A logistics company is comparing the delivery times of two different routes. They believe that route $1$ is generally slower than or equal to route $2$ in terms of average time, but a manager claims that route 1 actually has a higher average delivery time (${\mu }_1>{\mu }_2$).Test the claim at a significance level of $\alpha =0.01$. Assume the samples are random and independent, the populations are normally distributed, and the population variances are equal (${\sigma }_1^2={\sigma }_2^2$). Sample statistics are:
Route 1: ${\bar{x}}_1=52.4$, $s_1=4.15$, $n_1=15$
Route 2: ${\bar{x}}_2=47.8$, $s_2=3.90$, $n_2=15$