Question

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

Using the Motor Trend Cars Data Set, you would like to determine if there is a relationship between MPG (miles per gallon) and specific variables included in the data set.

1. Develop a correlation matrix using MPG (miles per gallon), HP (horsepower), WT (weight). Is the correlation between the variables MPG v HP significant? Perform the appropriate test and use a level of significance of 0.05
2. Fit a multiple regression model using MPG (miles per gallon) as the dependent variable and HP (horsepower), and WT (weight) as the independent variables. State your regression model.
3. Is the overall regression model significant? Conduct the appropriate hypothesis test for the significance of the overall regression model. Use a 0.05 level of significance.
4. Are the individual independent variables significant? Conduct the hypothesis tests for the significance of each independent variable at the= 0.05 level of significance.
5. What proportion of the variation in MPG is explained by the independent variables HP and WT?
6. Graph the residuals from the regression you developed in part (a). Do the regression assumptions appear to be appropriate given the residual plots? Explain your answer.

Answer 1

I have done the analysis in R and posted the screenshot of the code along with the output below. I have also provided the statistical concepts behind each part.

Please upvote and provide feedback if this answer helped you.
This would help me improve and better my solutions.
I will be happy to answer your doubts, if any in the comment section below. Thanks! :)

The hypothesis can be stated as follows:

Under the Null Hypothesis, the test-statistic is F-distributed

URSS - Unrestricted Residual Sum of Squares

RRSS - Unrestricted Residual Sum of Squares

k - Number of parameters in the Unrestricted model = 3

- Number of restrictions in the restricted model as compared to the unrestricted model = 2

- The coefficient of multiple determination = 0.8268

The observed F-statistic is

The critical value of the F-statistic at level of significance is

Since the observed F-statistic is more than the critical F-statistic

Reject the NULL hypothesis. The model is overall significant at 0.05 level of significance

The hypothesis for the significance of the parameter can be stated as:

So we will perform a two tailed t-test

Under the null hypothesis, the test statistic is t-distributed with n-1 = 31 degrees of freedom

The critical value of the test-statistic at level of significance is  

Hence we can calculate the t-values for each of the three hypothesis tests

For the intercept term

For the HP term

For the WT term

(f). The residual plot indicates some presence of heteroskedasticity as the residuals don't show an equal variance accross the fitted values. Further tests like bruesh pagan test need to be conducted
to test the significance. The assumption of homoskedasticity i.e. equal variance of the errors seems to be violated here