STATS 100C Midterm Exam 2 Information (Spring 2009)
•Midterm exam 2: Monday, May 18, 2-2:50pm in MS 5128.
•This is a closed-book exam, but you can use a calculator and the formulas provided.
•The z, t and F tables will be provided if necessary.
•Material: Chapters 1-4, HW 1-6.
Formulas for STATS 100C Midterm Exam 2 (Spring 2009)
Chapter 2: Simple Linear Regression
The pdf of N(µ, σ2) is f(x) = 1
√2πσ exp[−(x−µ)2
2σ2].
Model: yi=β0+β1xi+#i. LSE: ˆ
β1=sxy
sxx ,ˆ
β0= ¯y−ˆ
β1¯x, ˆµ0=ˆ
β0+ˆ
β1x0.
Variance: V(ˆ
β1) = σ2/sxx,V(ˆ
β0) = σ2(1
n+¯x2
sxx ), V(ˆµ0) = σ2(1
n+(x0−¯x)2
sxx ).
Notation: r=sxy/√sxx syy ,sxx =!(xi−¯x)2=!x2
i−1
n(!xi)2
sxy =!(xi−¯x)(yi−¯y) = !(xi−¯x)yi=!xiyi−1
n(!xi)(!yi)
SST =!(yi−¯y)2,SSR =!(ˆµi−¯y)2=ˆ
β2
1sxx and SSE =!(yi−ˆµi)2.
Chapter 3: Random Vectors Suppose E(y) = µand V(y) = Σ.
E(Ay +b) = Aµ +b, V (Ay +b) = AΣA#, V (y) = E[(y−µ)(y−µ)#], cov(a#y, b#y) = a#Σb.
The pdf of a n-variate normal distribution N(µ, Σ) is
f(x) = (2π)−n
2|Σ|−1
2exp[−1
2(x−µ)#Σ−1(x−µ)]
Chapter 4: Multiple Linear Regression Model
Model: y=Xβ+#,#∼N(0,σ2I)
LSE ˆ
β= (X#X)−1X#y,s2=S(ˆ
β)/(n−p−1) = (y−Xˆ
β)#(y−Xˆ
β)/(n−p−1).
Fitted values ˆµ=Xˆ
β=Hy; residuals e=y−ˆµ=y−Hy = (I−H)y; where H=X(X#X)−1X#.
Variance V(ˆ
β) = σ2(X#X)−1,V(ˆµ) = σ2H,V(e) = σ2(I−H), V(a#ˆ
β) = σ2a#(X#X)−1a.
Properties: ˆ
βand eare independent; S(ˆ
β)/σ2=e#e/σ2∼χ2
n−p−1; LSE is BLUE.
CI for βi:ˆ
βi±t(1 −α/2, n −p−1)s√vii ; CI for a#β:a#ˆ
β±t(1 −α/2, n −p−1)s"a#(X#X)−1a
F statistic for testing H0:Aβ= 0:
F=$ˆµ−ˆµA$2/l
S(ˆ
β)/(n−p−1) =(S(ˆ
βA)−S(ˆ
β))/l
S(ˆ
β)/(n−p−1)
Joint confidence region for β: ( ˆ
β−β)#(X#X)( ˆ
β−β)/[(p+ 1)s2]≤F(1 −α, p + 1, n −p−1).
SST =!(yi−¯y)2,SSE =!(yi−ˆµi)2,SSR =!(ˆµi−¯y)2=ˆ
β#X#y−n¯y2=ˆ
β#X#Xˆ
β−n¯y2.
R2=SSR/SST = 1 −SSE/SST
3