Question

17. 1. A soft drink bottler is analyzing the vending machine serving routes in his distribution system. He is interested in predicting the time required by the distribution driver to service the vending machines in an outlet. It has been suggested that the two most important variables influencing delivery time (y in min) are the number of cases of product stocked (x₁), the distance walked by the driver (x₂ in feet), and the delivery charges (x₃ in OMR). 29 observations on delivery times, cases stocked, walking times and delivery charges have been recorded. Based on the following multiple regression summary output,

SUMMARY OUTPUT
Regression Statistics
Multiple R	?
R Square	0.84869
Adjusted R Square	0.83054
Standard Error	0.43435
Observations	?
ANOVA
	df	SS	MS	F
Regression	3	26.4558	?	?
Residual	?	4.7166	?
Total	?	?
	Coefficients	Standard Error	t Stat	P-value
Intercept	-0.5753	0.4786	-1.2020	0.2406
Number of cases	-0.0028	0.1674	-0.0165	0.9870
Distance	0.1006	0.1423	0.7069	0.4862
Delivery charges	1.0481	0.1192	8.7940	0.0000

1. Find the missing entries denoted by “?” in the above table.
2. Write down the fitted multiple regression equation.
3. Use the p-value approach to test the significance of the individual regression coefficients at the 5% level of significance.
4. Test the regression hypothesis by the F-value of the ANOVA table.   
5. Interpret R² and Adjusted R²

Answer 1

1. nd the missing entries denoted by “?” in the above table.

Multiple R	0.92124372
R Square	0.84869
Adjusted R Square	0.83054
Standard Error	0.43435
Observations	29
ANOVA
	df	SS	MS	F
Regression	3	26.456	8.8186	46.7423568
Residual	25	4.7166	0.1887
Total	28	?
	Coefficients	Standard Error	t Stat	P-value
Intercept	-0.5753	0.4786	-1.202	0.2406
Number of cases	-0.0028	0.1674	-0.017	0.987
Distance	0.1006	0.1423	0.7069	0.4862
Delivery charges	1.0481	0.1192	8.794	0

1. Write down the fitted multiple regression equation.

y = -0.5753 - 0.0028*x1 + 0.1006*x2 + 1.0481*x3

1. Use the p-value approach to test the significance of the individual regression coefficients at the 5% level of significance.

The hypothesis being tested is:

H0: β1 = 0

H1: β1 ≠ 0

The p-value is 0.987.

Since the p-value (0.987) is greater than the significance level (0.05), we fail to reject the null hypothesis.

Therefore, we cannot conclude that the slope is significant.

The hypothesis being tested is:

H0: β2 = 0

H1: β2 ≠ 0

The p-value is 0.4862.

Since the p-value (0.4862) is greater than the significance level (0.05), we fail to reject the null hypothesis.

Therefore, we cannot conclude that the slope is significant.

The hypothesis being tested is:

H0: β3 = 0

H1: β3 ≠ 0

The p-value is 0.0000.

Since the p-value (0.000) is less than the significance level (0.05), we can reject the null hypothesis.

Therefore, we can conclude that the slope is significant.

1. Test the regression hypothesis by the F-value of the ANOVA table.   

The hypothesis being tested is:

H0: β1 = β2 = β3 = 0

H1: At least one βi ≠ 0

The p-value is 0.0000.

Since the p-value (0.000) is less than the significance level (0.05), we can reject the null hypothesis.

Therefore, we can conclude that the model is significant.