To find the mean of the data summarized in the given frequency distribution, we'll follow these steps:
1. Identify the midpoints of each temperature range:
The midpoint of a range is calculated by averaging the upper and lower bounds of the range.
[tex]\[
\text{Midpoint} = \frac{ \text{Lower Bound} + \text{Upper Bound} }{2}
\][/tex]
For each range, the midpoints are:
- [tex]$40 - 44$[/tex]: [tex]$\frac{40 + 44}{2} = 42$[/tex]
- [tex]$45 - 49$[/tex]: [tex]$\frac{45 + 49}{2} = 47$[/tex]
- [tex]$50 - 54$[/tex]: [tex]$\frac{50 + 54}{2} = 52$[/tex]
- [tex]$55 - 59$[/tex]: [tex]$\frac{55 + 59}{2} = 57$[/tex]
- [tex]$60 - 64$[/tex]: [tex]$\frac{60 + 64}{2} = 62$[/tex]
2. Sum of frequencies ([tex]\(n\)[/tex]):
Add all the frequencies together:
[tex]\[
n = 1 + 5 + 10 + 5 + 1 = 22
\][/tex]
3. Calculate the weighted sum of midpoints:
Multiply each midpoint by its corresponding frequency and add all these products together:
[tex]\[
\text{Weighted sum} = (42 \times 1) + (47 \times 5) + (52 \times 10) + (57 \times 5) + (62 \times 1) = 42 + 235 + 520 + 285 + 62 = 1144
\][/tex]
4. Compute the mean:
The mean ([tex]\(\mu\)[/tex]) is found by dividing the weighted sum by the sum of the frequencies ([tex]\(n\)[/tex]):
[tex]\[
\mu = \frac{ \text{Weighted sum} }{n} = \frac{1144}{22} = 52.0
\][/tex]
5. Comparison with the actual mean:
- Computed mean: [tex]\( 52.0 \)[/tex]
- Actual mean: [tex]\( 52.2 \)[/tex]
The computed mean is 52.0 degrees and the actual mean is 52.2 degrees. The computed mean is very close to the actual mean.
Therefore, the mean of the frequency distribution is [tex]\(\boxed{52.0}\)[/tex] degrees.
