Page 31 - Mechanical Engineers' Handbook (Volume 2)

20 Instrument Statics

^
b + t b
^
^
^
^ y = a + bx
^ ^
b – t b
^ ^
b – t b
^
^ b + t b

Figure 4 Schematic illustrating confidence limits on a regression line.

ˆy t( , )ˆ yi (47)
yi
i
and
ˆ ˆ 1
X
2
i
2
2
n X i
y i y,x 2 (48)
At the centroid where X 0, Eq. (47) reduces to the result for y given in Eq. (45).
i
Line as a Whole
More uncertainty is involved when all points for the line are taken collectively. The ‘‘price’’
paid for the additional uncertainty is given by replacing t(v, ) in the confidence interval
relation by 2F , where F F (2, , ) obtained from an ‘‘F’’ table statistical distribution.
Thus the uncertainty interval for placement of the ‘‘line as a whole’’ confidence limit at x
x is found by the formula
i
1 X
ˆy 2Fˆ i (49)
y,x
i
Line
n X 2
i
Future Estimate of a Single Point at x x
i
This gives the expected confidence limits on a future estimate value of y at x x in a
i
relation to a prior regression estimate. This confidence limit can be found from the relation
ˆy t( , )ˆ (50)
y i yi
where ˆy is the best estimate of y from the regression line and
i
ˆ ˆ 1
2
X
i
2
y i 2 y,x 1 2 (51)
n X i
If one uses 0.99, nearly all (99%) of the observed data points should be within the
uncertainty limits calculated by Eq. (51).