The median of a set of `9` distinct observations is `20.5`. If each of the largest `4` observations of the set is increased by `2`, then the median of the new set

Options

A. is incrased by `2`
B. is decreased by `2`
C. is two times the original median
D. remains the same as that of the original set

Explanation

To solve the problem step by step, we need to find the new median after increasing the largest four observations of a set of 9 distinct observations by 2.

### Step-by-Step Solution:

1. **Understanding the Median**:
The median of a set of observations is the middle value when the observations are arranged in ascending order. For a set of 9 observations, the median is the 5th observation.

2. **Identifying the Original Median**:
Given that the median of the original set of 9 observations is 20.5, this means that the 5th observation (when arranged in ascending order) is 20.5.

\[ A_5 = 20.5 \]

3. **Increasing the Largest 4 Observations**:
The largest 4 observations in the set are \( A_6, A_7, A_8, A_9 \). We are instructed to increase each of these observations by 2. Therefore, the new observations will be:

\[ A_6 + 2, A_7 + 2, A_8 + 2, A_9 + 2 \]

4. **Arranging the New Observations**:
After increasing the largest 4 observations, the new set of observations will be:

\[ A_1, A_2, A_3, A_4, A_5, A_6 + 2, A_7 + 2, A_8 + 2, A_9 + 2 \]

Since \( A_6, A_7, A_8, A_9 \) are the largest observations, adding 2 to each of them will not affect the position of \( A_1, A_2, A_3, A_4, A_5 \) in the ordered set.

5. **Finding the New Median**:
The median of the new set will still be the 5th observation in the ordered list. Since \( A_5 \) remains unchanged, the new median will also be:

\[ \text{New Median} = A_5 = 20.5 \]

### Conclusion:
The new median after increasing the largest 4 observations by 2 remains **20.5**.

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