Question 1\ Suppose we have the following model,\

Y_(i)=\\\\beta _(0)+\\\\beta _(1)x_(i)+\\\\epsi lon_(i),i=1,2,dots,n

\ where

\\\\beta _(0)

and

\\\\beta _(1)

are unknown constants and

\\\\epsi lon_(i)^(id)N(0,\\\\sigma ^(2)),\\\\sigma >0

holds true. The distance from any point

y_(i)

in a collection of data, to the mean of the data

/bar (y)

, is the deviation, written as

y_(i)(-)/(b)ar (y)

. The analysis of variance is based on a partitioning of total variability in

y

.\

y_(i)(-)/(b)ar (y)=(hat(y)_(i)-(/bar (y)))+(y_(i)-hat(y)_(i))

\ a. Show that

ubrace(\\\\sum_(i=1)^n (y_(i)-(/bar (y)))^(2)ubrace)_(SS_(T))=ubrace(\\\\sum_(i=1)^n (hat(y)_(i)-(/bar (y)))^(2)ubrace)_(SS_(R))+ubrace(\\\\sum_(i=1)^n (y_(i)-hat(y)_(i))^(2)ubrace)_(SS_(Res ))(5pts)

\ b. Show that\

E(SS_(R))=\\\\sigma ^(2)+\\\\beta _(1)^(2)S_(xx)

\ where

S_(xx)=\\\\sum_(i=1)^n (x_(i)-(\\\\bar{x} ))^(2)*(5pts)

\ c. Show that\

E(SS_(Res ))=\\\\sigma ^(2)(n-2)

\ (5 pts)\ d. Show that the coefficient of determination\

R^(2)=(r^(2))/(r^(2)+(S(S_(Res ))/(S)S_(T)))

\ where

r

is the sample correlation between

x

and

Y

. (5 pts)

Question 1 Suppose we have the following model, Yi=0+1Xi+ii=1,2,,n where 0 and 1 are unknown constants and iiidN(0,2),>0 holds true. The distance from any point yi in a collection of data, to the mean of the data y, is the deviation, written as yiy. The analysis of variance is based on a partitioning of total variability in y. yiy=(y^iy)+(yiy^i) a. Show that SSTi=1n(yiy)2=SSRi=1n(y^iy)2+SSResi=1n(yiy^i)2(5pts) b. Show that E(SSR)=2+12SXX where SXX=i=1n(xix)2.(5pts) c. Show that E(SSRes)=2(n2) (5 pts) d. Show that the coefficient of determination R2=r2+(SSRes/SST)r2 where r is the sample correlation between X and Y. (5 pts)