How do you find the linear regression equation for the data?

Question

age(months) weight(pounds)

19 22

21 23

24 25

27 28

29 31

31 28

34 32

38 34

43 39

a.)find the linear regression equation for the data

b.) find the slope. what does it reppresent

c.) superimpose the graph on a scatter plot of the data

d.) use regression equation to predict approximate weight of 30 month girl

please explain

Answer 1

I'm guessing this is a HS math question.  Usually this would be solved using a built in Linear Regression function on a graphing calculator.  If that is the case you may want to reference the manual for your specific graphing calculator.  

I provide the solution that that function would generate:

This is a "Least-Squares" problem.  We want to find a linear equation of the form ax+b that satisfies the given data.  This corresponds to 9 linear equations of the following form:

19a + b = 22

21a + b = 23

...

38a + b = 34

43a + b = 39

We wish to solve for a and b.  If we attempted to solve this system by usual methods we would quickly realize that no solution exists because the system is overdetermined (it has more equations than unknowns).  Thus our objective now is to obtain the "best" solution.

Next we fashion a matrix A in the following form:

[1  19]

[1  21]

[1  24]

[1  27]

[1  29]

[1  31]

[1  34]

[1  38]

[1  43]

A unique solution is obtained if (A'A) is invertible. (A' represents the transpose of A).  In this case A'A is in fact invertible and our solution k is given by:

k = ((A'A)^-1)A'j

where j =(22 23 25 28 31 28 32 34 39)'

(It is highly recommended that these calculations be done on a graphing calculator)

Plugging in we obtain:

k = (9.0125 0.6800)'

Thus a = 9.0125 b = .680

So our best fit is y = .6800 + 9.0125x