Question

2. Regression review II, problem 6 Aa Aa E This question aims to review the La Quinta Inn model building exercise as discussed in lecture and described in the course packet. The data collected can be obtained in this Excel file. (Note: Reload this file each time you go through this problem.) The coefficient of multiple determination for the regression of profit margin (MARGIN) on all the other variables is The adjusted coefficient of multiple determination is From the regression output, at a 10% level of significance, which of the following variables are insignificant individually? Check all that apply. ROOMS O INCOME O OFFICE I DISTTWN ONEAREST D COLLEGE Perform a partial F-test to see whether these variables are really useless as a group in explaining the behavior of profit margin. The F-test statistic for testing the significance of the subset variables is . The p-value of the test statistic for testing the significance of the subset variables is . At a 5% level of significance, these variables significant as a group. Therefore, you choose as your final model. Now, using this final model, predict the profit margin in each of the following locations: Ann Arbor 2,672 1.3 Characteristic Rooms Competitor distance Office space ('000s) Students ('000s) Income ('000s) Distance to downtown Predicted profit margin Bloomington 2,500 1.2 604 21 37 952 Champaign 2,300 0.5 1,430 45 33.5 1.4 D- 42 35 3.4 - 4.5 1 C C Flach player MAC 32 270

INN	MARGIN	ROOMS	NEAREST	OFFICE	COLLEGE	INCOME	DISTTWN
1	44.2	3471	2.1	523	12	35	9.4
2	29.8	3567	1.8	140	13.5	42	5.7
3	38.4	3264	1.6	404	22.5	45	4.4
4	54.4	3234	1.1	649	19.5	35	6.5
5	34.5	2730	4	171	17	41	10.5
6	44.9	3003	3.4	402	15.5	37	4.6
7	46	2341	2	580	23	45	7.4
8	50.2	3021	2.6	572	8.5	33	9.3
9	46	2655	3.2	666	22	40	6.7
10	45.5	2691	1.1	519	13.5	28	9.1
11	55.5	3203	0.1	549	8	37	12.1
12	33.8	2810	1.5	496	17.5	39	0.4
13	49	2890	1.9	254	20	39	12.2
14	31.9	3422	1	434	15.5	36	2.7
15	57.4	2687	3.4	678	15.5	32	7.9
16	49	3759	1.4	635	19	41	4
17	46	2341	2	580	23	45	7.4
18	50.2	3021	2.6	572	8.5	33	9.3
19	46	2655	3.2	666	22	40	6.7
20	45.5	2691	1.1	519	13.5	28	9.1
21	41.1	3098	2	449	11.5	40	7.1
22	35.7	3591	2.9	270	9.5	31	9.8
23	58.5	2210	1.6	611	17.5	39	12.1
24	53.5	3285	2.3	548	23.5	32	11.9
25	54.2	3018	1.2	529	20	35	6.6
26	43.3	3383	2.1	790	11.5	41	5.3
27	59	2382	1.1	484	12	35	13.5
28	38.1	3568	2	365	16.5	37	11.1
29	45.1	2172	2.9	198	7	39	5.6
30	42.8	3003	2.1	457	19.5	35	2.8
31	51.4	2640	0.5	431	19.5	37	7.2
32	45.4	2443	4.2	515	10.5	38	7.3
33	48.2	2429	2	498	20	41	7.2
34	40.2	3670	1.5	240	17.5	37	4.9
35	36.8	3006	1.8	456	15.5	37	10.6
36	54.2	2484	1.5	443	16	41	14.2
37	36.9	2923	3.1	449	14	34	7.2
38	53	2844	0.8	406	13.5	44	4
39	39.3	3159	1.6	392	11	40	14.6
40	54.4	3241	2.1	578	21.5	30	9.7
41	43.7	2915	2.7	620	13.5	38	9.5
42	51.5	3069	1	512	20.5	37	7.1
43	43.5	2910	2.9	418	12.5	39	13
44	48.8	2935	0.4	551	20	39	8.7
45	36.1	2962	3.4	164	12.5	33	6.2
46	33.2	2629	2.9	359	20	39	7.6
47	41.9	3517	1.4	516	6	36	5.7
48	49.8	2859	2.6	544	14.5	35	11.8
49	45	3697	2.2	236	20.5	34	5.8
50	43.2	2724	3.1	597	14.5	32	14.5
51	38.8	3330	1.6	410	13.5	33	10.8
52	37.8	2980	1.1	427	12	42	8.2
53	39	2846	1.2	231	20.5	43	6
54	48.8	2825	3.2	774	13	34	5.8
55	34.3	3548	1.8	449	13.5	43	6.5
56	49.6	2922	2.4	459	19.5	42	10.4
57	49.9	3255	2.1	551	22.5	38	6.1
58	39.2	2996	3	359	15.5	36	6.3
59	42.5	2403	3.3	345	19	42	6.6
60	52.6	3013	0.6	830	20	33	0.7
61	47.3	2642	2.5	450	9	44	10
62	41	3480	2.4	265	18	43	3.8
63	51.1	3227	2.4	711	11	37	8.8
64	52.1	3140	2.1	433	23.5	40	11.1
65	47.6	2751	1.3	426	10.5	38	5.7
66	40.8	2676	1	300	25	33	9.7
67	52.2	2879	2.2	747	19.5	38	11.9
68	60.1	2619	1.4	683	12	43	8
69	52	3354	2.3	614	26.5	37	7.6
70	51.2	3082	2.8	637	22	34	3.6
71	51.2	2775	1.8	497	15	35	9.2
72	44	2813	0.5	233	15	41	3
73	49.6	3359	2.7	473	10.5	30	12.2
74	47.5	3080	2.4	488	13.5	31	6.7
75	54.4	2756	1.1	832	14.5	35	6.9
76	46.2	2244	0.7	496	15.5	38	8.9
77	54.1	2862	1.4	809	16.5	33	3
78	43.5	3198	2.5	548	17	38	10.2
79	52.7	3378	1	815	10	40	7.2
80	49.5	2726	3.1	464	19	38	5.5
81	43.4	3597	0.4	616	15.5	39	1.6
82	33.5	3657	2.8	372	18	46	11.3
83	46.3	2554	2.2	585	10	42	4.9
84	42.9	3838	1.9	875	10	34	11.9
85	27.3	4214	1.9	358	16	40	9.2
86	41.6	2776	2.2	360	16	43	7.5
87	49	1998	0.7	461	19	33	9.9
88	55.4	2790	2.7	645	12	37	5.7
89	54.1	2432	1.4	370	17.5	39	4.5
90	32.4	3124	2.7	349	17	45	7.5
91	47.3	2761	0.9	485	16	35	8.2
92	30.8	3622	2	347	15	40	7.2
93	49	2826	0.1	390	17	40	13.6
94	57.3	2601	1.2	520	18.5	35	3.8
95	60.5	2932	2.5	469	19.5	35	8.7
96	53.4	2772	3.5	622	14.5	29	3.5
97	35.9	2786	2.4	511	15.5	37	6.3
98	40	3397	2.7	855	19.5	42	11.7
99	39.8	3823	0.7	202	17	36	10
100	30.1	3251	2.6	275	13	39	10.5

