Question

Using the mtcars dataset, answer the following questions:

Fill in the following table:

Variable	Correlation with mpg
cyl	-0.85216
disp	-0.84755
hp	-0.77617
drat	0.681172
wt	-0.86766
qsec	0.418684
vs	0.664039
am	0.599832
gear	0.480285
carb	-0.55093

	mpg	cyl	disp	hp	drat	wt	qsec	vs	am	gear	carb
Mazda RX4	21	6	160	110	3.9	2.62	16.46	0	1	4	4
Mazda RX4 Wag	21	6	160	110	3.9	2.875	17.02	0	1	4	4
Datsun 710	22.8	4	108	93	3.85	2.32	18.61	1	1	4	1
Hornet 4 Drive	21.4	6	258	110	3.08	3.215	19.44	1	0	3	1
Hornet Sportabout	18.7	8	360	175	3.15	3.44	17.02	0	0	3	2
Valiant	18.1	6	225	105	2.76	3.46	20.22	1	0	3	1
Duster 360	14.3	8	360	245	3.21	3.57	15.84	0	0	3	4
Merc 240D	24.4	4	146.7	62	3.69	3.19	20	1	0	4	2
Merc 230	22.8	4	140.8	95	3.92	3.15	22.9	1	0	4	2
Merc 280	19.2	6	167.6	123	3.92	3.44	18.3	1	0	4	4
Merc 280C	17.8	6	167.6	123	3.92	3.44	18.9	1	0	4	4
Merc 450SE	16.4	8	275.8	180	3.07	4.07	17.4	0	0	3	3
Merc 450SL	17.3	8	275.8	180	3.07	3.73	17.6	0	0	3	3
Merc 450SLC	15.2	8	275.8	180	3.07	3.78	18	0	0	3	3
Cadillac Fleetwood	10.4	8	472	205	2.93	5.25	17.98	0	0	3	4
Lincoln Continental	10.4	8	460	215	3	5.424	17.82	0	0	3	4
Chrysler Imperial	14.7	8	440	230	3.23	5.345	17.42	0	0	3	4
Fiat 128	32.4	4	78.7	66	4.08	2.2	19.47	1	1	4	1
Honda Civic	30.4	4	75.7	52	4.93	1.615	18.52	1	1	4	2
Toyota Corolla	33.9	4	71.1	65	4.22	1.835	19.9	1	1	4	1
Toyota Corona	21.5	4	120.1	97	3.7	2.465	20.01	1	0	3	1
Dodge Challenger	15.5	8	318	150	2.76	3.52	16.87	0	0	3	2
AMC Javelin	15.2	8	304	150	3.15	3.435	17.3	0	0	3	2
Camaro Z28	13.3	8	350	245	3.73	3.84	15.41	0	0	3	4
Pontiac Firebird	19.2	8	400	175	3.08	3.845	17.05	0	0	3	2
Fiat X1-9	27.3	4	79	66	4.08	1.935	18.9	1	1	4	1
Porsche 914-2	26	4	120.3	91	4.43	2.14	16.7	0	1	5	2
Lotus Europa	30.4	4	95.1	113	3.77	1.513	16.9	1	1	5	2
Ford Pantera L	15.8	8	351	264	4.22	3.17	14.5	0	1	5	4
Ferrari Dino	19.7	6	145	175	3.62	2.77	15.5	0	1	5	6
Maserati Bora	15	8	301	335	3.54	3.57	14.6	0	1	5	8
Volvo 142E	21.4	4	121	109	4.11	2.78	18.6	1	1	4	2
correlation	-0.85216		-0.84755	-0.77617	0.681172	-0.86766	0.418684	0.664039	0.599832	0.480285	-0.55093

Which of the variables is the best predictor of mpg? Justify your answer using the correlation coefficient and a scatterplot.

Fit a regression model based on your answer to question 2. Write your model below.

Is the slope significant? Justify your answer.

Interpret the slope of your regression model.

Interpret the r² value.

Do you believe your model is a good model for predicting mpg? Justify your answer.

Answer 1

Best Predictor

Since mpg has highest correlation with Weight. The correlation is neagative which indiacates that as weight increases Mileage of vehicle decreases which is obvious also.

So, weight(wt) can be considered as best predictior.

Scatter plot

From above plot it is clear that mpg and weight are highly neagtively correlatd and we can fit linear regression model to it.

Model Fitting

We need to fit model between mpg and wt . I am using Excel to fot linear regression model.

Excel Output:

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.867659377
R Square	0.752832794
Adjusted R Square	0.744593887
Standard Error	3.045882125
Observations	32
ANOVA
	df	SS	MS	F	Significance F
Regression	1	847.72525	847.72525	91.375325	0.00000000013
Residual	30	278.3219375	9.277397918
Total	31	1126.047188
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	37.28512617	1.877627337	19.85757526	0.0000000000000000008	33.45049959	41.11975275
wt	-5.344471573	0.559101045	-9.559044147	0.0000000001293958701	-6.486308234	-4.202634912

Regression Model:

mpg = 37.285 - 5.3444 * wt

Slope:

Yes the slope is significant at 5 % significance level since p-value < 0.05

Interpreting slope:

The negative value of slope indicates that for each unit increase in weight mileage decreases by 5.344 units.

The decrease in mileage of vehicle with inrcease in its wieght is obvious

Interpreting R²

The R-square has value of 0.752 which indicates that 75% of variance in mpg is captured just by Weight of vehicle.

This also shows that Weight variable is good in determining the Mileage.

If we add more variables which also strong correlation with mpg like disp and cyl our R² will increase significantly

Model

With R² of 0.752 our model can be considered as decent. As we can see there are few outliers, so treating them will increase model efficieny little bit.

There are few other variables like disp and cyl which also has very high correlation with Mpg. Adding them while building model will increase model accuracy.

Overall, Model is good