Question

MPG	Horsepower	Weight
43.1	48	1985
19.9	110	3365
19.2	105	3535
17.7	165	3445
18.1	139	3205
20.3	103	2830
21.5	115	3245
16.9	155	4360
15.5	142	4054
18.5	150	3940
27.2	71	3190
41.5	76	2144
46.6	65	2110
23.7	100	2420
27.2	84	2490
39.1	58	1755
28.0	88	2605
24.0	92	2865
20.2	139	3570
20.5	95	3155
28.0	90	2678
34.7	63	2215
36.1	66	1800
35.7	80	1915
20.2	85	2965
23.9	90	3420
29.9	65	2380
30.4	67	3250
36.0	74	1980
22.6	110	2800
36.4	67	2950
27.5	95	2560
33.7	75	2210
44.6	67	1850
32.9	100	2615
38.0	67	1965
24.2	120	2930
38.1	60	1968
39.4	70	2070
25.4	116	2900
31.3	75	2542
34.1	68	1985
34.0	88	2395
31.0	82	2720
27.4	80	2670
22.3	88	2890
28.0	79	2625
17.6	85	3465
34.4	65	3465
20.6	105	3380

1. Determine the regression coefficient, b0, b1, and b2. State the multiple regression equation.
2. Interpret the meaning of b0, b1, and b2.
3. Explain why the regression coefficient b0 has no practical meaning in the context of this problem.
4. Predict the miles per gallon for cars that have 60 horsepower and weight 2,000 pounds.
5. If you were consulting for this organization and were provided these data to make a preliminary analysis, what would be your recommended next steps for the organization? (100 words)

Answer 1

I am using Excel to solve this problem.
First you copy data into excel then run regression

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.865689
R Square	0.749417
Adjusted R Square	0.738754
Standard Error	4.176602
Observations	50
ANOVA
	df	SS	MS	F	Significance F
Regression	2	2451.974	1225.987	70.28128	7.51E-15
Residual	47	819.8681	17.444
Total	49	3271.842
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	58.15708	2.658248	21.87797	2.76E-26	52.80938	63.50479	52.80938	63.50479
Horsepower	-0.11753	0.032643	-3.60028	0.000763	-0.1832	-0.05186	-0.1832	-0.05186
Weight	-0.00687	0.001401	-4.90349	1.16E-05	-0.00969	-0.00405	-0.00969	-0.00405

1) So from the summary we can see that regression coefficients are as below:

b0 = 58.157082, b1 = -0.117525, b2 = -0.006871

So multiple regression equation is :

MPG = 58.157082 - 0.117525 * Horsepower - 0.006871 * Weight

2) b0 ---> When horsepower and weight both are 0, the MPG value is 58.157

b1 ---> For every one unit increase in Horsepower, the MPG value is expected to go down by 0.117525

b2 ---> For every one unit increase in Weight, the MPG value is expected to go down by 0.006871

3) Here the regression coefficent b0 has no practical meaning as this is practically impossible to have a car of zero weight and zero horsepower.

4) To predict using the above model, we can make use of the predict() function as below:

predict(model,newdata = data.frame(Horsepower = 60,Weight = 2000))

= 37.36427

So the model predicts 37.36427 mpg for a car having 60 horsepower and weighing 2000 pounds.

5) Next Step is:

We can use this model for prediction of miles per gallon for cars using horsepower and weight

As value of  horsepower and weight increases value of miles per gallon expected to go down.

Here the regression coefficent b0 has no practical meaning as this is practically impossible to have a car of zero weight and zero horsepower.

This two variable  horsepower and weight explains 74.94% variability in  horsepower and weight but still 25% variability unexplained.

So we can use other information to improve it further like age of car .Also we can use variable selection to improve model further like stepwise regression to choose best set of variable.based on adjusted R^2

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