Chapter 8, Problem 53E

To determine

To explain what “least squares” means, using these data as a specific example.

Explanation of Solution

It is given in the question that, we should consider the four points:

  $(10,10),(20,50),(40,20),(50,80)$

And the least square line for this is as:

  $\widehat{y}=7+1.1x$

As the line is given and the points are given, so we used the formula of the line to calculate the predicted values and then find the residual by the formula that,

  $\text{Residual}=\text{Actual}-\text{Predicted}$

Thus, by using excel we have that,

x	y	Predicted y	Residual	Square of residual
10	10	=7+(1.1*AQ83)	=AR83-AS83	=AT83*AT83
20	50	=7+(1.1*AQ84)	=AR84-AS84	=AT84*AT84
40	20	=7+(1.1*AQ85)	=AR85-AS85	=AT85*AT85
50	80	=7+(1.1*AQ86)	=AR86-AS86	=AT86*AT86
			Total=	=SUM(AU83:AU86)

Thus, the result is as follows:

x	y	Predicted y	Residual	Square of residual
10	10	18	-8	64
20	50	29	21	441
40	20	51	-31	961
50	80	62	18	324
			Total=	1790

Therefore, we can say that in this context “least-square” means that this is the best fit line for the equation because the sum of the squared vertical distances to any other line would be greater than $1790$ .