The number of work-related injuries per week that occur at the manufacturing plant of the Excelsior Corporation is a Poisson-distributed random variable with mean \(\lambda \geq 3\), according to company analysts. In an attempt to lower insurance costs, the Corporation institutes a program of intensive safety instruction for all employees. Upon completion of the program, a 12 -week period produced an average of two accidents per week.

(a) Design a uniformly most powerful level 10 test of the null hypothesis \(H_{0}: \lambda \geq 3\) versus the alternative hypothesis \(H_{a}: \lambda<3\) having size as close to .10 as possible without exceeding . 10 .

(b) Test the null hypothesis using the test you defined in part (a). What can you say about the effectiveness of the safety program?