To find the speed of the runner, we can use the formula for speed, which is:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]
In this case, the distance is \( \frac{1}{2} \) mile and the time is \( \frac{1}{15} \) hour.
Substituting the values into the formula:
\[ \text{Speed} = \frac{\frac{1}{2} \text{ mile}}{\frac{1}{15} \text{ hour}} \]
To divide by a fraction, we can multiply by its reciprocal:
\[ \text{Speed} = \frac{1}{2} \times \frac{15}{1} = \frac{15}{2} \text{ miles per hour} \]
This simplifies to:
\[ \text{Speed} = 7.5 \text{ miles per hour} \]
Therefore, the runner's speed is 7.5 miles per hour.