The mass of a high-precision turbine blade is modelled by a normal distribution with a mean of $450.0$ g and a standard deviation of $12.4$ g. After a change in the manufacturing process, a quality control engineer suspects that the mean mass of the blades has decreased. A random sample of $50$ blades is measured, and the sample mean mass is found to be $446.2$ g. Carry out a hypothesis test at the $2\%$ significance level to investigate the engineer's suspicion.