To solve the problem step by step, we will follow the reasoning presented in the video transcript while providing a clear breakdown of each step.
### Step-by-Step Solution:
1. **Understanding the Set of Observations**:
We have a set of \(2n\) distinct observations, which we can denote as \(X_1, X_2, \ldots, X_{2n}\).
2. **Finding the Median**:
Since there are \(2n\) observations, the median is calculated as the average of the \(n\)-th and \((n+1)\)-th observations:
\[
\text{Median} = \frac{X_n + X_{n+1}}{2}
\]
3. **Classifying Observations**:
- Observations below the median: \(X_1, X_2, \ldots, X_n\)
- Observations above or equal to the median: \(X_{n+1}, X_{n+2}, \ldots, X_{2n}\)
4. **Modifying the Observations**:
- Each observation below the median is increased by 5:
\[
\text{New observations below median} = X_1 + 5, X_2 + 5, \ldots, X_n + 5
\]
- Each observation above or equal to the median is decreased by 3:
\[
\text{New observations above median} = X_{n+1} - 3, X_{n+2} - 3, \ldots, X_{2n} - 3
\]
5. **Calculating the New Mean**:
- The total of the new observations below the median:
\[
\text{Total below median} = (X_1 + 5) + (X_2 + 5) + \ldots + (X_n + 5) = (X_1 + X_2 + \ldots + X_n) + 5n
\]
- The total of the new observations above the median:
\[
\text{Total above median} = (X_{n+1} - 3) + (X_{n+2} - 3) + \ldots + (X_{2n} - 3) = (X_{n+1} + X_{n+2} + \ldots + X_{2n}) - 3n
\]
6. **Combining the Totals**:
- The total sum of the new observations is:
\[
\text{Total new observations} = \left( \sum_{i=1}^{n} X_i + 5n \right) + \left( \sum_{i=n+1}^{2n} X_i - 3n \right)
\]
- Simplifying this, we find:
\[
\text{Total new observations} = \sum_{i=1}^{2n} X_i + 2n
\]
7. **Finding the New Mean**:
- The new mean is given by the total of the new observations divided by the total number of observations:
\[
\text{New Mean} = \frac{\sum_{i=1}^{2n} X_i + 2n}{2n}
\]
- This can be rewritten as:
\[
\text{New Mean} = \frac{\sum_{i=1}^{2n} X_i}{2n} + \frac{2n}{2n} = \text{Old Mean} + 1
\]
### Conclusion:
Thus, the new mean of the modified set of observations is the old mean plus 1.
