In a crew escape system that uses a solid propellant, the burn rate of the propellant is an important quality characteristic. According to the specification, the mean burn rate should be $50$ cm$/$s$,$ and the population standard deviation is known to be $2$ cm$/$s$.$ A sample of $25$ propellants was taken, and the sample mean burn rate was found to be $51\mathrm{.}3$ cm$/$s$.$

$(1)$ Find the $95\%$ confidence interval for the population mean burn rate.

$(2)$ Determine whether this confidence interval includes the specified mean burn rate of $50$ cm$/$s$.$