Question

In a regression analysis involving 30 observations, the following estimated regression equation was obtained.

ŷ = 17.6 + 3.2x₁ − 2.2x₂ + 7.6x₃ + 2.7x₄

1. Interpret b₁ in this estimated regression equation. (choose an option)

a)b₁ = −2.2 is an estimate of the change in y corresponding to a 1 unit change in x₁ when x₂, x₃, and x₄ are held constant.

b)b₁ = 7.6 is an estimate of the change in y corresponding to a 1 unit change in x₃ when x₁, x₂, and x₄ are held constant.

c)b₁ = 3.2 is an estimate of the change in y corresponding to a 1 unit change in x₂ when x₁, x₃, and x₄ are held constant.

d)b₁ = 2.7 is an estimate of the change in y corresponding to a 1 unit change in x₄ when x₁, x₂, and x₃ are held constant

e)b₁ = 3.2 is an estimate of the change in y corresponding to a 1 unit change in x₁ when x₂, x₃, and x₄ are held constant.

2. Interpret b₂ in this estimated regression equation. (choose an option)

a)b₂ = −2.2 is an estimate of the change in y corresponding to a 1 unit change in x₁ when x₂, x₃, and x₄ are held constant.

b)b₂ = 7.6 is an estimate of the change in y corresponding to a 1 unit change in x₃ when x₁, x₂, and x₄ are held constant.

c)b₂ = −2.2 is an estimate of the change in y corresponding to a 1 unit change in x₂ when x₁, x₃, and x₄ are held constant.

d)b₂ = 3.2 is an estimate of the change in y corresponding to a 1 unit change in x₁ when x₂, x₃, and x₄ are held constant.

e)b₂ = 2.7 is an estimate of the change in y corresponding to a 1 unit change in x₄ when x₁, x₂, and x₃ are held constant.

3. Interpret b₃ in this estimated regression equation. (choose an option)

a)b₃ = 3.2 is an estimate of the change in y corresponding to a 1 unit change in x₃ when x₁, x₂, and x₄ are held constant.

b)b₃ = −2.2 is an estimate of the change in y corresponding to a 1 unit change in x₁ when x₂, x₃, and x₄ are held constant.

c)b₃ = 7.6 is an estimate of the change in y corresponding to a 1 unit change in x₂ when x₁, x₃, and x₄ are held constant.

d)b₃ = −2.7 is an estimate of the change in y corresponding to a 1 unit change in x₄ when x₁, x₂, and x₃ are held constant.

e)b₃ = 7.6 is an estimate of the change in y corresponding to a 1 unit change in x₃ when x₁, x₂, and x₄ are held constant.

4.Interpret b₄ in this estimated regression equation. (choose an option)

a)b₄ = −2.2 is an estimate of the change in y corresponding to a 1 unit change in x₂ when x₁, x₃, and x₄ are held constant.

b)b₄ = 7.6 is an estimate of the change in y corresponding to a 1 unit change in x₂ when x₁, x₃, and x₃ are held constant.

c)b₄ = 3.2 is an estimate of the change in y corresponding to a 1 unit change in x₄ when x₁, x₂, and x₃ are held constant.

d)b₄ = 2.7 is an estimate of the change in y corresponding to a 1 unit change in x₃ when x₁, x₂, and x₄ are held constant.

e)b₄ = 2.7 is an estimate of the change in y corresponding to a 1 unit change in x₄ when x₁, x₂, and x₃ are held constant.

5. Predict y when x₁ = 10, x₂ = 5, x₃ = 1, and x₄ = 2.

Answer 1

Solution:
Given in the question
Multiple regression equation is
Y^ = 17.6 + 3.2X1 - 2.2 X2 + 7.6X3 + 2.7X4
Solution(a)
Here b1 is the slope of X1 i.e. b1 = 3.2 that means if we change one unit in X1 than Y will change by 3.2 units if all other variables kept constant.
So its correct answer is e i.e. b1 = 3.2, is an estimate of the change in y corresponding to a 1 unit change in X1 when X2, X3 and X4 are held constant.
Solution(b)
Here b2 is the slope of X2 i.e. b2 = -2.2 that means if we change one unit in X2 than Y will change by 2.2 units if all other variables kept constant.
So its correct answer is c i.e. b₂ = −2.2 is an estimate of the change in y corresponding to a 1 unit change in x₂ when x₁, x₃, and x₄ are held constant.
Solution(c)
Here b3 is the slope of X3 i.e. b3 = 2.7 that means if we change one unit in X3 than Y will change by 2.7 units if all other variables kept constant.
So its correct answer is e i.e. b₃ = 7.6 is an estimate of the change in y corresponding to a 1 unit change in x₃ when x₁, x₂, and x₄ are held constant.
Solution(d)
Here b4 is the slope of X4 i.e. b4 = 7.6 that means if we change one unit in X4 than Y will change by 7.6 units if all other variables kept constant.
So its correct answer is e i.e. b₄ = 2.7 is an estimate of the change in y corresponding to a 1 unit change in x₄ when x₁, x₂, and x₃ are held constant.
Solution(e)
Y^ = 17.6 + 3.2X1 - 2.2 X2 + 7.6X3 + 2.7X4
Here X1 = 10, X2 = 5, X3 = 1 and X4 = 2
So Y^ = 17.6 + 3.2*10 - 2.2*5 + 7.6*1 + 2.7*2 = 17.6 + 32 - 11 + 7.6 + 5.4 = 51.6