What is driving the Cost of Cars?

Manufacturers are interested in understanding the driving factors of car prices. They want to make the best decisions in terms of car designs. One way to justify increasing the price of cars is to adjust the size of the car. But size is a function of several other components. Before investing in exotic materials and different sizes, manufacturers want evidence of the benefit. We will answer this question using multiple regression. Data collected on several new cars, on which various variables have been measured, are presented in the JMP file Cars.jmp and a glossary of the tabulated variables is given below. Response Variable: MidPrice (in $1,000, average price between the basic version of this model and the price for the premium version) Explanatory variables: EngineSize (in liters) Horsepower (number of) REV (engine revolutions per mile, in highest gear) FuelTank (capacity, in gallons) Length of the car (in inches) Wheelbase (distance between a car's front and rear wheels, in inches) Width of the car (in inches) Weight(pounds) - 1. Make a scatterplot matrix and a correlation matrix of all variables. To get this go to Analyze Multivariate Methods Multivariate. Then put all variable names into Y columns and click OK. Keep this scatterplot matrix and the correlation matrix for a reference but do not worry about turning this output in. I recommend putting MidPrice in first and then putting the rest of the explanatory variables below. This makes referencing the variables much easier. (a) Which explanatory variable(s) has(have) have a positive correlation with the response? MidPrice EngineSize Horsepower REV FuelTank Length Wheelbase Width Weight NONE (b) Which explanatory variable(s) has(have) a strong correlation (|r|> 0.7) with the response? MidPrice EngineSize Horsepower REV FuelTank Length Wheelbase Width Weight NONE (c) Which pair(s) of explanatory variables have a very strong correlations (i.e. |r|> 0.9)? In the upper diagonal of the matrix below, put a mark for all pairs with the strength of the correlations (VS for very strong). Also record the pair with the strongest correlation. Horsepower REV FuelTank Length Wheelbase Width Weight Engine Horsepower X Rev X X FuelTank X X X Length X X X X Wheelbase X X X X X Width X X X X X X

2. Using Fit Model, fit a multiple regression model that predicts MidPrice based on all the explanatory variables in this dataset. We will call this our Full Model. To save the VIF values hold the cursor over the t-Ratio in the Parameter Estimates part of the output. Next right click and select Columns. Choose option VIF. Record all requested summaries in Table 1 on page 3. Save the largest VIF value for this Full Model. (a) Is the Full Model Useful? Answer this question with a hypothesis test evaluating for the overall utility of the model. (b) We discussed several tools (procedures) that are used to detect indicators of multicollinearity. For each parts read each description below. Select all variables (if any) which suggest a problem with multicollinearity in the full model that is consistent with the provided description. i. An opposite sign of the estimated slope for an explanatory variable compared to the sign of the correlation between this explanatory variable and the response variable. Here the concern is related to inconsistencies when the original correlation is moderate or strong (i.e. |r|> 0.3). EngineSize Horsepower REV FuelTank Length Wheelbase Width Weight NONE ii. Observing very strong correlations (i.e. |r|> 0.9) between explanatory variables. EngineSize Horsepower REV FuelTank Length Wheelbase Width Weight NONE iii. Observing very large Variance Inflation Factors (i.e. > 10) in an estimated model. EngineSize Horsepower REV FuelTank Length Wheelbase Width Weight NONE iv. Observing at least a moderately large Variance Inflation Factors (i.e. > 5) in an estimated model. EngineSize Horsepower REV FuelTank Length Wheelbase Width Weight NONE (c) Based on the output for the full model, which explanatory variable would you eliminate first if you were to perform the Backwards Elimination procedure using a Prob to Leave of 0.05. EngineSize Horsepower REV FuelTank Length Wheelbase Width Weight NONE

3. Run Forward Selection in JMP. Go to Fit Model. Put all explanatory variables into the Construct Model Effects box. Put the response variables into the Y box. Select Personality and choose Stepwise. Click Run. Next change the Stopping rule to P-value threshold. Use Prob to Enter as 0.05. Make sure you start with no variables checked. Click Go. Next click Run Model. Record the R2and other summaries in Table 1 below under the column labeled Forward Selection. You will come back to this model in a little bit, so do not close the window just yet.

