1. The response variableRentalRate is the rental rate for 81 suburban commercial properties in a large metropolitan area. The predictor variables are Age (the age of property),Expenses (the operating expenses and taxes),Vacancy (the vacancy rate), andArea (the total square footage).
   1. Fit a multiple linear regression model usingRentalRate as the response variable andAge,Expenses,Vacancy, andArea as the predictor variables. Perform a hypothesis test at significance level 0.05 to determine if at least one of the predictors in this model is useful in predictingRentalRate. State your null and alternative hypotheses in terms of the regression coefficients (b's), the test statistic value with calculations shown (i.e., how the relevant number in the Anova table is calculated from other numbers in the Anova table), the decision rule and the conclusion.
   2. Use a partial F-test to determine if the predictor variablesVacancy andAreacan be deleted from the model you fit in part (a) while retaining the two remaining predictor variablesAge andExpenses. Again, state your null and alternative hypotheses in terms of regression coefficients, show your work in calculating the test statistic (i.e., using sequential sums of squares), state the decision rule and the conclusion.
   3. Confirm the value of the partial F-statistic from part (b) by calculating the F-statistic using the general linear F-test formula. State the full and reduced models andshow your work in calculating the test statistic. [There may be round-off error in your calculation so that the values may not match exactly.]
   4. Perform a hypothesis test to determine ifVacancycan be dropped from the four-predictor model you fit in part (a) by using:
      1. an F-statistic
      2. a t-statistic

In each case, state your null and alternative hypotheses in terms of the regression coefficients, the test statistic value, the decision rule and the conclusion.

Is there any relationship between the two test statistics in (i) and (ii) above?

1. Fit a multiple linear regression model usingRentalRate as the response variable andAge,Expenses, andArea as the predictor variables and write down the fitted regression equation.
2. Calculate the value of the coefficient of partial determination R²_Y,3|1,2 for the model you fit in part (e) and explain in words what it measures. [Here "Y refers to the response,RentalRate, "3" refers to the third predictor,Area, and "1" and "2" refer toAge andExpenses.]

The CommercialProperties data set for the above question is below:

Rental Rate	Age	Expenses	Vacancy	Area
13.5	1	5.02	0.14	123000
12	14	8.19	0.27	104079
10.5	16	3	0	39998
15	4	10.7	0.05	57112
14	11	8.97	0.07	60000
10.5	15	9.45	0.24	101385
14	2	8	0.19	31300
16.5	1	6.62	0.6	248172
17.5	1	6.2	0	215000
16.5	8	11.78	0.03	251015
17	12	14.62	0.08	291264
16.5	2	11.55	0.03	207549
16	2	9.63	0	82000
16.5	13	12.99	0.04	359665
17.225	2	12.01	0.03	265500
17	1	12.01	0	299000
16	1	7.99	0.14	189258
14.625	12	10.33	0.12	366013
14.5	16	10.67	0	349930
14.5	3	9.45	0.03	85335
16.5	6	12.65	0.13	235932
16.5	3	12.08	0	130000
15	3	10.52	0.05	40500
15	3	9.47	0	40500
13	14	11.62	0	45959
12.5	1	5	0.33	120000
14	15	9.89	0.05	81243
13.75	16	11.13	0.06	153947
14	2	7.96	0.22	97321
15	16	10.73	0.09	276099
13.75	2	7.95	0	90000
15.625	3	9.1	0	184000
15.625	3	12.05	0.03	184718
13	16	8.43	0.04	96000
14	16	10.6	0.04	106350
15.25	13	10.55	0.1	135512
16.25	1	5.5	0.21	180000
13	14	8.53	0.03	315000
14.5	3	9.04	0.04	42500
11.5	15	8.2	0	30005
14.25	1	6.13	0	60000
15.5	15	8.32	0	73521
12	1	4	0	50000
14.25	15	10.1	0	50724
14	3	5.25	0.16	31750
16.5	3	11.62	0	168000
14.5	4	5.31	0	70000
15.5	1	5.75	0	27000
16.75	4	12.46	0.03	129614
16.75	4	12.75	0	129614
16.75	2	12.75	0	130000
16.75	2	11.38	0	209000
17	1	5.99	0.57	220000
16	2	11.37	0.27	60000
14.5	3	10.38	0	110000
15	15	10.77	0.05	101206
15	17	11.3	0	288847
16	1	7.06	0.14	105000
15.5	14	12.1	0.05	276425
15.25	2	10.04	0.06	33000
16.5	1	4.99	0.73	210000
19.25	0	7.33	0.22	240000
17.75	18	12.11	0	281552
18.75	16	12.86	0	421000
19.25	13	12.7	0.04	484290
14	20	11.58	0	234493
14	18	11.58	0.03	230675
18	16	12.97	0.08	296966
13.75	1	4.82	0	32000
15	2	9.75	0.03	38533
15.5	16	10.36	0.02	109912
15.9	1	8.13	0.23	236000
15.25	15	13.23	0.05	243338
15.5	4	10.57	0.04	122183
14.75	20	11.22	0	128268
15	3	10.34	0	72000
14.5	3	10.67	0	43404
13.5	18	8.6	0.08	59443
15	15	11.97	0.14	254700
15.25	11	11.27	0.03	434746
14.5	14	12.68	0.03	201930