Question

An assistant in the district sales office of a national cosmetics firm obtained data on advertising expenditures and sales last year in the district’s 44 territories.

X1: expenditures for point-of-sale displays in beauty salons and department stores (X$1000).

X2: expenditures for local media advertising.

X3: expenditures for prorated share of national media advertising.

Y: Sales (X$1000).

y	x1	x2	x3
12.85	5.6	5.6	3.8
11.55	4.1	4.8	4.8
12.78	3.7	3.5	3.6
11.19	4.8	4.5	5.2
9	3.4	3.7	2.9
9.34	6.1	5.8	3.4
13.8	7.7	7.2	3.8
8.79	4	4	3.8
8.54	2.8	2.3	2.9
6.23	3.2	3	2.8
11.77	4.2	4.5	5.1
8.04	2.7	2.1	4.3
5.8	1.8	2.5	2.3
11.57	5	4.6	3.6
7.03	2.9	3.2	4
0.27	0	0.2	2.7
5.1	1.4	2.2	3.8
9.91	4.2	4.3	4.3
6.56	2.4	2.2	3.7
14.17	4.7	4.7	3.4
8.32	4.5	4.4	2.7
7.32	3.6	2.9	2.8
3.45	0.6	0.8	3.4
13.73	5.6	4.7	5.3
8.06	3.2	3.3	3.6
9.94	3.7	3.5	4.3
11.54	5.5	4.9	3.2
10.8	3	3.6	4.6
12.33	5.8	5	4.5
2.96	3.5	3.1	3
7.38	2.3	2	2.2
8.68	2	1.8	2.5
11.51	4.9	5.3	3.8
1.6	0.1	0.3	2.7
10.93	3.6	3.8	3.8
11.61	4.9	4.4	2.5
17.99	8.4	8.2	3.9
9.58	2.1	2.3	3.9
7.05	1.9	1.8	3.8
8.85	2.4	2	2.4
7.53	3.6	3.5	2.4
10.47	3.6	3.7	4.4
11.03	3.9	3.6	2.9
12.31	5.5	5	5.5

1. Test the regression relation between sales and the three predictor variables. State the hypotheses, test statistic and degrees of freedom, the p-value, the conclusion in words.

2. Determine whether the linear regression model is appropriate by using the “usual” plots (scatterplot, residual plots, histogram/QQ plot). Explain in detail whether or not each assumption appears to be substantially violated.

Answer 1

An assistant in the district sales office of a national cosmetics firm obtained data on advertising expenditures and sales last year in the district’s 44 territories.

X1: expenditures for point-of-sale displays in beauty salons and department stores (X$1000).

X2: expenditures for local media advertising.

X3: expenditures for prorated share of national media advertising.

Y: Sales (X$1000).

1. Test the regression relation between sales and the three predictor variables. State the hypotheses, test statistic and degrees of freedom, the p-value, the conclusion in words.

Ho: The regression model is not significant.

Ho: The regression model is significant.

Calculated F= 38.28, P=0.0000 which is < 0.05 level of significance. Ho is rejected.

The regression model is significant.

2. Determine whether the linear regression model is appropriate by using the “usual” plots (scatterplot, residual plots, histogram/QQ plot). Explain in detail whether or not each assumption appears to be substantially violated.

Residual plots and histogram/QQ plot shows the normality assumption and homogeneity assumption is not violated.

Residual vs predictors plot shows that the linearity assumption is not violated.

Excel Addon Megastat used.

Menu used: correlation/Regression ---- Regression Analysis

Regression Analysis
	R²	0.742
	Adjusted R²	0.722	n	44
	R	0.861	k	3
	Std. Error	1.825	Dep. Var.	y
ANOVA table
Source	SS	df	MS	F	p-value
Regression	382.6588	3	127.5529	38.28	0.0000
Residual	133.2863	40	3.3322
Total	515.9451	43
Regression output					confidence interval
variables	coefficients	std. error	t (df=40)	p-value	95% lower	95% upper
Intercept	1.0233	1.2029	0.851	.4000	-1.4078	3.4544
x1	0.9657	0.7092	1.362	.1809	-0.4677	2.3991
x2	0.6292	0.7783	0.808	.4237	-0.9438	2.2022
x3	0.6760	0.3557	1.900	.0646	-0.0430	1.3950