1. **State the problem:** We have the data set $\{19, 10, 12, 12, 13, 13, 14, 15, 16, m, n\}$ with unknowns $m$ and $n$. Both the mean and median of this data are 14. We need to determine which of the statements I. $m>14$, II. $n\leq16$, III. $m+n=30$ are true. 2. **Recall formulas and rules:** - Mean formula: $$\text{mean} = \frac{\text{sum of all data}}{\text{number of data points}}$$ - Median: The middle value when data is sorted. Since there are 11 data points, the median is the 6th value in sorted order. 3. **Calculate mean condition:** Sum of known values: $19 + 10 + 12 + 12 + 13 + 13 + 14 + 15 + 16 = 124$ Number of data points: 11 Mean is 14, so: $$14 = \frac{124 + m + n}{11} \implies 124 + m + n = 154 \implies m + n = 30$$ This confirms statement III is true. 4. **Analyze median condition:** Sort the known values (excluding $m,n$): $$10, 12, 12, 13, 13, 14, 15, 16, 19$$ We insert $m$ and $n$ into this sorted list and find the 6th value must be 14. 5. **Position of median:** The 6th value in sorted order is the median. To have median 14, the 6th smallest number must be 14. 6. **Check possible values for $m$ and $n$:** Since $m+n=30$, and median is 14, $m$ and $n$ must be placed so that the 6th value is 14. 7. **Check statement I ($m>14$):** If $m \leq 14$, then to keep median 14, $m$ would be at or before the 6th position, pushing 14 to a higher position, which contradicts median = 14. Thus, $m > 14$ is necessary. 8. **Check statement II ($n \leq 16$):** If $n > 16$, then $n$ would be greater than 16, but since $m + n = 30$ and $m > 14$, $n$ must be less than or equal to 16 to satisfy the sum. 9. **Conclusion:** Statements I, II, and III are all true. **Final answer:** D. I, II and III