Question 4 Cans of black olives are filled by two machines (A and B) in a food processing factory. The distribution of the gross mass is known to be normal with mean 500 g and standard deviation 5.3 g for machine A, and normal with mean 504 g and standard deviation 4.8 g for machine B. A batch of canned black olives have been filled by the same machine, but it is not known which machine was used. As some substandard olives have accidently been used by machine A, it is decided to test Ho: batch is from machine A, against HI: batch is from machine B By weighing a random sample of 6 cans and rejecting Ho if the mean mass exceeds a pre- determined mass of k g. (a) Determine constant k such that the risk of type I error is 5%. What is the corresponding risk of type II error? (9 marks) (b ) Apply and carry out the test for a sample of 6 cans with masses 51 1, 499, 500, 498, 507, 495 g. (3 marks) (c) It is decided to increase the sample size before reaching a final conclusion. Determine how many more cans are needed to be weighed if the risk of type I error is to remain at 5%, but the risk of type II error is to be reduced to just under 1%? What is the decision rule in this case? (8 marks)