Answer 1

Using Mini tab software , we perform multiple regression anylysis

Our output is

Analysis of Variance

Source	DF	Adj SS	Adj MS	F-Value	P-Value
Regression ROOMS NEAREST OFFICE COLLEGE INCOME DISTTWN	6 1 1 1 1 1 1	2798.94 812.16 220.67 1076.22 57.78 237.48 33.59	466.49 812.16 220.67 1076.22 57.78 237.48 33.59	15.11 26.30 7.15 34.85 1.87 7.69 1.09	0.000 0.000 0.009 0.000 0.175 0.007 0.300
Error Lack-of-Fit Pure Error	93 89 4	2871.69 2871.69 0.00	30.88 32.27 0.00	*	*
Total	99	5670.62

Model Summary

      S      R-sq        R-sq(adj)    R-sq(pred)
5.55683 49.36%     46.09%      41.55%

Answer

i) The coefficient of multiple determination for the regression of profit margin (MARGIN) on all other variable

          (R² ) is given by R^{2 =}1 - SS_{RES /} SS_T

Calculation R^{2 =}1 - SS_{RES /} SS_T = 1 -(2871.69 / 5670.62) =1-0.5064 = 0.4936

Thus ,The coefficient of multiple determination for the regression of profit margin (MARGIN) on all other variable is 49.36%

Coefficient of multiple determination(R²) is 49.36% . Thus , it the measures the variation in the dependent variable MARGIN that can be predict from independent variables , so we conclude that 49.36% variation in the dependent variable (MARGIN) can be explained by independent variables

ii) The adjusted coefficient of multiple determination : R²_adj = 1- [ SS_RES/(n-p) ) / ( SS_T/(n-1) ]

                                                                                           = 1 - [ (2871.69/93) / ( 5670.62/99)

