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11 A study was conducted to investigate lengths of stay, in days, of short-term stay patients in a particular hospital. Independent random samples of 40 male patients and 35 female patients were selected and the lengths of stay of these patients are given in the following tables: Male Female 10 LA A 10 -NUON W OOD A LI WINE LA Male: Er = 215 Er- = 1,481 Female: Er = 168 Er = 1,026 The male observations are assumed to be normally distributed with mean u, and standard deviation G1 , and independently the female observations are assumed to be normally distributed with mean uz and standard deviation oz. (i) Suppose that it is known that of = 3.0 days and o, = 2.5 days. (a) Construct a 95% confidence interval for the difference between the mean length of stay for males and the mean length of stay for females, that is for uj - M2 (b) Comment briefly on any implications of this confidence interval. [6] (ii) Suppose now that of and 62 are unknown. (a) Perform a two-sample test to investigate whether there is a difference between the mean length of stay for males and the mean length of stay for females, assuming that o, and on are equal. (b) Show that the variances in the male and female samples are not significantly different at the 5% level, and comment briefly with reference to the validity of the test conducted in (ii)(a). (9) Suppose you are not prepared to assume more than you feel is absolutely necessary - in particular you do not want to assume that 6, and 62 are equal, nor that the observations necessarily come from normal populations. Perform an alternative (large-sample) test to that conducted in part (ii)(a). "to investigate whether there is a difference between the mean length of stay for males and the mean length of stay for females", and compare the results of the test with the results of the test obtained in part (ii)(a). [11] [Total 17]Outside an apartment block there is a small car park with three parking spaces. A prospective purchaser of an apartment in the block is concerned about how often he would return in his car to find that there was no empty parking space available. He decides to model the number of parking spaces free at any time using a time homogeneous Markov Jump Process where: . The probability that a car will arrive seeking a parking space in a short interval de is A de + old(). . For each car which is currently parked, the probability that its owner drives the car away in a short interval or is B.di + o(di). where A. B > 0. (i) Specify the state space for the above process. [1] (ii] Draw a transition graph of the process. iid Write down the generator matrix for the process. [2] (iv) Derive the probability that, given all the parking spaces are full, they will remain full for at least the next two hours. [2] (v) Explain what is meant by a jump chain. [1] (vi) Specify the transition matrix for the jump chain associated with this process [2] Suppose there are currently two empty parking spaces. (vii) Determine the probability that all the spaces become full before any cars are driven away. (vili) Derive the probability that the car park becomes full before the car park becomes empty. [3] (ix) Comment on the prospective purchaser's assumptions regarding the arrival and departure of cars. (3] [Total 17]
