To solve this question, let's carefully follow these steps:
1. Determine the total sum of all 30 observations.
We know that the mean (average) of 30 observations is 4. The formula for the mean is:
[tex]\[
\text{Mean} = \frac{\text{Sum of all observations}}{\text{Number of observations}}
\][/tex]
Rearranging this formula to find the total sum of the observations gives us:
[tex]\[
\text{Sum of all observations} = \text{Mean} \times \text{Number of observations}
\][/tex]
Given the mean is 4 and the number of observations is 30, we get:
[tex]\[
\text{Sum of all observations} = 4 \times 30 = 120
\][/tex]
2. Assume the value of the largest observation.
Let's denote the value of the largest observation as [tex]\( L \)[/tex]. This value needs to be provided or hypothesized.
3. Subtract the largest observation from the total sum.
Once the largest observation [tex]\( L \)[/tex] is removed, the new sum of the remaining 29 observations is:
[tex]\[
\text{New sum} = 120 - L
\][/tex]
4. Determine the number of remaining observations.
Since we have removed one observation, the number of remaining observations is:
[tex]\[
30 - 1 = 29
\][/tex]
5. Calculate the new mean.
The new mean of the remaining 29 observations can be determined using the formula for the mean:
[tex]\[
\text{New mean} = \frac{\text{New sum}}{\text{Number of remaining observations}}
\][/tex]
Plugging in the values we have:
[tex]\[
\text{New mean} = \frac{120 - L}{29}
\][/tex]
6. Conclusion:
The new mean of the remaining distribution, after removing the largest observation, is given by:
[tex]\[
\text{New mean} = \frac{120 - L}{29}
\][/tex]
To find the exact numerical value of the new mean, you would need to know or be given the specific value of the largest observation [tex]\( L \)[/tex].
