Page 29 - Mechanical Engineers' Handbook (Volume 2)

18 Instrument Statics
x z
P z (39)
/ n
It shows a probability or ‘‘confidence level’’ that the experimental value of z will be
between z obtained from the Gaussian distribution table. For a 95% confidence level,
z 1.96 from the Gaussian table and P[ 1.96 z 1.96] , where z
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(x )/( / n ). Therefore, the expression for the 95% confidence limits on is

x 1.96 (40)
n
In general

x k (41)
n
where k is found from the Gaussian table for the specified value of confidence, .
If the population variance is not known and must be estimated from a sample, the
statistic (x )/( / n ) is not distributed normally but follows the t-distribution. When n
is very large, the t-distribution is the same as the Gaussian distribution. When n is finite, the
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value of k is the ‘‘t’’ value obtained from the t-distribution table. The probabilistic statement
then becomes
P t x t
/ n (42)

and the inequality yields the expression for the confidence limits on :

x t (43)
n
If the effects of resolution and significant digits are included, the expression becomes as
previously indicated in Eq. (29):
x t R
5
n 10 m (44)

3.10 Confidence Limits on Regression Lines
The least-squares method is used to fit a straight line to data that are either linear or trans-
formed to a linear relation. 9,14 The following method assumes that the uncertainty in the
variable x is negligible compared to the uncertainty in the variable y and that the uncertainty
in the variable y is independent of the magnitude of the variable x. Figure 3 and the defi-
nitions that follow are used to obtain confidence levels relative to regression lines fitted to
experimental data:
a intercept of regression line
b slope of regression line, XY/ X 2
y value of y from data at x x i
i
ˆ y i value of y from regression line at x ˆx ; i note that ˆy a bˆx i for a straight
i
line