Question

You may need to use the appropriate technology to answer this question.

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

ŷ = 17.6 + 3.8x₁ − 2.3x₂ + 7.6x₃ + 2.7x₄

For this estimated regression equation, SST = 1,835 and SSR = 1,790.

(a)

At α = 0.05, test the significance of the relationship among the variables.

State the null and alternative hypotheses.

H₀: One or more of the parameters is not equal to zero.
H_a: β₀ = β₁ = β₂ = β₃ = β₄ = 0

H₀: β₀ = β₁ = β₂ = β₃ = β₄ = 0
H_a: One or more of the parameters is not equal to zero.    

H₀: One or more of the parameters is not equal to zero.
H_a: β₁ = β₂ = β₃ = β₄ = 0

H₀: β₁ = β₂ = β₃ = β₄ = 0
H_a: One or more of the parameters is not equal to zero.

Find the value of the test statistic. (Round your answer to two decimal places.)

Find the p-value. (Round your answer to three decimal places.)

p-value =

State your conclusion.

Reject H₀. We conclude that the overall relationship is significant.

Reject H₀. We conclude that the overall relationship is not significant.    

Do not reject H₀. We conclude that the overall relationship is not significant.

Do not reject H₀. We conclude that the overall relationship is significant.

Suppose variables x₁ and x₄ are dropped from the model and the following estimated regression equation is obtained.

ŷ = 11.1 − 3.6x₂ + 8.1x₃

For this model, SST = 1,835 and SSR = 1,735.

(b)

Compute SSE(x₁, x₂, x₃, x₄)

SSE(x₁, x₂, x₃, x₄) =

(c)

Compute SSE(x₂, x₃).

SSE(x₂, x₃) =

(d)

Use an F test and a 0.05 level of significance to determine whether x₁ and x₄ contribute significantly to the model.

State the null and alternative hypotheses.

H₀: β₁ = β₄ = 0
H_a: One or more of the parameters is not equal to zero.

H₀: β₂ = β₃ = 0
H_a: One or more of the parameters is not equal to zero.    

H₀: One or more of the parameters is not equal to zero.
H_a: β₁ = β₄ = 0

H₀: One or more of the parameters is not equal to zero.
H_a: β₂ = β₃ = 0

Find the value of the test statistic. (Round your answer to two decimal places.)

Find the p-value. (Round your answer to three decimal places.)

p-value =

State your conclusion.

Reject H₀. We conclude that x₁ and x₄ contribute significantly to the model.

Do not reject H₀. We conclude that x₁ and x₄ contribute significantly to the model.    

Do not reject H₀. We conclude that x₁ and x₄ do not contribute significantly to the model.

Reject H₀. We conclude that x₁ and x₄ do not contribute significantly to the model.

Answer 1

H₀: β₀ = β₁ = β₂ = β₃ = β₄ = 0
H_a: One or more of the parameters is not equal to zero.   

SSE =SST-SSR=		45.0
MSR=SSR/k=		447.5
MSE=SSE/(n-k-1)=		1.8
F =MSR/MSE =		248.61
p value =0.000

Reject H₀. We conclude that the overall relationship is significant.

b)

SSE(x1,x2,x,3,x4) =

45

c)

SSE(x2x3) =

100

d)

H₀: β₁ = β₄ = 0
H_a: One or more of the parameters is not equal to zero.

F =	((100-45)/2)/(45/25)=		15.28
p value =0.000

Reject H₀. We conclude that x₁ and x₄ contribute significantly to the model.