Question

The manager of a car plant wishes to investigate how the plant’s electricity usage depends upon the plant’s production. Month Production (Million) Electricity Usage (Million kWh) January 4.51 2.48 February 3.58 2.26 March 4.31 2.47 April 5.06 2.77 May 5.64 2.99 June 4.99 3.05 July 5.29 3.18 August 5.83 3.46 September 4.70 3.03 October 5.61 3.26 November 4.90 2.67 December 4.20 2.53 Determine the equation of least squares regression line. Compute the residuals and verify that they add to zero.

Answer 1

Answer: -

Regression Line = 0.4954+0.4982x

&

Residuals is 0.

Step by Step explanation:-

Let month production (million) be X and electricity Usage be Y.

The Least Squares Linear regression line is y=alpha+beta×X where

beta= Sxy/Sxx

alpha= Ybar - beta×Xbar

Sxy = sigmaXY - sigmaX - sigmaY / n  

Sxx = sigmaX2 - n×X-2

{Here Sigma = addition}

Since  

sigmaX=58.62

sigmaY=34.15

sigmaX2=291.231

sigmaY2=98.6947

sigmaXY=169.2506

Now Xbar= 58.62/12 = 4.885

Ybar= 34.15/12 = 2.84583

Sxy= 169.2506-58.62×34.15/12 = 2.42785

Sxx= 292.231-(58.62)2/12 = 4.8723

beta= 2.42785/4.8723= 0.4982

alpha= (34.15-0.4982×58.62)× 1÷12= 0.4954

Hence the regression line of Y on X:Y = 0.4954 + 0.4982X

-------------------------------

Now The residuals are calculated as  

eCAP = y1-y1CAP

For January

Yjan = 2.48  

By substituting value of x in the regression line  

yCAPjan = 0.4954+0.4982×4.51= 2.70

eCAP= 2.48-2.70= -0.22

Similarly you can find for other months and IT CAN BE SEEN SUM OF ALL RESIDUALS IS 0.