Sharp JX-9400 Technical Information Seite 170

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Seitenansicht 169
Confidence in the coefficients
The variances on the linear coefficients of the regression of the first kind are
usually estimated using the following relations, which assume that the dis-
persion around the line is Gaussian with a constant standard deviation and is
the result of the measurement errors:
s
n
¼
1
s
x
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
s
2
y
ns
xy
N 2
s
and s
a
¼ s
n
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
N
X
N
i ¼1
x
2
i
v
u
u
t
ð7:17Þ
If T(P;) is the significance limit of the two-sided Student distribution for a
probability, P , for degree of freedom, , then the confide nce levels on the
coefficients are:
I
a
¼ s
a
TðP; N 2Þð7:18Þ
I
n
¼ s
n
TðP; N 2Þð7:19Þ
This means that with a probability, P, the coefficient, a, lies in the in terval
½a I
a
; a þ I
a
and the same for n.
The estimate of the variance around the regression line at abscissa x is:
s
y
ðxÞ¼s
n
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðN 1Þ
N
s
2
x
þðx
xxÞ
2
r
ð7:20Þ
and the confidence interval in the estimate of y using the regression line for any
x is:
I
y
ðxÞ¼s
y
ðxÞTðP ; N 2Þð7:21Þ
The values of the two-sided Student distribution are given in Table 7.6. The
relation (Equation 7.17) can be used to obtain the confidence intervals for a
0
and n
0
in the second kind regression if the roles of x and y are permuted.
Table 7.6 Two-sided confidence limits TðP; N 2Þ for a Student distribution
TðP; N 2Þ for probability P ¼
N 2 0.8 0.9 0.95 0.9 9 0.995 0.999
1 3.078 6.3138 12.706 63.657 127.32 636.619
2 1.886 2.9200 4.3027 9.9248 14.089 31.598
3 1.638 2.3534 3.1825 5.8409 7.4533 12.924
4 1.533 2.1318 2.7764 4.6041 5.5976 8.610
5 1.476 2.0150 2.5706 4.0321 4.7733 6.869
6 1.440 1.9432 2.4469 3.7074 4.3168 5.959
7 1.415 1.8946 2.3646 3.4995 4.0293 5.408
8 1.397 1.8595 2.3060 3.3554 3.8325 5.041
9 1.383 1.8331 2.2622 3.2498 3.6897 4.781
10 1.372 1.8125 2.2281 3.1693 3.5814 4.5787
Common Methods and Techniques 149
Seitenansicht 169
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