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: y 1274 1701 2695 3760 4848 6805 7560 7474 | 7772 8477 8568 8574 9031 * 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 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 62. 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). (c) Estimate the number of patients in the list in 2016. (d) Calculate the fitted value of y corresponding to x= 2010. Find the corresponding residual. Round your answer to the nearest integer (e.g. 9876). y = Round your answer to the nearest integer (e.g. 9876). e =

Answer 1

estimate for error variance is given by,

SSE c? = n = samplesize n-2


XY SSE = SSyy - Sxx

We'll Calculate the quantities as follows:

х у 2002 1274 2003 1701 2004 2695 2005 3760 2006 4848 2007 6805 2008 7560 2009 7474 2010 7772 2011 8477 2012 8568 2013 8574 2014 9031 The independent variable is x, and the dependent variable is y. In order to compute the regression coefficients, the following table needs to be used:


х y x*y x2 y2 2002 1274 2550548 4008004 1623076 2003 1701 3407103 4012009 2893401 2004 2695 5400780 4016016 7263025 2005 3760 7538800 4020025 14137600 2006 4848 9725088 4024036 23503104 2007 6805 13657635 4028049 46308025 2008 7560 15180480 4032064 57153600 2009 7474 15015266 4036081 55860676 2010 7772 15621720 4040100 60403984 2011 8477 17047247 4044121 71859529 2012 8568 17238816 4048144 73410624 2013 8574 17259462 4052169 73513476 2014 9031 18188434 4056196 81558961 Sum 26104 78539 157831379 52417014 569489081 Based on the above table, the following is calculated: n 1 26104 X = X; = = 2008 n 13 i=1


n 78539 Y - Y; 6041.4615384615 n 13 i=1 2 SSxx - Σ ! (ΣΑ) = 52417014 – 26104? /13 - 182 2 η n 1 SSYy = ΣΥ? ΣΥ, 569489081 – 78539? /13 5? /13 = 94998733.230769 n 1=1 i=1 SSXY - Σκη (Στ) (Σ) 157831379 – 26104 x 78539/13 125067 Therefore, based on the above calculations, the regression coefficients (the slope m, and the y- intercept n) are obtained as follows: m = SSXY SSxx 125067 182 687.1813 η =Y – Χ. m = 6041.4615384615 – 2008 x 687.1813 - -1373818.6264 Therefore, we find that the regression equation is: y = -1373818.6264 + 687.18130

So , SSE = 94998733.230769 - (125067)²/ 182 = 9055027.247
Therefore, $\widehat{\sigma}^2$ = SSE/(n-2) = 9055027.247/(13-2) = 823184.(nearest integer)

b)
$\widehat{\beta} = coefficient \: of \: x = m = 687.1813$ = 687

c)
$\widehat{y}$ for 2016 is
$\widehat{y}$ = −1373818.6264 + 687.1813*2016 = 11538.87

d)
$\widehat{y}$ for 2010 is
$\widehat{y}$ = −1373818.6264 + 687.1813*2010 = 7415.7866 = 7416(nearest integer)
= y - $\widehat{y}$ = 7772 - 7415.7866 = 356.21 = 356(nearest integer)