Certainly! Let's calculate the mean height step-by-step using the given data:
### Step-by-Step Solution
1. Data Tables:
We are given the heights and their corresponding frequencies.
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline \text{Height} & 4 & 5 & 6 & 7 & 8 \\
\hline \text{Frequency} & 5 & 8 & 10 & 12 & 7 \\
\hline
\end{array}
\][/tex]
2. Calculate the Total Number of Data Points:
The total number of data points is the sum of all frequencies.
[tex]\[
\text{Total Frequency} = 5 + 8 + 10 + 12 + 7 = 42
\][/tex]
3. Calculate the Sum of Heights Multiplied by Their Frequencies:
We need to find the sum of the product of each height and its corresponding frequency.
[tex]\[
\sum (\text{Height} \times \text{Frequency}) = (4 \times 5) + (5 \times 8) + (6 \times 10) + (7 \times 12) + (8 \times 7)
\][/tex]
Calculate each product:
[tex]\[
4 \times 5 = 20
\][/tex]
[tex]\[
5 \times 8 = 40
\][/tex]
[tex]\[
6 \times 10 = 60
\][/tex]
[tex]\[
7 \times 12 = 84
\][/tex]
[tex]\[
8 \times 7 = 56
\][/tex]
Sum these products:
[tex]\[
20 + 40 + 60 + 84 + 56 = 260
\][/tex]
4. Calculate the Mean Height:
The mean height is the total sum of heights multiplied by their frequencies divided by the total frequency.
[tex]\[
\text{Mean Height} = \frac{\sum (\text{Height} \times \text{Frequency})}{\text{Total Frequency}} = \frac{260}{42}
\][/tex]
Calculate the division:
[tex]\[
\text{Mean Height} \approx 6.19
\][/tex]
### Final Answer
The mean height is approximately [tex]\( 6.19 \)[/tex].