4. For all parts below read each description below. Circle all explanatory variables which are consistent with each description's indicator of multicollinearity. In each case, reference the model produced using Forward Selection. (a) Observing large Variance Inflation Factors (i.e. > 10) in an estimated model. EngineSize Horsepower REV FuelTank Length Wheelbase Width Weight NONE (b) Observing p values > 0.05 for coefficient hypothesis tests along with strong correlations (i.e. |r|> 0.7) between the explanatory variable and the response. EngineSize Horsepower REV FuelTank Length Wheelbase Width Weight NONE (c) An opposite sign of the estimated slope for an explanatory variable compared to the sign of the correlation between this explanatory variable and the response variable. Here the concern is related to inconsistencies when the original correlation is moderate or strong (i.e. |r|> 0.3). EngineSize Horsepower REV FuelTank Length Wheelbase Width Weight NONE

5. Similarly, run Backward Elimination in JMP. Change the Direction to Backward. Use the stopping rule Prob to Leave equal to 0.05. In this case you have to hit the Enter All button so that all variables are check marked before hitting GO. After you hit Run Model, record the R2and other summaries in Table 1 below under the column labeled Backward Elimination. You will come back to this model in a little bit, so do not close the model fit window just yet.

6. Similarly, run Mixed Selection in JMP. Change the direction to Mixed, Prob to Enter (0.10) and Prob to Leave (0.10). Hit the button Remove All before hitting GO. After you run the model, record the R2and other summaries in Table 1 below under the column labeled Mixed (0.10).

7. Similarly, run Mixed Selection in JMP. Change the direction to Mixed, Prob to Enter (0.05) and Prob to Leave (0.05). Hit the button Remove All before hitting GO. After you run the model, record the R2and other summaries in Table 1 below under the column labeled Mixed (0.05). Table 1: Table of Summaries for the models fit in this lab. Full Model Forward Backward Mixed Mixed Selection Elimination 0.10 0.05 R2 RMSE Largest VIF Largest p-value for all individual coefficient t-tests: i = 0 vs i 6= 0

8. Explain in own words what the various summaries in the table above imply about the different tools available for model building. Do all of the summaries suggest the same preferred model? Comments should include discussion related to R2, p-value and VIF.

Here is the JMP to use to get the correct data.

