To solve the problem of which change cannot affect the value of the median in a set of 9 randomly selected numbers, we will analyze each option step by step.
### Step-by-Step Solution:
1. **Understanding the Median**:
- The median is the middle value of a sorted list of numbers. For a set of 9 numbers, the median is the 5th number when arranged in ascending order.
2. **Consider a Sample Set**:
- Let's take a sample set of 9 numbers: {1, 2, 3, 4, 5, 6, 7, 8, 9}. The median of this set is 5 (the 5th number).
3. **Option A: Subtracting 2 from Each Number**:
- New set: {1-2, 2-2, 3-2, 4-2, 5-2, 6-2, 7-2, 8-2, 9-2} = {-1, 0, 1, 2, 3, 4, 5, 6, 7}.
- New median: 3 (the 5th number).
- **Conclusion**: The median changes.
4. **Option B: Decreasing the Largest Number Only**:
- Original set: {1, 2, 3, 4, 5, 6, 7, 8, 9}.
- Decrease the largest number (9) by 3: New set = {1, 2, 3, 4, 5, 6, 7, 8, 6}.
- Sort the new set: {1, 2, 3, 4, 5, 6, 6, 7, 8}.
- New median: 5 (the 5th number).
- **Conclusion**: The median remains unchanged.
5. **Option C: Decreasing the Smallest Number Only**:
- Original set: {1, 2, 3, 4, 5, 6, 7, 8, 9}.
- Decrease the smallest number (1) by 1: New set = {0, 2, 3, 4, 5, 6, 7, 8, 9}.
- Sort the new set: {0, 2, 3, 4, 5, 6, 7, 8, 9}.
- New median: 5 (the 5th number).
- **Conclusion**: The median remains unchanged.
6. **Option D: Increasing the Largest and Smallest Number Only**:
- Original set: {1, 2, 3, 4, 5, 6, 7, 8, 9}.
- Increase the smallest number (1) by 1 and the largest number (9) by 1: New set = {2, 2, 3, 4, 5, 6, 7, 8, 10}.
- Sort the new set: {2, 2, 3, 4, 5, 6, 7, 8, 10}.
- New median: 5 (the 5th number).
- **Conclusion**: The median remains unchanged.
### Final Conclusion:
- The changes that do not affect the value of the median are **Option B (Decreasing the largest number only)**, **Option C (Decreasing the smallest number only)**, and **Option D (Increasing the largest and smallest number only)**. However, the only option that does not change the median is **Option C: Decreasing the smallest number only**.
