HP 48gII - Manual (Page 627)

Page 18-56
Example 3 – Test of significance for the linear regression. Test the null
hypothesis for the slope H
0
: Β = 0, against the alternative hypothesis, H
1
: Β ≠
0, at the level of significance α = 0.05, for the linear fitting of Example 1.
The test statistic is t
0
= (b - Β
0
)/(s
e
/ √S
xx
) = (3.24-0)/( √0.18266666667/2.5) =
18.95. The critical value of t, for ν = n – 2 = 3, and α/2 = 0.025, was
obtained in Example 2, as t
n-2, α /2
= t
3,0.025
= 3.18244630528. Because, t
0
>
t
α /2
, we must reject the null hypothesis H
1
: Β ≠ 0, at the level of significance α
= 0.05, for the linear fitting of Example 1.
Multiple linear fitting
Consider a data set of the form
x
1
x
2
x
3
… x
n
y
x
11
x
21
x
31
… x
n1
y
1
x
12
x
22
x
32
… x
n2
y
2
x
13
x
32
x
33
… x
n3
y
3
. . . . .
. . . . . .
x
1,m-1
x
2,m-1
x
3,m-1
… x
n,m-1
y
m-1
x
1,m
x
2,m
x
3,m
… x
n,m
y
m
Suppose that we search for a data fitting of the form y = b
0
+ b
1
⋅x
1
+ b
2
⋅x
2
+
b
3
⋅x
3
+ … + b
n
⋅x
n
. You can obtain the least-square approximation to the
values of the coefficients b = [b
0
b
1
b
2
b
3
… b
n
], by putting together the
matrix X:
_ _
1 x
11
x
21
x
31
… x
n1
1 x
12
x
22
x
32
… x
n2
1 x
13
x
32
x
33
… x
n3
. . . . .
. . . . . .
1 x
1,m
x
2,m
x
3,m
… x
n,m
_ _