Question

A student used multiple regression analysis to study how family spending (y) is influenced by income (x1), family size (x2), and additionsto savings(x3). The variables y, x1, and x3 are measured in thousands of dollars. The following results were obtained. ANOVA df SS Regression 3 45.9634 Residual 11 2.6218 Total Coefficients Standard Error Intercept 0.0136 x1 0.7992 0.074 x2 0.2280 0.190 x3 -0.5796 0.920

a. Write out the estimated regression equation for the relationship between the variables. (1 mark)

b. Compute coefficient of determination. What can you say about the strength of this relationship?

c. Carry out a test to determine whether y is significantly related to the independent variables. Use a 5% level of significance.

d. Carry out a test to see if x3 and y are significantly related. Use a 5% level of significance.

Answer 1

from the given data,

a.

Considering the providing data, the estimated regression equation for the relationship between the variables can be stated as;

Here, b0 is the intercept and b1, b2 and b3 are the coefficients of x1, x2 and x3.

The ANOVA table can be completed as;

b.

Coefficient of Determination:

It is denoted by r2, here r represents the correlation between the two variables.

The r2 value represents the proportion of variation in the dependent variable explained by the independent variable.

Consider, the coefficient of determination is 0.9460 or 94.60% and it is calculated below:

That is, 94.60% of the variation in the dependent variable.

c.

Here calculated F is 64.281 and the critical value at 5% level of 3, 11 degrees of freedom (from F-table) we get it as 3.587.

The test statistics is greater than the critical value; we reject the null hypothesis. At 5% level of significance we can conclude that y is significantly related to the independent variables.

d.

For x3 the value of test statistics t is

t = -0.5796 / 0.920

   = -0.63

The critical t for 14 degrees of freedom is given as (-/+) 2.145. The absolute value of test statistics is less the critical value (0.63 < 2.145); we fail to reject the null hypothesis.

At 5% level of significance we cannot conclude that x3 and y are significantly related.