Question

The Cronch Café, located at the Gulf of Mexico, has an increase in business during the summer vacation season. The owner hires a large number of servers as seasonal help. When he interviews a prospective server, he would like to provide data on the amount a server can earn in tips. He believes that the amount of the bill and the number of diners are both related to the amount of the tip. He gathered this sample information. 1) Develop a multiple regression equation with the amount of tips as the dependent variable and the amount of the bill and the amount of diners as independent variables. Write out the regression equation. How much does another diner add to the amount of the tips? 2) Conduct a global test of hypothesis to determine if at least one of the independent variables is significant What is your conclusion? 3) Conduct an individual test on each of the variables. Should one or the other be deleted? Plot the residuals against the fitted values. Is it reasonable to conclude they are random?

Customer	Amount of Tip ($)	Amount of Bill ($)	Diners
1	5.15	74.5	2
2	4.5	28.23	4
3	1	10.65	1
4	2.4	19.82	3
5	5	28.62	3
6	4.25	24.83	2
7	0.5	6.25	1
8	6	49.2	4
9	5	43.26	3
10	4.65	62.23	1
11	5.6	84.81	1
12	6	34.99	3
13	4	33.91	4
14	3.35	23.06	2
15	0.75	4.65	1
16	3.3	23.59	2
17	3.5	22.3	2
18	3.25	32	2
19	5.4	50.02	4
20	2.25	17.6	3
21	4.35	63.16	6
22	3	20.27	2
23	1.25	19.53	2
24	3.25	27.03	3
25	3	21.28	2
26	6.25	43.38	4
27	5.6	28.12	4
28	2.5	26.25	2
29	6.85	53.08	7
30	8.6	87.85	8

Answer 1

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.864506
R Square	0.747371
Adjusted R Square	0.728658
Standard Error	0.991806
Observations	30
ANOVA
	df	SS	MS	F	Significance F
Regression	2	78.57234	39.28617	39.93801	8.58E-09
Residual	27	26.55932	0.983679
Total	29	105.1317
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	0.850209	0.401147	2.119443	0.043394	0.027123	1.673294	0.027123	1.673294
Amount of Bill ($)	0.05208	0.009705	5.366394	1.14E-05	0.032167	0.071992	0.032167	0.071992
Diners	0.449504	0.122111	3.681115	0.001022	0.198953	0.700055	0.198953	0.700055

Regression equation
Amount of Tip = 0.85 + 0.05*Amount_of_Bill +0.45*Diners
How much does another diner add to the amount of the tips? = Coefficient of Diners = 0.449504

2) Conduct a global test of hypothesis
For a global test of, we need to check if the F value of the model is significant

F	Significance F
39.93801	8.58E-09

Since the F-value is greater than the critical value and the significance is lesser than the confidence interval of 10%, 5% or 1%, the model is significant and at least one of the independent variables is significant.

3) For individual test, t-test must be carried out on each of the variable and the P value shall be less than the confidence level. In the results above, all the variables have p-value less than 0.05, therefore, are significant and need not be dropped.

As the residuals are randomly distributed along the horizontal axis, it is reasonable to assume that they are random.