Sure! Let's go through the steps to solve for the new mean of the remaining distribution after removing the largest value.
1. Determine the initial total sum of observations:
- Given: Mean of initial observations ([tex]\( \overline{x}_{initial} \)[/tex]) = 5
- Number of initial observations ([tex]\( n_{initial} \)[/tex]) = 30
- We use the formula for the mean: [tex]\( \overline{x}_{initial} = \frac{\text{Sum of observations}}{n_{initial}} \)[/tex]
[tex]\[
\text{Sum of initial observations} = \overline{x}_{initial} \times n_{initial} = 5 \times 30 = 150
\][/tex]
2. Subtract the largest value from the total sum:
- The largest value to be deleted from the observations = 34
- New sum of observations after removing the largest value:
[tex]\[
\text{Total sum remaining} = 150 - 34 = 116
\][/tex]
3. Calculate the new number of observations:
- As we have deleted one value, the new number of observations:
[tex]\[
n_{remaining} = n_{initial} - 1 = 30 - 1 = 29
\][/tex]
4. Calculate the new mean:
- The new mean of the remaining distribution using the total sum of remaining observations and the new number of observations:
[tex]\[
\overline{x}_{remaining} = \frac{\text{Total sum remaining}}{n_{remaining}} = \frac{116}{29} = 4.0
\][/tex]
Therefore, the mean of the remaining distribution after deleting the largest observed value of 34 is 4.0.
