To find the mean (average) age of the 45 children who attended the birthday party, we need to follow these steps:
1. Identify the Age Groups and Their Frequencies:
We have the age groups: 3, 4, 5, 6, 7, 8, 9, and 10 years.
The corresponding frequencies (number of children in each age group) are: 2, 6, 5, 4, 6, 9, 8, and 5.
2. Verify the Total Number of Children:
Adding up the frequencies:
[tex]\[
2 + 6 + 5 + 4 + 6 + 9 + 8 + 5 = 45
\][/tex]
This confirms that the total number of children is 45.
3. Calculate the Sum of Ages Multiplied by Their Frequencies:
- For age 3: \(3 \times 2 = 6\)
- For age 4: \(4 \times 6 = 24\)
- For age 5: \(5 \times 5 = 25\)
- For age 6: \(6 \times 4 = 24\)
- For age 7: \(7 \times 6 = 42\)
- For age 8: \(8 \times 9 = 72\)
- For age 9: \(9 \times 8 = 72\)
- For age 10: \(10 \times 5 = 50\)
Adding these up:
[tex]\[
6 + 24 + 25 + 24 + 42 + 72 + 72 + 50 = 315
\][/tex]
4. Calculate the Mean (Average) Age:
The mean age is computed by dividing the total sum of ages by the total number of children:
[tex]\[
\text{Mean Age} = \frac{\text{Sum of Ages}}{\text{Total Number of Children}} = \frac{315}{45} = 7.0
\][/tex]
5. Summary:
- Total number of children: 45
- Sum of ages multiplied by their frequencies: 315
- Mean (average) age: 7.0
Therefore, the mean age of the children who attended the birthday party is 7.0 years.
