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3. An assembly line returns a nished item every 10 seconds. Let 23- be equal to 1 if the ith item is faulty, or 0 otherwise. The random variables Z1, Z2, . .. are assumed independent and identically distributed, with unknown fault probability p > 0. We are interested in characterising the mean 9 : 1/1) of the geometric population X1, X2, . . . , which denote the number of items up to and including the next faulty item. The corresponding probability mass function is: 1 111 fx($l6)=(1g) :1:=1.2,3.... Note that, since :1: > 1, also 6' 2 1. An observed sample X1,...,X50 gives the average value 53 = 13.7. (a) Write down the likelihood and log-likelihood for 9. [10] (b) Derive a 95% approximate condence interval for (9 centred around its max- imum likelihood estimator. You should show all your workings and give the nal numerical solution. [10] (c) We are interested in whether the assumed value 0 = 10 should be rejected in favour of 0 = 15. Define a rejection criterion based on the observed value of x. [10] (d) Make a decision on whether 0 = 10 should be rejected in favour of 0 = 15, using significance level 0.05. [10]