Statistics Questions on Distributions, Probability, and Testing - Solutions Provided

Question 1 Three distributions each of 100 members and standard deviation 4.5 units are located with means at 12.1, 17.1, 22.1 units respectively. Showing details in the working, determine the standard deviation of the distribution obtained by combining the three.  Question 2 In a certain institute, it is known that 15% of all student score grade “A” in Statistics 101. Four male students and five female students are selected at random from the Statistics 101 class. With appropriate assumption(s), compute the probability that in the selected group (a) 3 students in the selected group score “A”; (b) 1 male and 2 female students score “A”; (c) at most 2 female students and at least 1 male student score “A”.

Question 3 Certain devices are received at an assembly plant in batches of 50. The sampling scheme used to test all batches has been set up in the following way. One of the 50 devices is chosen randomly and tested. If it is defective, all the remaining 49 items in that batch are returned to the supplier for individual testing; if the tested device is not defective, another device is chosen randomly and tested. If the second item is not defective, the complete batch is accepted without any more testing; if the second device is defective, a third device is chosen randomly and tested. If the third device is not defective, the complete batch is accepted without any more testing, but the one defective device is replaced by the supplier. If the third device is defective, all remaining 47 items in that batch are returned to the supplier for individual testing. The receiver pays for all initial single-item tests. However, whenever the remaining devices in a batch are returned to the supplier for individual tests, the costs of this extra testing are paid by the supplier. If a batch is returned to the supplier, the superintendent must ensure that the receiver is sent 50 items which have been tested and shown to be good. We ignore cost of subsequent tests for replacement items. Assume that the superintendent accepts the results of the receiver’s tests. Each device is worth $60 and the cost of testing is $10 per device. Consider a particular batch which contains 12 defective items. Showing your full workings for the following clearly, (a) compute the probability that the batch will be accepted; (b) compute the expected cost to the supplier of the testing and of replacing defectives; (c) of the 12 defective items in the batch, derive the expected number which will be accepted.