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 1316 1848 2851 3819 47806835 7703 7601 782984938459 8680 9056 * 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 o. 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. y = (d) Calculate the fitted value of y corresponding to x = 2013. 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

Here, We have the data of Liver Transplantation waiting list (LTWL)

yi - No. of patients (the response variable)

xi - the corresponding year ( the independant variable)

a) we have to estimate  $\widehat{\sigma ^2}$  

  $\widehat{\sigma ^2}$ =  $\frac{1}{n-2} (S_y_y-\frac{S_x_y^2}{S_x_x})$

Lets find , =  $\sum x_i^2 -n*\overline{x}^2$ == 182 n = 13

       =   $\sum y_i^2 -n*\overline{y}^2$ == 93505394.8

      =       $\sum x_i y_i -n*\overline{xy}$ == 123918

hence , from the formula we get,

     $\widehat{\sigma ^2}$ = 830325

b) Hence we heve the simple linear regression model is

$\ y_i = \beta _0 +\beta _1 + \epsilon_i$

Now , we have to find $\widehat{\beta _1}$ = ?

We know    $\widehat{\beta _1}$ =  $\frac{S_x_y}{S_x_x}$ == 681

Hence, the LTWL size change in avarage for a year is 680.8681

Lets find the   $\beta_0$ ?

$\beta_0$ = $\overline{y} - \widehat{\beta}_1* \overline{x}$

we have   $\overline{y}$ = 6097.692 &   $\overline{x}$ = 2010

$\beta_0$ = -1362447   

c) To find : The no. of patients in the list in 2017

that is , x= 2017 putting in to the regression model

we ger y = 10863.77

The no. of patients who need Liver tranpantation in the year 2017 is 10863.77  

d) now the fitted value of  $\widehat{y}$ for x = 2013

  $\widehat{y}$ = 8140

and the corresponding residual is

e = 269