Question

Assume you are interested in the average income of aboriginals 18 years and up in Canada. Let income represent the wage of an aboriginal person 18 years and up living in Canada. For ease of presentation assume n, the sample size equals 5. Two possible estimator are proposed income1 +income2++incomen ncome _ 7L incomei +income2 + incomes +incomes +incomes an income1 income ncome +2ncome5 Note: the second estimator only uses information from the first and last person drawn in the sample. a) Are both estimators random variables? Justify your answer. b) Calculate the mean of each estimator c) Calculate the variance of each estimator. Note: since one is dealing with a random sample, there is no link between each draw. As such Var(incomeincome) Var(income + Var(income2) d) which estimator is preferable? Justify your answer e) Assume the following sample, income,: i 20000, 100000, 80000, 60000,40000) Calculate the estimate of each estimator

Answer 1

a) Yes, both the estimators are random variables. As the sample is a random sample, the income included in the formulae are random and can be a certain value. Hence, the two estimators can only be attached a probability to their values and are random variables

b) Mean of first estimator =( mean(income1) + mean(income 2) +.....+ mean(income n) )/ n

Mean of second estimator = ( mean(income 1) + mean(income n) )/ 2

c) variance of first estimator = ( variance(income 1) + variance(income 2) +....+ variance(income n) )/ n²

Variance of second estimator = ( variance(income 1) + variance(income n) ) / 4

d) The first estimator is preferable as it is more representative of the sample becuase it includes the income of everyone in the sample and also has a lesser variance than the second estimator