Certainly! Let's solve the problem step-by-step.
1. Understanding the given data:
- We have a total of 9 numbers.
- The mean of the first three numbers is 8.
- The mean of the last six numbers is 4.
2. Calculating the sum of the first three numbers:
- The mean is calculated by dividing the sum of the numbers by the number of terms.
- Let the sum of the first three numbers be [tex]\( S_1 \)[/tex].
- Given: [tex]\( \text{Mean of the first three numbers} = 8 \)[/tex].
- Thus, [tex]\( S_1 / 3 = 8 \)[/tex].
- Rearranging to find [tex]\( S_1 \)[/tex]: [tex]\( S_1 = 8 \times 3 = 24 \)[/tex].
3. Calculating the sum of the last six numbers:
- Similarly, let the sum of the last six numbers be [tex]\( S_2 \)[/tex].
- Given: [tex]\( \text{Mean of the last six numbers} = 4 \)[/tex].
- So, [tex]\( S_2 / 6 = 4 \)[/tex].
- Rearranging to find [tex]\( S_2 \)[/tex]: [tex]\( S_2 = 4 \times 6 = 24 \)[/tex].
4. Calculating the total sum of all nine numbers:
- The total sum of all the numbers in the data set is the sum of the first three numbers plus the sum of the last six numbers.
- Thus, [tex]\( \text{Total Sum} = S_1 + S_2 = 24 + 24 = 48 \)[/tex].
5. Calculating the mean of all nine numbers:
- The mean of all the numbers is the total sum divided by the total number of numbers.
- Here, the total number of numbers is 9.
- So, [tex]\( \text{Overall Mean} = \text{Total Sum} / 9 = 48 / 9 = 5.33 \)[/tex].
Therefore, the mean of all the numbers in the data set is approximately [tex]\( 5.33 \)[/tex].
