Question

Multiple regression analysis was used to study the relationship between a dependent variable, y, and four independent variables; x₁, x₂, x_3, and x₄. The following is a partial result of the regression analysis involving 31 observations.

	Coefficients	Standard Error
Intercept	18.00	6.00
x1	12.00	8.00
x2	24.00	48.00
x3	-36.00	36.00
x4	16.00	2.00
ANOVA
	df	SS	MS	F
Regression			125
Error
Total		760

a) Compute the multiple coefficient of determination.

b) Perform a t test and determine whether or not β₁ is significantly different from zero (α = .05).

c) Perform a t test and determine whether or not β₄ is significantly different from zero (α = .05).

d) At α = .05, perform an F test and determine whether or not the regression model is significant.

Answer 1

Solution: To find the answers to part a through d, we need to complete the given tables. The tables are filled as:

ANOVA:

The p values are found using the t distribution for

a) Compute the multiple coefficient of determination.

Answer: The multiple coefficient of determination is given below:

rounded to four decimal places

b) Perform a t test and determine whether or not β1 is significantly different from zero (α = .05).

Answer: The null and the alternative hypotheses are:

The test statistic from the above table is:

And the is:

Since the p-value is greater than the significance level 0.05, therefore we fail to reject the null hypothesis and conclude that β1 is not significantly different from zero

c) Perform a t test and determine whether or not β4 is significantly different from zero (α = .05).

Answer: The null and the alternative hypotheses are:

The test statistic from the above table is:

And the is:

Since the p-value is less than the significance level 0.05, therefore we reject the null hypothesis and conclude that β4 is significantly different from zero.

d) At α = .05, perform an F test and determine whether or not the regression model is significant.

Answer: The null and the alternative hypotheses are:

At least one coefficient is different from 0.

The test statistic from the above ANOVA table is:

And the using the F distribution for is:

Since the p-value is less than the significance level 0.05, therefore we reject the null hypothesis and conclude at least one coefficient is significantly different from zero.