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 estimators are random variables as outcome of both estimators depend on random sample set of 5 (18 and Up) individuals. With change in sample, value of estimators will differ.

B) Mean of first estimator will be (Income 1 + Income 2 + Income 3 + Income 4 + Income 5)/5, Let it be x₅

Mean of second estimator will be (Income 1 + Income 5)/2. Let it be x₂

c) variance formula is (_i=1ⁿ_$\sum$(x_i-x_n)^{^2})/N

For estimator 1, Variance will be ((Income_{1 -} x5_)^{^2} + (Income_{2 -} x5_)^{^2} + (Income_{3 -} x5_)^{^2} + (Income_{4 -} x5_{)^{^2 +}} + (Income_{5 -} x5_)^{^2}) / 5

For estimator 2, Variance will be Variance will be ( ((Income_{1 -} x5_)^{^2} + (Income_{5 -} x5_)^{^2} )/2

d) Estimator 1 is more preferable than Estimator 2 as it is taing into consideration larger sample (5) than Estimator 2 (Out of 5, only 2 which are 1st and last are considered). Larger sample provides better results as larger samples are more uniform and they try to cover all variety of observations of population. Also since, mean and variance are more sensitive to highest and lowest value and in smaller samples it tends to move more towards high and low value.

e)First Estimator: (20,000+ 100,000+ 80,000 + 60,000 + 40,000)/ 5 = 60,000

Second Estimator: (20,000+ 40,000)/2 = 30,000

The results of point e more justify that Estimator 1 is better, Since second estimator is considering two smallest oservation of sample, value is coming half of considering all five observations. It is because observaions used in second estimator are not uniform and they donot fully represent actiual population.

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