Question

In a regression analysis involving 27 observations, the following estimated regression equation was developed. ŷ = 25.2 + 5.5x₁ For this estimated regression equation SST = 1,550 and SSE = 530.

(a) At α = 0.05, test whether x₁  is significant.State the null and alternative hypotheses.

H₀: β₁ ≠ 0
H_a: β₁ = 0

H₀: β₀ ≠ 0
H_a: β₀ = 0   

H₀: β₀ = 0
H_a: β₀ ≠ 0

H₀: β₁ = 0
H_a: β₁ ≠ 0

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

F =

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

p-value =

Is x₁ significant?

Do not reject H₀. We conclude that x₁ is significant

Do not reject H₀. We conclude that x₁ is not significant.

Reject H₀. We conclude that x₁ is significant.

Reject H₀. We conclude that x₁ is not significant.

Suppose that variables x₂ and x₃ are added to the model and the following regression equation is obtained.ŷ = 16.3 + 2.3x₁ + 12.1x₂ − 5.8x₃ For this estimated regression equation SST = 1,550 and SSE = 100.

(b) Use an F test and a 0.05 level of significance to determine whether x₂ and x₃  contribute significantly to the model. State the null and alternative hypotheses.

H₀: β₁ ≠ 0
H_a: β₁ = 0

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

  H₀: β₁ = 0
H_a: β₁ ≠ 0

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

Find the value of the test statistic.

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

p-value =

Is the addition of x₂ and x₃ significant?

Do not reject H₀. We conclude that the addition of variables x₂ and x₃ is not significant.

Do not reject H₀. We conclude that the addition of variables x₂ and x₃ is significant.   

Reject H₀. We conclude that the addition of variables x₂ and x₃ is significant.

Reject H₀. We conclude that the addition of variables x₂ and x₃ is not significant.

Answer 1

Source	SS	df	MS	F
regression	1020.00	1	1020.000	48.113
error	530.00	25	21.200
total	1550.00	26

a)

H₀: β₁ = 0
H_a: β₁ ≠ 0

value of the test statistic =48.11

p-value.=0.000

Reject H₀. We conclude that x₁ is significant.

b)

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

Partial F=((SSEr-SSEc)/(c-r))/(SSEc/(n-c-1)) =

49.45

p-value =0.000

Reject H₀. We conclude that the addition of variables x₂ and x₃ is significant.