To determine if Emma's estimate of the mean number of workers (30) is correct, let's calculate the actual mean number of workers using the information given about the distribution of workers across the offices.
Here is the information provided:
1. Number of workers in the range 1 to 20: 9 offices
2. Number of workers in the range 21 to 40: 8 offices
3. Number of workers in the range 41 to 60: 2 offices
4. Number of workers in the range 61 to 80: 1 office
We will follow these steps to calculate the actual mean number of workers:
1. Find the midpoint of each range to use as a representative number of workers for that range.
2. Multiply the midpoint by the number of offices in each range to find the total number of workers in that range.
3. Sum the total number of workers for all ranges.
4. Sum the number of offices.
5. Divide the total number of workers by the total number of offices to find the mean number of workers per office.
Step 1: Find midpoints
- Midpoint of 1 to 20: [tex]\(\frac{1 + 20}{2} = 10.5\)[/tex]
- Midpoint of 21 to 40: [tex]\(\frac{21 + 40}{2} = 30.5\)[/tex]
- Midpoint of 41 to 60: [tex]\(\frac{41 + 60}{2} = 50.5\)[/tex]
- Midpoint of 61 to 80: [tex]\(\frac{61 + 80}{2} = 70.5\)[/tex]
Step 2: Calculate total workers in each range
- Total workers in the range 1 to 20: [tex]\(10.5 \times 9 = 94.5\)[/tex]
- Total workers in the range 21 to 40: [tex]\(30.5 \times 8 = 244\)[/tex]
- Total workers in the range 41 to 60: [tex]\(50.5 \times 2 = 101\)[/tex]
- Total workers in the range 61 to 80: [tex]\(70.5 \times 1 = 70.5\)[/tex]
Step 3: Sum the total number of workers
[tex]\[
94.5 + 244 + 101 + 70.5 = 510
\][/tex]
Step 4: Sum the number of offices
[tex]\[
9 + 8 + 2 + 1 = 20
\][/tex]
Step 5: Calculate the mean number of workers
[tex]\[
\text{Mean} = \frac{\text{Total number of workers}}{\text{Total number of offices}} = \frac{510}{20} = 25.5
\][/tex]
Thus, the actual mean number of workers per office is 25.5.
Conclusion:
Emma's estimate of 30 is not correct since the calculated mean number of workers per office is 25.5.
