student GPA (y) ACT (x) y-avg(y) x-avg(x) (y-avg(y))*(x-avg(x)) (x-avg(x))^2 estimated y residual
1 2.8 21 -0.33 -4.93 1.64 24.34 2.67 0.13
2 3.4 24 0.27 -1.93 -0.52 3.74 2.95 0.45
3 3.0 26 -0.13 0.07 -0.01 0.00 3.14 -0.14
4 3.5 27 0.37 1.07 0.39 1.14 3.23 0.27
5 3.6 29 0.47 3.07 1.43 9.40 3.42 0.18
6 3.0 25 -0.13 -0.93 0.12 0.87 3.04 -0.04
7 2.7 25 -0.43 -0.93 0.40 0.87 3.04 -0.34
8 3.7 30 0.57 4.07 2.30 16.54 3.52 0.18
9 3.2 23 0.07 -2.93 -0.20 8.60 2.86 0.34
10 3.0 28 -0.13 2.07 -0.28 4.27 3.33 -0.33
11 3.5 30 0.37 4.07 1.49 16.54 3.52 -0.02
12 2.5 20 -0.63 -5.93 3.76 35.20 2.57 -0.07
13 3.8 32 0.67 6.07 4.04 36.80 3.71 0.09
14 2.6 26 -0.53 0.07 -0.04 0.00 3.14 -0.54
15 2.7 23 -0.43 -2.93 1.27 8.60 2.86 -0.16
n sum(y) sum(x) sum sum
15 47 389 15.83 166.93
avg(y) avg(x)
3.13 25.93 estimated
beta_0 0.67
beta_1 0.09
y = 0.0948x + 0.6736
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
G
P
A
ACT
observations estimated model
student GPA (y) ACT (x) vector y
1 2.8 21 2.8 1 21 1 1 1 1 1 1 1 1 1
2 3.4 24 3.4 1 24 21 24 26 27 29 25 25 30 23
3 3.0 26 3.0 1 26
4 3.5 27 3.5 1 27
5 3.6 29 3.6 1 29 15 389 Hint for Windows: Expand to the size of the matrix & Press F2, then Ctrl+Shift+En
6 3.0 25 3.0 1 25 389 10255 Hint for Mac: Expand to the size of the matrix & press CONTROL+U, and then pre
7 2.7 25 2.7 1 25
8 3.7 30 3.7 1 30
9 3.2 23 3.2 1 23 4.10 -0.16 Hint for Windows: Expand to the size of the matrix & Press F2, then Ctrl+Shift+En
10 3.0 28 3.0 1 28 -0.16 0.01 Hint for Mac: Expand to the size of the matrix & press CONTROL+U, and then pre
11 3.5 30 3.5 1 30
12 2.5 20 2.5 1 20 X'y estim.
13 3.8 32 3.8 1 32 47 beta_0 0.67
14 2.6 26 2.6 1 26 1235 beta_1 0.09
15 2.7 23 2.7 1 23
matrix X X'
X'X
(X'X)^(-1)
estimated y residual
1 1 1 1 1 1 2.67 0.13
28 30 20 32 26 23 2.95 0.45
3.14 -0.14
3.23 0.27
ss F2, then Ctrl+Shift+Enter (use "MMULT") 3.42 0.18
NTROL+U, and then press +RETURN 3.04 -0.04
3.04 -0.34
3.52 0.18
ss F2, then Ctrl+Shift+Enter (use "MINVERSE") 2.86 0.34
NTROL+U, and then press +RETURN 3.33 -0.33
3.52 -0.02
2.57 -0.07
3.71 0.09
3.14 -0.54
2.86 -0.16
y = 0.0948x + 0.6736
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
G
P
A
ACT
observations estimated model
y x y-avg(y) x-avg(x) (y-avg(y))*(x-avg(x)) (x-avg(x))^2 estimated y true model y residual disturbance
2 1 -2.67 -2.17 5.78 4.69 1.80 1.5 0.20 0.5
6 4 1.33 0.83 1.11 0.69 5.77 6.0 0.23 0.0
3 2 -1.67 -1.17 1.94 1.36 3.12 3.0 -0.12 0.0
8 5 3.33 1.83 6.11 3.36 7.09 7.5 0.91 0.5
5 3 0.33 -0.17 -0.06 0.03 4.45 4.5 0.55 0.5
4 4 -0.67 0.83 -0.56 0.69 5.77 6.0 -1.77 -2.0
sum(y) sum(x) sum sum
28 19 14.33 10.83
avg(y) avg(x)
4.67 3.17 estimated true
beta_0 0.48 0
beta_1 1.32 1.5
y = 1.3231x + 0.4769
1.5
2.5
3.5
4.5
5.5
6.5
7.5
8.5
0.5 1.5 2.5 3.5 4.5 5.5
y
X
observations estimated model true model
y x vector y estimated y true model y
2 1 2 1 1 1 1 1 1 1 1 1.80 1.5
6 4 6 1 4 1 4 2 5 3 4 5.77 6.0
3 2 3 1 2 3.12 3.0
8 5 8 1 5 7.09 7.5
5 3 5 1 3 6 19 1.09 -0.29 4.45 4.5
4 4 4 1 4 19 71 -0.29 0.09 5.77 6.0
X'y estim. true
28.0 beta_0 0.48 0
103.0 beta_1 1.32 1.5
(X'X)^(-1)
matrix X X'
X'X
residual disturbance
0.20 0.5
0.23 0.0
-0.12 0.0
0.91 0.5
0.55 0.5
-1.77 -2.0
y = 1.3231x + 0.4769
1.5
2.5
3.5
4.5
5.5
6.5
7.5
8.5
0.5 1.5 2.5 3.5 4.5 5.5
y
X
observations estimated model true model