Question

The department of health studied the number of patients who need liver transplantation. The following data are the Liver Transplantation Waiting List (LTWL), where y is the size (in number of patients) and x is the corresponding year: y1375 1803 2772 386348416935 7635|7555 776684638412 8568 8921 x2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 Round your intermediate answers to four decimal places (e.g. 98.7654) (a) Fit the simple linear regression model using the method of least squares. Find the estimate of σ. Round your answer to the nearest integer (e.g. 9876) (b) How does the LTWL size change in average for a year? Round your answer to the nearest integer (e.g. 9876) βι (c) Estimate the number of patients in the list in 2017 (d) Calculate the fitted value of y corresponding to 2012. Find the corresponding residual. Round your answer to the nearest integer (e.g. 9876) Round your answer to the nearest integer (e.g. 9876)

Answer 1

a. Standard Error of Y on X i.e Y = bo + b1 X
					Xi	Yi	Y'=-1338520.84+669.95*X	Y-Y'	(Y-Yi)^2
2001	1375	2051.11	-676.11	457124.732
2002	1803	2721.061	-918.061	842836
2003	2772	3391.012	-619.012	383175.856
2004	3863	4060.963	-197.963	39189.349
2005	4841	4730.914	110.086	12118.927
2006	6935	5400.865	1534.135	2353570.198
2007	7635	6070.816	1564.184	2446671.586
2008	7555	6740.767	814.233	662975.378
2009	7766	7410.718	355.282	126225.3
2010	8463	8080.669	382.331	146176.994
2011	8412	8750.62	-338.62	114663.504
2012	8568	9420.571	-852.571	726877.31
2013	8921	10090.522	-1169.522	1367781.708

Standard error = Sqrt( ( sum ( Y -Yi )^2/ n-2 )

sum ( Y -Yi )^2 = 9679376.606

Standard Error = 9679376.606

variance = standard error^2 =93690331480780

b.

Line of Regression Y on X i.e Y = bo + b1 X
					X	Y	(Xi - Mean)^2	(Yi - Mean)^2	(Xi-Mean)*(Yi-Mean)
2001	1375	36	22042303	28170
2002	1803	25	18206633	21335
2003	2772	16	10876297	13192
2004	3863	9	4870510	6621
2005	4841	4	1510252	2458
2006	6935	1	748358	-865
2007	7635	0	2449466	0
2008	7555	1	2205453	1485
2009	7766	4	2876677	3392
2010	8463	9	5726817	7179
2011	8412	16	5485324	9368
2012	8568	25	6240388	12490
2013	8921	36	8128639	17106

calculation procedure for regression

mean of X = sum ( X / n ) = 2007

mean of Y = sum ( Y / n ) = 6069.9231

sum ( (Xi - Mean)^2 ) = 182

sum ( (Yi - Mean)^2 ) = 91367117

sum ( (Xi-Mean)*(Yi-Mean) ) = 121931

b1 = sum ( (Xi-Mean)*(Yi-Mean) ) / sum ( (Xi - Mean)^2 )

= 121931 / 182

= 670

bo = sum ( Y / n ) - b1 * sum ( X / n )

bo = 6069.9231 - 670*2007 = -1338521

value of regression equation is, Y = bo + b1 X

Y'=-1338521+670* X

Answer:

beta1 =b1 =670

slope of equation =b1=670

c.

value of regression equation is, Y = bo + b1 X

Y'=-1338521+670* X

when X =2017 then

Y'=-1338521+(670* 2017)

Y' =12869

d.

Y'=-1338521+670* X

when X =2012 then

Y'=-1338521+(670*2012)

Y' =9519