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2 BLUE t F 1

3 1. TSS ESS RSS TSS = ESS + RSS R 2 := ESS/TSS = 1 RSS/TSS R 2 := 1 [RSS/(n k)]/[tss/(n 1)] 2. y i ŷ i y i x i 2 2

4 k F 1 F

5 pp. 60, pp. 63, F

6 1 1.1 pp. 60, 272 y i x i E(y i x i ) = α + βx i ŷ i := a + b x i e i := y i ŷ i 5

7 1. e i = 0 x i e i = 0 6

8 OLS min a,b and (y i a bx i ) 2 a, b R 1 (y i a b x i ) = 0 x i (y i a b x i ) = 0 7

9 1. (y 1,..., y n ) TSS := (y i ȳ) 2 2. (y 1,..., y n ) ESS := (ŷ i ȳ) 2 8

10 3. (y 1,..., y n ) 2 RSS := e 2 i 2. TSS = ESS + RSS 9

11 TSS := = = = (y i ȳ) 2 [(ŷ i ȳ) + e i ] 2 [ (ŷi ȳ) 2 + 2(ŷ i ȳ)e i + e 2 i ] (ŷ i ȳ) (ŷ i ȳ)e i + e 2 i 10

12 (ŷ i ȳ)e i = [(a + b x i ) (a + b x)]e i n = b (x i x)e i n = b x i e i b x = 0 e i 11

13 4. R 2 := ESS TSS R 2 = 1 RSS TSS 12

14 R 2 := 1 RSS/(n k) TSS/(n 1) 13

15 E ( 1 n 1 ) (y i ȳ) 2 = var(y i ) E ( 1 n k e 2 i ) = var(u i ) R 2 1 var(u i )/ var(y i ) 14

16 E ( ) ( n [1/(n k)] R2 = 1 E e2 i [1/(n 1)] n (y i ȳ) 2 1 E ( [1/(n k)] n ) e2 i E([1/(n 1)] n (y i ȳ) 2 ) = 1 var(u i) var(y i ) ) 15

17 1.3 pp. 63, y i ŷ i y i x i y i (x i,1,..., x i,k )

18 (ŷ 1,..., ŷ n ) 1 n ŷ i = 1 n (a + b x i ) = a + b x = ȳ 17

19 ((y 1, ŷ 1 ),..., (y n, ŷ n )) 1 n (y i ȳ)(ŷ i ȳ) = 1 n = 1 n = 1 n [(ŷ i ȳ) + e i ](ŷ i ȳ) (ŷ i ȳ) n (ŷ i ȳ) 2 e i (ŷ i ȳ) 18

20 ((y 1, ŷ 1 ),..., (y n, ŷ n )) (1/n) n (y i ȳ)(ŷ i ȳ) n (1/n) (y i ȳ) 2 (1/n) n (ŷ i ȳ) 2 (1/n) n = (ŷ i ȳ) 2 n (1/n) (y i ȳ) 2 (1/n) n = n (ŷ i ȳ) 2 n (y i ȳ) 2 (ŷ i ȳ) 2 = ESS TSS 19

21 k k N ( µ 1, σ 2),..., N ( µ k, σ 2) n 1,..., n k (y 1,1,..., y 1,n1 ),..., (y k,1,..., y k,nk ) n := n n k 7. µ 1,..., µ k µ := µ µ k k 20

22 8. µ 1,..., µ k h = 1,..., k α h := µ h µ h 3. 21

23 11. 1 h = 1,..., k, i = 1,..., n h y h,i = µ h + u h,i u h,i N ( 0, σ 2) α α k = 0 y h,i = µ + α h + u h,i u h,i N ( 0, σ 2) 22

24 h = 1,..., k, i = 1,..., n h x h,i := h y h,i = k j=1 µ j [x h,i = j] + u h,i 23

25 2.2 h = 1,..., k ȳ h := 1 n h n h y h,i ȳ := 1 n k h=1 n h y h,i 24

26 12. S := 13. S w := k h=1 k h=1 n h n h (y h,i ȳ) 2 (y h,i ȳ h ) 2 25

27 14. S b := k h=1 n h (ȳ h ȳ) 2 4. S = S w + S b 26

28 S = = k h=1 k h=1 n h n h = S w (y h,i ȳ h + ȳ h ȳ) 2 [ (yh,i ȳ h ) 2 + 2(y h,i ȳ h )(ȳ h ȳ) + (ȳ h ȳ) 2] k (ȳ h ȳ) h=1 n h (y h,i ȳ h ) + S b 27

29 2.3 F H 0 : µ 1 = = µ k vs H 1 : µ h µ for some h = 1,..., k 5. S w σ 2 χ2 (n k) 28

30 h = 1,..., k nh (y h,i ȳ h ) 2 σ 2 χ 2 (n h 1) k h=1 nh (y h,i ȳ h ) 2 σ 2 χ 2 ( k ) (n h 1) h=1 29

31 6. H 0 S σ 2 χ2 (n 1) H 0 {y h,i } IN ( µ, σ 2) 7. S w S b 30

32 1. 1. X + Y χ 2 (m) 2. X χ 2 (n) 3. X Y Y χ 2 (m n) 31

33 X Y t M X+Y (t) = M X (t)m Y (t) M Y (t) = M X+Y (t) M X (t) = χ 2 (m n) mgf (1 2t) m/2 (1 2t) n/2 = (1 2t) (m n)/2 32

34 8. H 0 S b σ 2 χ2 (k 1) H 0 S σ 2 χ2 (n 1) S w σ 2 χ2 (n k) S w S b S σ 2 S w σ 2 χ2 (k 1) 33

35 1. H 0 S b /(k 1) S w /(n k) F(k 1, n k) F k 1 S b S b /(k 1) [S b /(k 1)]/[S w /(n n k S w S w /(n k) n 1 S S/(n 1) 34

36 A, B A j = 1,..., J B k = 1,..., K j = 1,..., J, k = 1,..., K, i = 1,..., n j,k y j,k,i = µ + α j + β k + γ j,k + u j,k,i u j,k,i N ( 0, σ 2) α j, β k, γ j,k 35

37 j = 1,..., J, k = 1,..., K, i = 1,..., n j,k x j,k,i := j z j,k,i = k y j,k,i = µ + + J j =1 J j =1 α j [x j,k,i = j ] + K k =1 K k =1 β j [z j,k,i = k ] γ j,k [x j,k,i = j ][z j,k,i = k ] + u j,k,i 36

39 (1) (2) (3) (4) 38

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