Question

A regional express delivery service company recently conducted a study to investigate the relationship between the cost of shipping a package ($), the package weight (in pound) and the distance shipped (in miles). Twenty packages were randomly selected from among the large number received for shipment, and a detailed analysis of the shipping cost was conducted for each package. The data for this sample observations are given in the file Assignment 4 S1 2020.XLS.

a. Estimate a simple linear regression model involving shipping cost and package weight. Interpret the slope coefficient of the least squares line as well as the computed value of ? 2 . [4 marks]

b. Add another explanatory variable–distance shipped–to the regression model in part a. Estimate and interpret this expanded model. How does the ? 2value for this multiple regression model compare to that of the simple regression model estimated in part a? [5 marks]

c. Use the F test to determine the overall significance of the regression relationship for the expanded model. What is the conclusion at the 0.01 level of significance? [4 marks]

d. Use the t test to determine the significance of each independent variable. What is the conclusion for each test at the 0.01 level of significance? [4 marks]

Data

Cost of Shipment and Potentially Relevant Data
Cost_of_Shipment	Package_Weight	Distance_Shipped
$3.30	4.10	95
$2.00	0.30	160
$11.00	5.10	240
$2.60	5.90	47
$1.90	4.50	53
$8.00	3.50	250
$15.50	7.00	260
$5.00	2.40	209
$1.00	0.60	100
$4.40	0.75	280
$6.00	6.20	115
$1.70	1.10	90
$14.50	6.50	240
$14.00	7.50	190
$9.20	6.60	160
$1.10	2.70	45
$12.10	8.10	160
$1.50	0.70	80
$8.00	4.40	202
$3.90	3.20	145
$4.40	0.75	280
$16.50	7.20	280
$15.50	7.00	250
$14.00	7.50	190
$3.30	4.10	95
$2.20	1.50	160
$11.00	5.10	240
$1	0.6	100
$4	0.75	280
$2	0.7	80
$8	4.4	202
$2	4.5	52
$8.00	3.2	240
$15.50	7.6	270
$5.00	2.5	211
$1.00	7	98
$8.00	4.4	202
$3.90	3.2	145
$4.40	0.75	280
$5.00	2.4	209

Answer 1

using excel>data>data analysis>Regression

we have

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.742802
R Square	0.551755
Adjusted R Square	0.539959
Standard Error	3.354405
Observations	40
ANOVA
	df	SS	MS	F	Significance F
Regression	1	526.3139	526.3139	46.77501	4.06E-08
Residual	38	427.5771	11.25203
Total	39	953.891
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	0.781701	0.994461	0.786055	0.43671	-1.23148	2.794882
Package weight	1.472373	0.215284	6.839226	4.06E-08	1.036554	1.908192

a simple linear regression model involving shipping cost and package weight is

cost = 0.7817+1.4724 *Package weight

For every one one unit increase in weight , cost of shipment will increase by $1.4724

the ? 2value = 0.5518 , about 55.18 % variation in shipment cost is explained by  Package weight

using excel>data>data analsysis>Regression

we have

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.931476
R Square	0.867647
Adjusted R Square	0.860493
Standard Error	1.847203
Observations	40
ANOVA
	df	SS	MS	F	Significance F
Regression	2	827.6411	413.8206	121.2782	5.65E-17
Residual	37	126.2499	3.412159
Total	39	953.891
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	-5.08357	0.830332	-6.12233	4.29E-07	-6.76598	-3.40116
Package weight	1.347949	0.119289	11.29982	1.48E-13	1.106246	1.589653
Distance speed	0.036372	0.00387	9.397333	2.43E-11	0.02853	0.044214

b. the new regression model is

cost = -5.08357 +1.34795 *Package weight + 0.0364* Distance speed

For every one one unit increase in speed , cost of shipment will increase by $\0.0364

the ? 2value = 0.8677, about 86.67 % variation in shipment cost is explained by  Package weight and distance speed .

for this multiple regression model the value of R2 is more in compare to that of the simple regression model estimated in part a

c ) the value of F stat for overall model is 121.278

and p value is 0.0000 which is less than 0.01 so we conclude that model is significant to use .

d ) the explanatory variables package placed and distance speed are signficant because theri p value is less than 0.01