A sports scientist claims that a new stretching routine increases vertical jump height in athletes. The vertical jump heights (in centimeters) for $8$ athletes are measured before and after following the routine. The following are the differences: $2$, $3$, $1$, $4$, $2$, $3$, $2$, $5$. At $\alpha =0.05$, is there sufficient evidence to support the scientist's claim? Assume the samples are random, dependent, and the population is normally distributed.