The supervisor of a factory believes that method 1 takes a worker lesser time to perform a task than method 2, on the average. Two independent random samples of size 10 workers of the factory are selected and workers from different samples are asked to perform the task using different methods. The times used by the 20 workers are given below in seconds: Worker 1 2 3 4 5 6 7 8 9 10 Method 1 47 44 43 49 48 44.5 48 47 46 50 Method 2 45 43 42 45 46 45 44 45 44 47 a. If we assume that both population distributions are normal with unknown but equal variances, then at a level of significance of 0.01, can you conclude that the supervisor's belief is true? Use the 8-step rejection region approach. b. Use R to conduct the same test as in part (a). Attach the code and output. c. Use R and the sample data to draw 2 normal probability plots in the same picture. Attach the code and output. d. From the plots in part (c), can you justify the assumption in part a that both population distributions are normal? Why or why not? e. From the plots in part (c), can you justify the assumption in part (a) that population variances are equal? Why or why not?