Question

Multiple regression analysis was used to study how an individual's income (Y in thousands of dollars) is influenced by age (X₁ in years), level of education (X₂ ranging from 1 to 5), and the person's gender (X₃ where 0 =female and 1=male). The following is a partial result of a computer program that was used on a sample of 20 individuals.

	Coefficient	Standard Error
X1	0.6251	0.094
X2	0.9210	0.190
X3	-0.510	0.920
Analysis of Variance
Source of Variation	Degrees of Freedom	Sum of Squares	Mean Square	F
Regression		84
Error		112

Required:

a.	Compute the coefficient of determination.
b.	Perform a t test and determine whether or not the coefficient of the variable "level of education" (i.e., X2) is significantly different from zero. Let a = 0.05.
c.	At a = 0.05, perform an F test and determine whether or not the regression model is significant.
d.	As you note the coefficient of X3 is -0.510. Fully interpret the meaning of this coefficient.

Answer 1

[OR]

Solution:

a) coefficient of determination =SSR/SST =84/(112+84)=0.4286

b) Ho: b2=0 (i.e. null hypothesis)

Ha: b2 not equal to 0 (i.e. alterniatve hypothesis)

The test statistic is

t=coefficient/standard error

=0.921/0.19

=4.85

Given a=0.05, the critical values are t(0.025, df=n-2=18) =2.1 or -2.1 (from student t table)

Since t=4.85 is larger than 2.1, we reject the null hypothesis.

So we can conclude that the coefficient of the variable X2 is significant different from zero..

c) Ho: the regression model is not signficant

Ha: the regression model is signficant

The test statistic is
F= (SSR/degree of freedom )/ (SSE/degree of freedom)

=(84/3)/(112/16)

=4

Given a=0.05, the critical value is F(0.95, df1=3, df2=16) =3.24 (from F table)

Since F=4 is larger than 3.24, we reject the null hypothesis.

So we can conclude that the regression model is signficant...

d) When the variable X3 increases one unit and variables X1 and X2 fixed, the regression line decreases 0.51