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The data given below in Table i are 40 long-duration monthly average river discharge measure- ments, D, in cumecs (cubic metres per second) of a river basin system in the UK from 201013. Table 1. Monthly average river discharge measurements, D, in cumecs 11.825 12.75 13.38 14.575 14.95 14.95 15.525 15.825 16.25 16.275 16.575 16.575 17.15 17.475 18.20 18.35 18.525 18.525 18.9 19.2 19.325 19.52 19.55 19.8 20.325 20.325 20.4 20.475 20.775 21.3 21.3 21 .925 22.425 22.65 22.8 23.85 24.375 25.9 26.4 27.675 Answer the following questions: a) [7 marks] Let Y1,Y2, ...,.'l',1 be a sample of n iid random variables with N(y,62) distribution, show that the maximum-likelihood estimator (MLE) for the population mean, if 0'2 is assumed to be known, is equal to 25; Y,. b) [5 marks] Given that, for the data in Table 1, )3 Y,- = 766.875 and )3 17,2 = 15232764, i=1 i=1 calculate the sample mean and the sample standard deviation (unbiased). c) [3 marks] Construct a frequency table for the data, taking the 6 grouping intervals: (0,15], (15,17], (17,19], (19,21], (21,23], (23,00). d) [15 marks] Test the hypothesis, at 5% significance level, that the river discharge measure- ments have normal distribution, if the sample mean and the sample standard deviation equals to the values calculated in part b): i) using z-transform: Z = X: , find the lower and upper boundaries for each range for z; ii) find the expected frequencies using table for cumulative standard normal distribution; iii) find the value of the appropriate test statistics; iv) what is your conclusion? Comment on your