Question

he following ANOVA table was obtained when estimating a multiple linear regression model.

ANOVA	df	SS	MS	F	Significance F
Regression	2	22,009.81	11,004.905		0.0225
Residual	17	39,118.31	2,301.077
Total	19	61,128.12

a-1. How many explanatory variables were specified in the model?

a-2. How many observations were used?

b. Choose the hypotheses to determine whether the explanatory variables are jointly significant.

• H₀: β₁ = β₂ = 0; H_A: At least one β_j > 0

• H₀: β₁ = β₂ = 0; H_A: At least one β_j ≠ 0

• H₀: β₁ = β₂ = 0; H_A: At least one β_j < 0

c. Compute the value of the test statistic. (Round your answer to 3 decimal places.)

d-1. Find the p-value.

• p-value < 0.01

• p-value 0.10
• 0.01 p-value < 0.025
• 0.05 p-value < 0.10
• 0.025 p-value < 0.05

d-2. At the 5% significant level, what is the conclusion to the test?

• Reject H₀; the explanatory variables are jointly significant.

• Reject H₀; the explanatory variables are not jointly significant.

• Do not reject H₀; we can conclude the explanatory variables are jointly significant.

• Do not reject H₀; we cannot conclude the explanatory variables are not jointly significant.

Answer 1

Solution:

Given:

ANOVA	df	SS	MS	F	Significance F
Regression	2	22,009.81	MSregression=1,004.91		0.0225
Residual	17	39,118.31	MSresiduals = 2,301.08
Total	19	61,128.12

Part a-1. How many explanatory variables were specified in the model?

We have:

df_regression = k = 2

where k = number of independent / explanatory variables

thus number of explanatory variables = 2

Part a-2. How many observations were used?

N = total observations

We have :

df_total = N - 1 = 19

thus N = 19+1

N= 20

thus total observations = N = 20

Part b. Choose the hypotheses to determine whether the explanatory variables are jointly significant.

H₀: β₁ = β₂ = 0; H_A: At least one β_j ≠ 0

Here we chose  β_j ≠ 0, since we are asked to test the explanatory variables are jointly significant or not.

Part c. Compute the value of the test statistic.

F = MS_regression / MS_residuals

F = 1,004.91 / 2,301.08

F = 4.783

Part d-1. Find the p-value.

From table we have:  Significance F = 0.0225

Thus 0.0225 is between 0.01 and 0.025

thus correct answer is:

0.01 < p-value < 0.025

Part d-2. At the 5% significant level, what is the conclusion to the test?

Since level of significance is 0.05 and p-value is 0.0225 < 0.05 level of significance, we reject null hypothesis H0 in favor of H_A.

Thus we get:
Reject H₀; the explanatory variables are jointly significant.