To find the mean of the given frequency distribution, we follow these steps:
1. Determine the midpoints of each temperature interval:
Each temperature interval is given as:
[tex]\(40-44\)[/tex]
[tex]\(45-49\)[/tex]
[tex]\(50-54\)[/tex]
[tex]\(55-59\)[/tex]
* [tex]\(60-64\)[/tex]
The midpoints are calculated by averaging the endpoints of each interval. Therefore:
[tex]\[
\begin{align*}
\text{Midpoint of } 40-44 &: \frac{40 + 44}{2} = 42 \\
\text{Midpoint of } 45-49 &: \frac{45 + 49}{2} = 47 \\
\text{Midpoint of } 50-54 &: \frac{50 + 54}{2} = 52 \\
\text{Midpoint of } 55-59 &: \frac{55 + 59}{2} = 57 \\
\text{Midpoint of } 60-64 &: \frac{60 + 64}{2} = 62 \\
\end{align*}
\][/tex]
2. List the frequencies corresponding to each interval:
The given frequencies are:
[tex]\( f_1 = 1 \)[/tex] for [tex]\(40-44\)[/tex]
[tex]\( f_2 = 4 \)[/tex] for [tex]\(45-49\)[/tex]
[tex]\( f_3 = 9 \)[/tex] for [tex]\(50-54\)[/tex]
[tex]\( f_4 = 7 \)[/tex] for [tex]\(55-59\)[/tex]
* [tex]\( f_5 = 1 \)[/tex] for [tex]\(60-64\)[/tex]
3. Compute the product of each midpoint and its corresponding frequency:
[tex]\[
\begin{align*}
42 \times 1 &= 42 \\
47 \times 4 &= 188 \\
52 \times 9 &= 468 \\
57 \times 7 &= 399 \\
62 \times 1 &= 62 \\
\end{align*}
\][/tex]
4. Sum the frequencies and sum the products of midpoints and frequencies:
The sum of frequencies ([tex]\(\Sigma f\)[/tex]) is:
[tex]\[
1 + 4 + 9 + 7 + 1 = 22
\][/tex]
The sum of the products ([tex]\(\Sigma (f \cdot x)\)[/tex]) is:
[tex]\[
42 + 188 + 468 + 399 + 62 = 1159
\][/tex]
5. Calculate the mean ([tex]\(\mu\)[/tex]) using the formula for the mean of grouped data:
[tex]\[
\text{Mean} = \frac{\Sigma (f \cdot x)}{\Sigma f} = \frac{1159}{22} \approx 52.68
\][/tex]
6. Round the mean to the nearest tenth:
[tex]\[
\text{Mean} \approx 52.7
\][/tex]
So, the mean of the frequency distribution is [tex]\( 52.7 \)[/tex] degrees.
Comparing this computed mean to the actual mean of 52.8 degrees, the values are very close, showing that the calculation is accurate.
Thus, the mean of the frequency distribution is
[tex]\[
\boxed{52.7}
\text{ degrees (rounded to the nearest tenth).}
\][/tex]