                                                                                           = 1 - 0.5391

                                                                                           = 0.4609

Thus , adjusted coefficient of multiple determination R²_adj is 46.09%

iii) from regression output at 10% of level of significance to check which variable are insignificant

Our regression model is

MARGIN (y) = $\small \beta$ 0 + $\small \beta$ 1*ROOMS +$\small \beta$2*NEAREST+$\small \beta$3*OFFICE+$\small \beta$4*COLLEGE+$\small \beta$5*INCOME+$\small \beta$6*DISTTWN

i . e

MARGIN = 69.74 - 0.00690 ROOMS - 1.681 NEAREST + 0.02063 OFFICE + 0.180 COLLEGE
        - 0.391 INCOME + 0.189 DISTTWN

Hypothesis for testing significance of any individual regression coefficient is

      H0 : $\small \beta j$ = 0 ( i .e .given j variable in not significant or insignificant { j = 1,2,3,4,5,6 } )   

vs   H1 : $\small \beta j$ $\small \neq$ 0 ( i .e .given j variable in contributes significantly )

Test statistics

                       T_o = $\small \hat{\beta j}$ / SE( $\small \beta j$ )

where $\small \hat{\beta j}$ is coefficient of coresponding independent var

reject H0 is | To | >
_{a/2,n-k-1
   ,   where n = 100 , k = 6 ( no of independent variables)}

Term	Coef	SE Coef	T-Value	P-Value
Constant ROOMS NEAREST OFFICE COLLEGE INCOME DISTTWN	69.74 -0.00690 -1.681 0.02063 0.180 -0.391 0.189	7.98 0.00135 0.629 0.00349 0.131 0.141 0.181	8.73 -5.13 -2.67 5.90 1.37 -2.77 1.04	0.000 0.000 0.009 0.000 0.175 0.007 0.300

_a/2,n-k-1
= 1.661

Thus we reject H0 if |To | >1.661   i.e | T-Value | > 1.661

Thus we do not reject H0 for variables COLLAGE and DISTTWN

Therefore , from regression output, at 10% of level of significance these given variables are significant

ROOMS

NEAREST

OFFICE

INCOME

And , from regression output, at 10% of level of significance these given variables are Insignificant

1. COLLEGE
2. DISTTWN
        

iv) To see whether these variable are really useless as a group we perform partial F-test

H0 : $\small \beta$ ₁ = $\small \beta$ ₂=$\small \beta$₃=$\small \beta$₄=$\small \beta$₅=$\small \beta$₆ = 0

                                               Test Statistics : F0 = MS _R / MS _RES

We reject H0 if F0 >
_Fa,k,n-k-1

The F statistic for testing the significance of the subset variable ( for 5% level of significance ) is =
F 5,6,100-6-1

                                                                                                                                                       =   = 2.19768

F 5,6,100-6-1

Now we perform partial F test to see variable are really useless as group : -

Source	DF	Adj SS	Adj MS	F-Value	P-Value
Regression ROOMS NEAREST OFFICE COLLEGE INCOME DISTTWN	6 1 1 1 1 1 1	2798.94 812.16 220.67 1076.22 57.78 237.48 33.59	466.49 812.16 220.67 1076.22 57.78 237.48 33.59	15.11 26.30 7.15 34.85 1.87 7.69 1.09	0.000 0.000 0.009 0.000 0.175 0.007 0.300
Error Lack-of-Fit Pure Error	93 89 4	2871.69 2871.69 0.00	30.88 32.27 0.00	*	*

Total

99

5670.62

Thus we reject H0 if    F0 > 2.19768

The P - Value of tests statistic for testing the significance of the subset variable is given in above table.

At 5 % of level of significance we reject H0 is P-value < 0.05

Thus at 5% of level of significance these variables COLLAGE and DISTTWN are significant as a group

Our Model is

MARGIN = 69.74 - 0.00690 ROOMS - 1.681 NEAREST + 0.02063 OFFICE + 0.180 COLLEGE
           - 0.391 INCOME + 0.189 DISTTWN

To predict Profit margin in each of these case

For Ann Arbor , substituting coresponding values of independent variable in our model

We get our profit margin

MARGIN =     69.74 - 0.00690*2672 - 1.681*1.3 + 0.02063*952+ 0.180*42
                    - 0.391*35+ 0.189*3.4

MARGIN = 76.31766

Predicted profit margin for Ann Arbor is 76.31766

Similarly we calculate Predicted profit margin for Bloomington and Champaign

Thus

Predicted profit margin for Ann Arbor    is 76.31766

Predicted profit margin for Bloomington is 66.71332

Predicted profit margin for Champaign     is 90.6304