Provide solution to the following questions. Include explanations in your working.

A student who regularly goes tenpin bowling thinks that he scores higher with one particular bowling ball than he does with his other two. For the next 15 games that he bowls, he chooses a ball at random and notes down the scores as follows: Ball Score sum of squares 173, 166, 179, 183, 199 900 162,616 195, 206, 188, 195, 210, 221 1.215 246,771 169, 155, 172, 188 684 117.514 From the data obtained, calculate unbiased estimates of: (i) the overall mean, # (Hi) the "treatment effect" for each ball. r; -The following data relate to 12 patients who had an operation that was intended to correct a life-threatening condition, where time 0 is the start of the period of the investigation: Reason Patient Time of operation Time observation (in weeks) observation number ended (in weeks) ended 120 Censored 68 Death 40 Death 120 Censored 35 Censored 40 Death 120 Censored 115 Death 90 Death 98 Death 70 120 Death 80 110 Death You can assume that censoring was non-informative with regard to the survival of any individual patient. (i) Compute the Nelson-Aalen estimate of the cumulative hazard function, A(!), where / is the time since having the operation. [6] (ii) Using the results of part (i), deduce an estimate of the survival function for patients who have had this operation. [2] (iii) Estimate the probability of a patient surviving for at least 70 weeks after undergoing the operation. [1] [Total 9]