Question

Reserve Problems Chapter 11 Section 2 Problem 1 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: y | 1327 1898 2865 3870 4867 | 6806 7576 | 7495 | 7772 8553 8495 8704 9068 * 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 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 02. 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). BE (c) Estimate the number of patients in the list in 2018. v= (d) Calculate the fitted value of y corresponding to x = 2014. Find the corresponding residual. Round your answer to the nearest integer (e.g. 9876). V= Round your answer to the nearest integer (e.g. 9876). e =

Answer 1

	x	y	X-X	y-ý	(y-7) (x - 1)	(x - 2)2	(y – 5)2
	2004	1327	-6	-4772.69	28636.15	36	22778592
	2005	1898	-5	-4201.69	21008.46	25	17654218
	2006	2865	-4	-3234.69	12938.77	16	10463234
	2007	3870	-3	-2229.69	6689.077	9	4971528
	2008	4867	-2	-1232.69	2465.385	4	1519530
	2009	6806	-1	706.3077	-706.308	1	498870.6
	2010	7576	0	1476.308	0	0	2179484
	2011	7495	1	1395.308	1395.308	1	1946884
	2012	7772	2	1672.308	3344.615	4	2796613
	2013	8553	3	2453.308	7359.923	9	6018719
	2014	8495	4	2395.308	9581.231	16	5737499
	2015	8704	5	2604.308	13021.54	25	6782419
	2016	9068	6	2968.308	17809.85	36	8810851
Total	26130	79296	0	0	123544	182	92158441
Mean	2010	6099.692

The regression equation between two variables can be given by following equation

ỹ = 60 + b1 * *

The estimate of b0 and b1 can be obtained by least square method.

The least square estimate of b1 can be given by following expression

b1 : Σy - y) * (x – x) Σ(x – x)2

bl: 123544 182

b1 = 679

The estimate of b0 can be given by

b0 = 7 – b1*

b0 = 6099.692-(678.81) * 2010

b0 = -1358315

The final estimated equation of line can be given as

ŷ = -1358315 + 679* x

The ANOVA table of fitted regression is given by

Source	Df	SS	MS	F	p-value
Regression	1	b1*Sxy=679*123544          = 83863296	83863296	111.21	4.34E-07
Error	11	8295144	754104
Total	n-1=13-1=12	SYY=92158441

Now we can answer the given questions

Answer(a):

Here we have, MSE =754104 which is estimate of σ²

o2 = 754104

Answer(b):

B1 = 679

Answer(c):

The estimate of patients in 2018 can be obtained as below:

û = -1358315 + 679* 2018

y = 11530

Answer(d):

The fitted value of y for x=2014 is

û = -1358315 + 679* 2014

y = 8815

The residual can be obtained as below:

e = y-

8495 - 8815

e = 320