Let X be a random variable representing the number of years of education an indi

Question

Let X be a random variable representing the number of years of education an individual has, and let Y be a random variable representing an individual’s annual income. Suppose that the latest research in economics has concluded that: Y = 6X + Z (1) is the correct model for the relationship between X and Y, where Z is another random variable that is independent of X. Suppose Var(X) = 2 and Var(Y ) = 172.

a) Find Var(Z). (2 marks)

(b) Find Cov(X, Y ) and Corr(X, Y ). (3 marks)

(c) The variance in Y (income) comes from variance in X (education) and Z (other factors unobserved to us). What fraction of the variance in income is explained by variance in education? (2 marks)

(d) How does the fraction you found in (c) compare to Corr(Y, X)? (2 marks)

Explanation / Answer

We have given

X: number of years of education

Y: individuals annual income

Var(X) = 2

Var(Y) = 172

The model given for the relationship between X and Y is

Y= 6X+ z

Which is the linear regression equation of Y on X.

The general linear regression equation of Y on X is given as

Y= a+bX

Where, b is the regression coefficient of Y on X

Thus, from the given information, we get

b= 6

a) Var(Z) =?

Given equation is

Y= 6X+Z

Var(Y) = Var(6X+Z)

172= 36Var(X) + Var(Z)

Var(Z) = 172-36(2)

Var(Z) = 100

b) Cov(X, Y) = b. Var(X) = 6*2 = 12

Corr(X, Y) = Cov (X, Y) /(Var(X) *Var(Y))

Corr(X, Y) = 12/(2*172)

Corr(X, Y) = 0.6469

C) The fraction of the variance in income explained by variance in education is the regression coefficient of Y on X which is

b= 6

d) Corr(X, Y) = Corr(Y, X) = Cov(X, Y) /(var(X) *var(Y))

Corr(Y, X) = bVar(X) /(var(X) *(var(Y))

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