midpoint is [tex]\(\frac{40 + 50}{2} = 45 \, cm\)[/tex].
- For the interval [tex]\(50 \leqslant h < 60\)[/tex], the midpoint is [tex]\(\frac{50 + 60}{2} = 55 \, cm\)[/tex].

2. Calculate the Weighted Sum of the Midpoints: This is done by multiplying each midpoint by the frequency of its class interval and then summing these products.
- [tex]\(5 \, cm \times 1 = 5\)[/tex]
- [tex]\(15 \, cm \times 4 = 60\)[/tex]
- [tex]\(25 \, cm \times 7 = 175\)[/tex]
- [tex]\(35 \, cm \times 2 = 70\)[/tex]
- [tex]\(45 \, cm \times 3 = 135\)[/tex]
- [tex]\(55 \, cm \times 3 = 165\)[/tex]

When we sum these values, we get:
[tex]\[ 5 + 60 + 175 + 70 + 135 + 165 = 610 \][/tex]

3. Calculate the Total Frequency: Add up the frequencies for all class intervals.
[tex]\[ 1 + 4 + 7 + 2 + 3 + 3 = 20 \][/tex]

4. Compute the Mean Height: Divide the weighted sum by the total frequency.
[tex]\[ \text{Mean height} = \frac{610}{20} = 30.5 \, cm \][/tex]

Thus, the estimated mean height of the plants is [tex]\(30.5 \, cm\)[/tex].