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 income ¯ = income1 + income2 + · · · + incomen n = income1 + income2 + income3 + income4 + income5 5 (1) and income1 + incomen 2 = income1 + income5 2 (2) 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 V ar(income1 + income2) = V ar(income1 + V ar(income2) d) which estimator is preferable? Justify your answer. e) Assume the following sample, {incomei : i = 1, . . . , n} = {20000, 100000, 80000, 60000, 40000}. Calculate the estimate of each estimator.

Answer 1

Solution

Let X represent the income of aboriginals 18 years up in Canada.

We assume mean of X = µ, and V(X) = σ²   

Part (a)

So, the given data is: X₁, X₂, X₃, X₄, X₅, is a random sample from X.

Since both estimators are based on the above random sample, both are random variables. Answer

Part (b)

Mean of T₁ = E(T₁) = E{(1/5)∑(i = 1 to 5)(X_i)}

=(1/5) ∑(i = 1 to 5){E(X_i)}

= (1/5)∑(i = 1 to 5)µ

= µ

E(T₂) = E{(1/2)(X₁ + X₅}

= (1/2){E(X₁) + E(X₅)}

= (1/2)(µ + µ)

= µ

Thus, both estimators have the same mean µ. Answer

Part (c)

V(T₁) = V{(1/5)∑(i = 1 to 5)(X_i)}

= (1/25)∑(i = 1 to 5)V(X_i)}

= (1/25)∑(i = 1 to 5)σ²

= (1/25)5σ²

= (1/5)σ²

V(T₂) = V{(1/2)(X₁ + X₅}

= (1/4){V(X₁) + V(X₅)}

= (1/4)(2σ²)

= (1/2)σ²

Thus, V(T₁) = σ²/5 and V(T₂) = σ²/2 Answer

Part (D)

Since V(T₁) < V(T₂), T₁ is preferable as an estimator of µ. Answer

Part (e)

Given, {X₁ = 20000, X₂ = 100000, X₃ = 80000, X₄ = 60000, X₅ = 40000},

T₁ = (20000 + 100000 + 80000 + 60000 + 40000)/5

= 60000 Answer 1

T₂ = (20000 + 40000)/2

= 30000 Answer 2

DONE