High School

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Find the [tex]\sum(x \cdot f)[/tex], or the sum of the product of the midpoints and frequency.

[tex]
\[
\begin{array}{|c|c|}
\hline
\text{Age (in years)} & \text{Number of Teachers} \\
\hline
20-24 & 10 \\
\hline
25-29 & 28 \\
\hline
30-34 & 32 \\
\hline
35-39 & 50 \\
\hline
40-44 & 48 \\
\hline
45-49 & 35 \\
\hline
50-54 & 13 \\
\hline
\end{array}
\]
[/tex]

A. 37.9
B. 216
C. 8187

Answer :

To find [tex]\(\sum(x \cdot f)\)[/tex], which is the sum of the product of the midpoints and their corresponding frequencies, we can follow these steps:

1. Identify the Class Intervals:
- The age groups given are: [tex]\(20-24\)[/tex], [tex]\(25-29\)[/tex], [tex]\(30-34\)[/tex], [tex]\(35-39\)[/tex], [tex]\(40-44\)[/tex], [tex]\(45-49\)[/tex], and [tex]\(50-54\)[/tex].

2. Determine the Midpoints:
- The midpoint of a class interval is calculated by taking the average of the lower and upper bounds of the interval.
- For each age group:
- [tex]\(20-24\)[/tex]: Midpoint = [tex]\((20 + 24) / 2 = 22.0\)[/tex]
- [tex]\(25-29\)[/tex]: Midpoint = [tex]\((25 + 29) / 2 = 27.0\)[/tex]
- [tex]\(30-34\)[/tex]: Midpoint = [tex]\((30 + 34) / 2 = 32.0\)[/tex]
- [tex]\(35-39\)[/tex]: Midpoint = [tex]\((35 + 39) / 2 = 37.0\)[/tex]
- [tex]\(40-44\)[/tex]: Midpoint = [tex]\((40 + 44) / 2 = 42.0\)[/tex]
- [tex]\(45-49\)[/tex]: Midpoint = [tex]\((45 + 49) / 2 = 47.0\)[/tex]
- [tex]\(50-54\)[/tex]: Midpoint = [tex]\((50 + 54) / 2 = 52.0\)[/tex]

3. List the Frequencies:
- The frequencies corresponding to each age group are given as: 10, 28, 32, 50, 48, 35, and 13.

4. Calculate the Product of Midpoints and Frequencies:
- Multiply each midpoint by its corresponding frequency:
- [tex]\(22.0 \times 10 = 220\)[/tex]
- [tex]\(27.0 \times 28 = 756\)[/tex]
- [tex]\(32.0 \times 32 = 1024\)[/tex]
- [tex]\(37.0 \times 50 = 1850\)[/tex]
- [tex]\(42.0 \times 48 = 2016\)[/tex]
- [tex]\(47.0 \times 35 = 1645\)[/tex]
- [tex]\(52.0 \times 13 = 676\)[/tex]

5. Sum the Products to Find [tex]\(\sum(x \cdot f)\)[/tex]:
- Add all the products together:
[tex]\[
220 + 756 + 1024 + 1850 + 2016 + 1645 + 676 = 8187
\][/tex]

Thus, the sum of the product of the midpoints and frequencies, [tex]\(\sum(x \cdot f)\)[/tex], is 8187.

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