Question 1 [25 marks]Suppose we are interested in estimating p, the population proportion of left-handedpeople. To do this, we have randomly selected three independent samples of people. Letn1,n2 or n3 denote the sample size of each sample, where n1 < n2 < n3, and let X1,X2 and X3 denote the number of left-handed people in each sample. Finally, letb p1 = X1 n1 , b p2 = X2 n2 andb p3 = X3 n3 denote the sample proportions of left-handed people in each sample. (a). (3 points) Letb p12 = n1b p1+n2b p2 n1+n2 denote the combined proportion calculated using the first two samples. Show thatb p12 is an unbiased estimator of p. (b). (4 points) Calculate the variance ofb p12 = n1b p1+n2b p2 n1+n2 . (c). (2 points) Let ¯ p12 = b p1+b p2 2 . Show that ¯ p12 is an unbiased estimator of p. (d). (3 points) Calculate the variance of ¯ p12 = b p1+b p2 2 . (e). (5 points) Show that the variance ofb p12 = n1b p1+n2b p2 n1+n2 is less than the variance of ¯ p12 = b p1+b p2 2 . Show all working. (f). (2 points) If you could only choose one estimator betweenb p12 and ¯ p12, which one would you choose? Explain your answer.For parts (g) and (h), assume that n2 = 2n1 and n3 = 3n1.(g) (3 points) Calculate the variance of ¯ p123 = b p1+b p2+b p3 3 . Show all working. (h) (3 points) To estimate p, suppose that you can choose either ¯ p123 = b p1+b p2+b p3 3 or the combined proportion calculated using any two of the samples, i.e.,b p12 = n1b p1+n2b p2 n1+n2 , b p13 = n1b p1+n3b p3 n1+n3 , orb p23 = n2b p2+n3b p3 n2+n3 . Assuming that n2 = 2n1 and n3 = 3n1, order the estimators ¯ p123,b p12,b p13 andb p23 from best to worst. Explain your answer.2