please complete the questions attached

Q 4. ( 20 marks) Consider a population of 20 digits. 1, 2, 3, 4,5,6, 7,8,9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 (a) (3 marks) Suppose a systematic sample of size 5 was selected. (i) (1 mark) Explain how you would select that systematic sample from the list. (ii) (2 marks) If the part 55612, 78095, 83197, 03732, 05810 of a table of random numbers is used, which elements from the list Will be included in the sample? Explain your method of using the table. (b) (5 marks) The following table shows the ANOVA calculation for all possible systematic samples of size 1125 selected from the population of the 20 digits. Source df SS MS -- (i) (1 mark) Find the missed values marked XXX, XXXX in the table. (ii) (1 mark) Find the population variance, e.g 02. (iii) (2 marks) Find the value of the intracluster correlation (ICC), e.g p. (iv) (1 mark) Do you expect that the systematic sampling design to be better than the sample random sample design for this population, and Why? (0) (4 marks) Suppose the following systematic sample in (a) was obtained (digits numbers): 1, 5, 9, 13, 17. (i) (1 mark) Estimate the average digits number. Is this estimator unbiased? (ii) (1 mark) Using results in parts (b), nd the variance of the estimator in (i). (c) (4 marks) Suppose the following systematic sample in (a) was obtained (digits numbers): 1, 5, 9, 13, 17. (i) (1 mark) Estimate the average digits number. Is this estimator unbiased? (ii) (1 mark) Using results in parts (b), nd the variance of the estimator in (i). (iii) (2 marks) Can you estimate the variance of that estimator using this systematic sample? Explain, and do it, if you can. (d) (4 marks) For purpose of exercising in sampling, two more systematic samples of the same size as above were selected (without repetition). The following two sample totals were obtained, 50, 60. Combine those results with the original sample in (c) to again estimate the average digit number. Estimate the variance of your estimator. (e) (2 marks) Find an unbiased estimate of the variance of the sample mean from 1 in 4 systematic samples, using the previous three systematic samples in (d). (f) (2 marks) Use either your calculation from (c), or from (d) to estimate the intracluster correlation coefcient of 1 in 4 systematic samples