The lowest value in a set of ordered scores is x and the highest value is y. If each score is increased by k, then which of the following must be true of the revised set of scores?
I. The mean is increased by k.
II. The range is k.
III.The median remains unchanged.

Options

A. I only
B. II only
C. I and III only
D. II and III only

Answer

To solve the problem, we need to analyze each of the three statements given regarding the revised set of scores after each score is increased by \( k \).

### Step-by-Step Solution:

1. **Understanding the Initial Set of Scores:**
- Let the initial set of scores be represented as \( S = \{ x, a_1, a_2, \ldots, a_{n-2}, y \} \), where \( x \) is the lowest score and \( y \) is the highest score.

2. **Calculating the Mean of the Initial Set:**
- The mean of the initial set of scores is calculated as:
\[ \text{Mean}_{\text{initial}} = \frac{x + a_1 + a_2 + \ldots + a_{n-2} + y}{n} \]

3. **Calculating the Revised Set of Scores:**
- When each score is increased by \( k \), the new set of scores becomes:
\[ S' = \{ x + k, a_1 + k, a_2 + k, \ldots, a_{n-2} + k, y + k \} \]

4. **Calculating the Mean of the Revised Set:**
- The mean of the revised set of scores is:
\[ \text{Mean}_{\text{revised}} = \frac{(x + k) + (a_1 + k) + (a_2 + k) + \ldots + (a_{n-2} + k) + (y + k)}{n} \]
- This simplifies to:
\[ \text{Mean}_{\text{revised}} = \frac{x + a_1 + a_2 + \ldots + a_{n-2} + y + nk}{n} = \text{Mean}_{\text{initial}} + k \]
- Therefore, **Statement I** is true: The mean is increased by \( k \).

5. **Calculating the Range of the Initial Set:**
- The range of the initial set of scores is:
\[ \text{Range}_{\text{initial}} = y - x \]

6. **Calculating the Range of the Revised Set:**
- The range of the revised set is:
\[ \text{Range}_{\text{revised}} = (y + k) - (x + k) = y - x \]
- Therefore, the range remains unchanged and is not equal to \( k \). Thus, **Statement II** is false.

7. **Calculating the Median of the Initial Set:**
- The median is the middle value of the ordered set. If the number of scores \( n \) is odd, the median is the middle score. If \( n \) is even, it is the average of the two middle scores.

8. **Calculating the Median of the Revised Set:**
- When each score is increased by \( k \), the median will also increase by \( k \). Therefore, **Statement III** is false: The median does not remain unchanged.

### Conclusion:
- Only **Statement I** is true. Therefore, the answer is:
**Option A: Only Statement I is true.**