To determine if the mean of the set \( S = \{1, 2, 5, 7, x\} \) is greater than 4, we will analyze the two given statements step by step.
### Step 1: Calculate the Mean of Set S
The mean of set \( S \) is calculated as follows:
\[
\text{Mean} = \frac{1 + 2 + 5 + 7 + x}{5} = \frac{15 + x}{5}
\]
We need to check if this mean is greater than 4:
\[
\frac{15 + x}{5} > 4
\]
### Step 2: Solve the Inequality
To solve the inequality, we multiply both sides by 5 (since 5 is positive, the inequality remains the same):
\[
15 + x > 20
\]
Now, subtract 15 from both sides:
\[
x > 5
\]
This means that for the mean to be greater than 4, \( x \) must be greater than 5.
### Step 3: Analyze the First Statement
**Statement (1): The median of set S is greater than 2.**
To find the median, we need to consider different cases for \( x \):
1. **Case 1:** \( x \leq 1 \) → Median is 2 (not greater than 2).
2. **Case 2:** \( 1 < x \leq 2 \) → Median is 2 (not greater than 2).
3. **Case 3:** \( 2 < x \leq 5 \) → Median is 5 (greater than 2).
4. **Case 4:** \( x > 5 \) → Median is 5 (greater than 2).
From this analysis, the first statement does not provide a definitive answer about whether \( x > 5 \) since it only tells us that the median is greater than 2 when \( x \) is between 2 and 5 or greater than 5. Thus, **Statement (1) is insufficient.**
### Step 4: Analyze the Second Statement
**Statement (2): The median of set S is equal to the mean of set S.**
Let’s denote the median as \( M \) and the mean as \( \mu \):
\[
M = \mu = \frac{15 + x}{5}
\]
The median \( M \) can take values from the set depending on the value of \( x \):
1. If \( x \leq 1 \), \( M = 2 \)
2. If \( 1 < x \leq 2 \), \( M = 2 \)
3. If \( 2 < x \leq 5 \), \( M = 5 \)
4. If \( x > 5 \), \( M = 5 \) or \( x \)
To satisfy \( M = \mu \), we can analyze the cases:
- If \( x = 5 \), then \( M = 5 \) and \( \mu = 4 \) (not equal).
- If \( x = 10 \), then \( M = 5 \) and \( \mu = 5 \) (equal).
Thus, if \( M = \mu \), \( x \) must be a multiple of 5. The smallest valid \( x \) that satisfies both conditions is \( x = 10 \), which is greater than 5.
Therefore, **Statement (2) is sufficient** to conclude that the mean is greater than 4.
### Conclusion
- **Statement (1)** is insufficient.
- **Statement (2)** is sufficient.
Thus, the answer is that the second statement alone is sufficient to determine that the mean of set \( S \) is greater than 4.
