Step 1: Use the Empirical Relationship.
From the empirical relationship between the mean, median, and mode:
\[
\text{Mode} \approx 3 \cdot \text{Median} - 2 \cdot \text{Mean}
\]
Step 2: Given Information.
We are told that the median exceeds the mean by 3:
\[
\text{Median} = \text{Mean} + 3
\]
Step 3: Substitute into the Empirical Formula.
Substitute \( \text{Median} = \text{Mean} + 3 \) into the empirical formula:
\[
\text{Mode} \approx 3 \cdot (\text{Mean} + 3) - 2 \cdot \text{Mean}
\]
Simplify:
\[
\text{Mode} \approx 3 \cdot \text{Mean} + 9 - 2 \cdot \text{Mean}
\]
\[
\text{Mode} \approx \text{Mean} + 9
\]
Step 4: Determine How Much the Mode Exceeds the Mean.
From the equation \( \text{Mode} \approx \text{Mean} + 9 \), we see that the mode exceeds the mean by 9.
Step 5: Analyze the Options.
Option (1): 8 — Incorrect, as the mode exceeds the mean by 9.
Option (2): 9 — Correct, as this matches the calculated value.
Option (3): 10 — Incorrect, as the mode exceeds the mean by 9.
Option (4): 11 — Incorrect, as the mode exceeds the mean by 9.
Step 6: Final Answer.
\[
(2) 9
\]