Chapter 8, Problem 54E

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:

  $(200,1950),(400,1650),(600,1800),(800,1600)$

And the least square line for this is as:

  $\widehat{y}=1975-0.45x$

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
200	1950	=1975-(0.45*AQ83)	=AR83-AS83	=AT83*AT83
400	1650	=1975-(0.45*AQ84)	=AR84-AS84	=AT84*AT84
600	1800	=1975-(0.45*AQ85)	=AR85-AS85	=AT85*AT85
800	1600	=1975-(0.45*AQ86)	=AR86-AS86	=AT86*AT86
			Total=	=SUM(AU83:AU86)

Thus, the result is as follows:

x	y	Predicted y	Residual	Square of residual
200	1950	1885	65	4225
400	1650	1795	-145	21025
600	1800	1705	95	9025
800	1600	1615	-15	225
			Total=	34500

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 residual is $34500$ and the sum of the residuals on any other line would be greater than $34500$ .