MidPrice	EngineSize	Horsepower	REV	FuelTank	Length	Wheelbase	Width	Weight
15.9	1.8	140	2890	13.2	177	102	68	2705
33.9	3.2	200	2335	18	195	115	71	3560
29.1	2.8	172	2280	16.9	180	102	67	3375
37.7	2.8	172	2535	21.1	193	106	70	3405
30	3.5	208	2545	21.1	186	109	69	3640
15.7	2.2	110	2565	16.4	189	105	69	2880
20.8	3.8	170	1570	18	200	111	74	3470
23.7	5.7	180	1320	23	216	116	78	4105
26.3	3.8	170	1690	18.8	198	108	73	3495
34.7	4.9	200	1510	18	206	114	73	3620
40.1	4.6	295	1985	20	204	111	74	3935
13.4	2.2	110	2380	15.2	182	101	66	2490
11.4	2.2	110	2665	15.6	184	103	68	2785
15.1	3.4	160	1805	15.5	193	101	74	3240
15.9	2.2	110	2595	16.5	198	108	71	3195
16.3	3.8	170	1690	20	178	110	74	3715
16.6	4.3	165	1790	27	194	111	78	4025
18.8	5	170	1350	23	214	116	77	3910
38	5.7	300	1450	20	179	96	74	3380
18.4	3.3	153	1990	18	203	113	74	3515
15.8	3	141	2090	16	183	104	68	3085
29.5	3.3	147	1785	16	203	110	69	3570
9.2	1.5	92	3285	13.2	174	98	66	2270
11.3	2.2	93	2595	14	172	97	67	2670
13.3	2.5	100	2535	16	181	104	68	2970
19	3	142	1970	20	175	112	72	3705
15.6	2.5	100	2465	16	192	105	69	3080
25.8	3	300	2120	19.8	180	97	72	3805
12.2	1.5	92	2505	13.2	174	98	66	2295
19.3	3.5	214	1980	18	202	113	74	3490
7.4	1.3	63	3150	10	141	90	63	1845
10.1	1.8	127	2410	13.2	171	98	67	2530
11.3	2.3	96	2805	15.9	177	100	68	2690
15.9	2.3	105	2285	15.4	180	101	68	2850
14	2	115	2340	15.5	179	103	70	2710
19.9	3	145	2080	21	176	119	72	3735
20.2	3	140	1885	16	192	106	71	3325
20.9	4.6	190	1415	20	212	114	78	3950
8.4	1	55	3755	10.6	151	93	63	1695
12.5	1.6	90	3250	12.4	164	97	67	2475
19.8	2.3	160	2855	15.9	175	100	70	2865
12.1	1.5	102	2650	11.9	173	103	67	2350
17.5	2.2	140	2610	17	185	107	67	3040
8	1.5	81	2710	11.9	168	94	63	2345
10	1.8	124	2745	13.7	172	98	66	2620
10	1.5	92	2540	11.9	166	94	64	2285
13.9	2	128	2335	17.2	184	104	69	2885
28	3	185	2325	18.5	188	103	70	3510
35.2	3	225	2510	20.6	191	106	71	3515
34.3	3.8	160	1835	18.4	205	109	73	3695
36.1	4.6	210	1840	20	219	117	77	4055
8.3	1.6	82	2370	13.2	164	97	66	2325
11.6	1.8	103	2220	14.5	172	98	66	2440
16.5	2.5	164	2505	15.5	184	103	69	2970
19.1	3	155	2240	19.6	190	110	72	3735
32.5	1.3	255	2325	20	169	96	69	2895
31.9	2.3	130	2425	14.5	175	105	67	2920
14.1	1.6	100	2475	11.1	166	95	65	2450
14.9	3.8	140	1730	18	199	113	73	3610
10.3	1.5	92	2505	13.2	172	98	67	2295
26.1	3	202	2210	19	190	107	70	3730
11.8	1.6	110	2435	13.2	170	96	66	2545
15.7	2.4	150	2130	15.9	181	103	67	3050
19.1	3	151	2065	20	190	112	74	4100
21.5	3	160	2045	18.5	188	104	69	3200
13.5	2.3	155	2380	15.2	188	103	67	2910
16.3	2.2	110	2565	16.5	190	105	70	2890
19.5	3.8	170	1690	20	194	110	74	3715
20.7	3.8	170	1570	18	201	111	74	3470
14.4	1.8	92	2360	15.9	173	97	67	2640
9	1.6	74	3130	13.2	177	99	66	2350
11.1	2	110	2665	15.2	181	101	66	2575
17.7	3.4	160	1805	15.5	196	101	75	3240
18.5	3.4	200	1890	16.5	195	108	72	3450
24.4	3.8	170	1565	18	177	111	74	3495
28.7	2.1	140	2910	18	184	99	67	2775
11.1	1.9	85	2145	12.8	176	102	68	2495
8.4	1.2	73	2875	9.2	146	90	60	2045
10.9	1.8	90	3375	15.9	175	97	65	2490
19.5	2.2	130	2330	15.9	179	102	67	3085
8.6	1.3	70	3360	10.6	161	93	63	1965
9.8	1.5	82	3505	11.9	162	94	65	2055
18.4	2.2	135	2405	15.9	174	99	69	2950
18.2	2.2	130	2340	18.5	188	103	70	3030
22.7	2.4	138	2515	19.8	187	113	71	3785
9.1	1.8	81	2550	12.4	163	93	63	2240
19.7	2.5	109	2915	21.1	187	115	72	3960
20	2	134	2685	18.5	180	103	67	2985
23.3	2.8	178	2385	18.5	159	97	66	2810
22.7	2.3	114	2215	15.8	190	104	67	2985
26.7	2.4	168	2310	19.3	184	105	69	3245