Question

Given the following regression analysis output.

 

a. What is the sample size?
b. How many independent variables are in the study?
c. Determine the coefficient of determination.
d. Conduct a global test of hypothesis. Can you conclude at least one of the independent variables does not equal zero? Use the .01 significance level.
e. Conduct an individual test of hypothesis on each of the independent variables. Would you consider dropping any of the independent variables? If so, which variable or variables would you drop? Use the .01 significance level.

Answer 1

The available output is as shown in below,

ANOVA table
Source	SS	Df	MS	F
Regression	227.0928	4	56.7732	14.50
Residual	153.0739	24	6.1230
Total	380.1667	29

 

Regression output
Variables	Coefficients	Std. error	t(df – 28)
Intercept	68.3366	8.9752	7.614
X1	0.8595	0.3087	2.784
X2	-0.3380	0.8381	-0.403
X3	-0.8179	0.2749	-2.975
X4	-0.5824	0.2541	-2.292

 

a)

Calculate the sample size.

dfTotal    = N – 1

29 = N – 1

⇒ N = 29 + 1

⇒ N = 30

 

Hence, the required sample size is, 30.

 

b)

Find the independent variables in the study.

There are four independent variables are there in given study, that is, X1, X2, X3 and X4.

 

c)

Calculate the coefficient of determination.

R2 = SSRegression/SSTotal

      = 227.0928/380.1667

     = 0.5974

 

Hence, the required answer is, 0.5974.

 

d)

Null hypothesis, H0: β1 = β2 = β3 = β4

Alternative hypothesis, H1 not all βi is 0 (for all ii = 1, 2, 3, 4)

Level of significance, α = 0.01

 

Decision rule: Reject H0| When Fstatistic > 4.18 (From F – tabulated values)

 

Since, the F – test statistic(14.50) is greater than the F – critical(4.18), so we reject the null hypothesis and conclude that the not all βi are 0 at the 0.01 level of significance.

 

e)

Decision rule: Reject H0| when t > 2787 or t < -2.787 (From t-tabulated values)

 

By using 1% level of significance, drop variable 2 initially and then rerun. Perhaps you will delete variable 1 and variable 4, since both test statistic values are less than the critical values.

a) Hence, the required sample size is, 30.

b) There are four independent variables are there in given study, that is, X1, X2, X3 and X4.