Question

Exercise 11. A researcher wants to develop a mathematical model that relates fuel efficiency (MPG), y, to three variables: Engine size, x1, curb weight, x2, and horsepower, x3. Data are gathered from 13 randomly selected vehicles and listed in the table below. Find the multiple regression equation for the data.

MPG, y	Engine size, x1	Curb weight, x2	Horsepower, x3
24	2.4	3289	177
25	2.4	3263	158
24	2.5	3230	170
22	3.5	3580	272
18	2.8	3175	255
22	3.5	3643	263
20	3.5	3497	306
21	3.0	3340	230
19	3.6	3861	263
24	2.4	3287	173
21	3.3	3629	234
22	2.5	3270	170
22	3.5	3292	270

Answer 1

You can do this in two methods, one is you can simply input the data in R, as x1,x2,x3,y

and type in the command : lm(formula = y~x1+x2+x3)

this will give you the following result.

or you could estimate by hand.

let I be a matrix of order 13*1, and I'=[1,1,1,1,1,1,1,1,1,1,1,1,1]

so, X= [ I x1 x2 x3 ]

and by least square method,

using this, the same answer is achieved as above.

>1m (y-xl+x2 +x3) Call: 1m (formula y xl x2 x3) Coefficients: (Intercept) xl x2 x3 35.826758 4.803063 -0.003945 -0.065831

1 2.4 3289 177 X= |1 2.4 3263 158 1 3.5 3292 270

(X'X)X'Y