#### Answer

The seven numbers in the set must all be in the range (between 4 and 19). Also, the average of the numbers between the minimum and $Q_{1}$ is 7. Additionally, the average of the numbers between the maximum and $Q_{3}$ is 14.
One dataset that has the five number summary is as follows: $4, 5, 9, 12, 13, 15, 19$.

#### Work Step by Step

$Q_{1}$ is the middle number between the minimum and the dataset's median. Since $Q_{1} = 7$, the two numbers between the minimum and the dataset's median must add to 14 (and thus have an average of 7). This same logic follows for $Q_{3}$.