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 1484 20743097 4104 5193 7195 8006 7891 8034 8656 8702 8963 9156 * 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 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). B (c) Estimate the number of patients in the list in 2017. ỹ = (d) Calculate the fitted value of y corresponding to x= 2012. 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

Obs Y X? X; - Y; 1 2004 1484 4016016 2202256 2973936 2 2005 2074 4020025 4301476 4158370 3 2006 3097 4024036 9591409 6212582 4 2007 4104 4028049 16842816 8236728 5 2008 5193 4032064 26967249 10427544 6 2009 7195 4036081 51768025 14454755 7 2010 8006 4040100 64096036 16092060 8 2011 7891 4044121 62267881 15868801 9 2012 8034 4048144 64545156 16164408 10 2013 8656 4052169 74926336 17424528 11 2014 8702 4056196 75724804 17525828 12 2015 8963 4060225 80335369 18060445 13 2016 9156 4064256 83832336 18458496 16605848 Sum = 26130 82555 52521482 61740114 9 The sum of squares obtained from the table above are: η n SSxx ΣΧ? η 1=1 =1 1 52521482 (26130)? 13 182 2 η SSYY ΣΥ? Y, η =1 i=1 1 617401149 (82555)? 13 93145147.0769 1 SSXY ΣΧΥ (Σκ) (Σ) - 166058481 (26130) (82555) 13 - 122931

The slope and y-intercept coefficients are computed using the following formulas: B = SSXY SSxx 122931 182 = 675.4451 Bo = Y - i * 6350.3846 - 675.4451 x 2010 - 1351294.1758 Therefore, the regression equation is: û = -1351294.1758 +675.4451X Now, we know that the total sum of squares is: SSTotal = SSyy = 93145147.0769 Also, the regression sum of squares is computed as follows: SSR B1 SSxy 675.4451 x 122931 83033136.0495 Since we know that SSTotal = SSRegression + SSError, then we can compute the sum of squared errors as follows: SSE SSTotal - SSR 93145147.0769 - 83033136.0495 10112011.0275 Hence, the residual sum of squares for the provided data is SSE = 10112011.0275.

(a)

estimate of variance,  = SSE/(n-2) = 10112011.0275 / (13-2) = 919273.72 ~ 919274

(b)

estimated slope ß1 = SSxy/SSxx = 122931 / 182 = 675.4451 ~ 676

(c)

Number of patients in the list in 2017,

Ŷ = -1351294.1758 + 675.4451(2017)

= 11078.5909

~ 11079

(d)

For x = 2012

Ŷ = -1351294.1758 + 675.4451(2012)

= 7701.3654

~ 7701

e = actual y - predicted y = 8034 - 7701 = 333

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