Solutions to the application examples
Solution to Example 1b:
a) Under H 0 we have t 0.4902 8.2804 . 0.0592 H 0 is rejected at the 0.05 level of significan ce.
60 Xb
70
80
90
19. Nov. 2009
W4479 - Ch 0-5-Applications: , Prof Dr. Y. Feng
6
Solutions to the application examples
Solution to Example 2:
ˆ * 0.7361 and ˆ * 1.5531 a) 2 1
19. Nov. 2009
W4479 - Ch 0-5-Applications: , Prof Dr. Y. Feng
7
Solutions to the application examples
Income & expenditure on food
o o o o o o o o o o o o o o o o o o o o o o o o oo o o o oo oo o o o o oo o oo o o o oo o o o o o o o o o o o oo o o o o o o o o o o o o oo oo o o o o o o o o o o o o o o o oo o o oo o o o o o o o o o o o o o o o o o oo o o o o o o o o o o o o o oo o o o o oo o o o o o o o o o o o o o o o o o o o o o o o o o o o oo o o o o oo o o o o o o o o o o oo o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o ooo o o oo o o o o o o oo o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o oo o o o oo o o o o o o oo o o o o o o o o o o o o o o o o o o o o o oo o o o oo oo o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o oo o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o oo o o o oo o o o o 0.5 1.0 1.5 2.0 Income(X) 2.5 3.0




0
0
0

The 95% - CI for Y0 is : 74.852 2*8.376 Y0 74 .852 2 * 8.376 58 .100 Y0 91 .604
19. Nov. 2009
W4479 - Ch 0-5-Applications: , Prof Dr. Y. Feng

We will reject H 0 , because n is lareg and t is negative with absolute value bigger tha n 1.8.
19. Nov. 2009
W4479 - Ch 0-5-Applications: , Prof Dr. Y. Feng
CI for E(Y0)
60
{
80
o ooo o oo oo o o o o o oo o
40
CI for Y0
20
40 X X
60
X 75
80
About 5% of the observations are outside the confidence range
19. Nov. 2009 W4479 - Ch 0-5-Applications: , Prof Dr. Y. Feng 4
Further details (not a part of the solutions)
Data/regression, 95% CI for mean & individual values
100
CI for X=75
oo o oo o o o oo o o o oo
0
20
o
o oo o o o o oo o o o o o o o o o o o o o o o o oo o o oo o o oo oo o o o o o o o oo o o o o oo o o oo o o o o o o o o o o o o o
2 i i 2 i 2 2 i i i i i i
ˆ 2
X Y n X Y 1 X X
i i i 2 i 2
1
i

62998.42 0.8784 71716.97
n
i
ˆ Y ˆ X 5220.2/104 - 0.8784 * 4880.6/104 8.9719 1 2 ˆ 8.9719 0.8784*X The fitted regression function is : Y i i b) TSS yi2 Yi 2
Solutions to the application examples
Solution to Example 1:
a)
x X X / n 300757.9 - 4880.6 / 104 71716.97 x y X Y X Y / n 307976.4 - 4880.6 * 5220.2 / 104 62998.42
ˆ 9056 .78 / 11644 .74 0.7778 2,2 ˆ Y ˆ * X 3259.3/43 - 0.7778 * 3113 .3 / 43 19 .483 2,1 2 2,2 2
19. Nov. 2009
W4479 - Ch 0-5-Applications: , Prof Dr. Y. Feng

ˆ2 68 .74 ˆ var 0.0009585 2 2 xi 71716 .97 ˆ 8.9719 0.8784 * 75 74 .852 c) The point prediction of Y0|X 0 75 is Y 0 X 4880.6 / 104 46 .93 1 X 0 X 2 2 2 ˆ | X 75 ˆ var Y0 Y 1 68 .74 * [1 1 / 104 (75 46 .93) / 71716.97] 70 .156 0 0 2 xi n ˆ se Y Y | X 75 70 .156 8.376
5
Solutions to the application examples
Data/regression funct. for group 1
50
Data/regression funct. for group 2
40
Ya
30
Yb
20
10
20
30 Xa
40
50
60
50
60
70
80
90
30
40
50
Y 50.194 , var(Y ) 605 .313, se(Y ) 24.603 95% - CI for Y (in any country) : 50.194 2* 24.603 Y0 50.194 2 * 24.603 0.988 Y0 9o the application examples
Solution to Example 1a:
a) TSS yi2 Yi 2
Y / n 324371.2 - 5220.2
2 i
2
/104 62347.28
ˆ 2 x 2 0.8784 2 * 71716.97 55335 .85 ESS 2 i RSS TSS - ESS 7011 .43 b) xi2 X i2 ˆ2
Y / n 324371.2 - 5220.2
2 i
2
/104 62347.28
ESS TSS - RSS 62347.28 - 7007.51 55339 .77 R 2 ESS/TSS 55337.77/6 2347.28 0.8876 , r R 2 0.9421 c) The hypotheses to test are : H 0 : 2 * 1 vs H1 : 2 1. ˆ * 0.8784 1 We have : t 2 3.923 ˆ 0.0310 se 2
This provides no information on Yi in the i-th country.
3) var(Y0-Ŷ0)→var(ui), as n→∞.
19. Nov. 2009 W4479 - Ch 0-5-Applications: , Prof Dr. Y. Feng 3
Solutions to the application examples
2 2 i
/ 43 237054.9 - 3113 .32 / 43 11644.74 x X X i Y 50 Yi 50 Yi 50 i xi yi X iYi X i Yi /43 9056.78