Page 32 - Mechanical Engineers' Handbook (Volume 2)

3 Statistics in the Measurement Process 21

Example 6 Confidence Limits on a Regression Line. Calibration data for a copper–
constantan thermocouple are given:
Temperature: x [0, 10, 20, 30, 40 50, 60, 70, 80, 90, 100] C
Voltage: y [ 0.89, 0.53, 9.15, 0.20, 0.61, 1.03, 1.45, 1.88, 2.31,

2.78, 3.22] mV
If the variation of y is expected to be linear over the range of x and the uncertainty in
temperature is much less than the uncertainty in voltage, then:
1. Determine the linear relation between y and x.
2. Test the goodness of fit at the 95% confidence level.
3. Determine the 90% confidence limits of points on the line by the slope-centroid
technique.
4. Determine the 90% confidence limits on the intercept of the regression line.
5. Determine the 90% confidence limits on a future estimated point of temperature at
120 C.
6. Determine the 90% confidence limits on the whole line.
7. How much data would be required to determine the centroid of the voltage values
within 1% at the 90% confidence level?
The calculations are as follows:
x 50.0 y 1.0827 y a bx
x 550 y 11.91 xy 1049.21
i
i
i i
(x ) 38,500.0 (y ) 31.6433
2
2
i
i
2
2
2
2
(x x) X 11,000.00 (y y) Y 18.7420
i
i
i
i
(x x)(y y) XY 453.70
i
i i
i
2
(y ˆy ) 0.0299
i
i
1. b XY/ X 453.7/11,000 0.0412
2
a y bx 1.0827 (0.0412)(50.00) 1.0827 2.0600
0.9773
2
2
2
2. r (ˆy y)/ (y y) 1 (y ˆy )/ (y y) 2
exp i i i i
XY/ X Y 0.998
2
2
r Table r( , ) r(0.05,9) 0.602
P[r exp r Table ] with H of no correlation; therefore reject H and infer
0
0
significant correlation
3. t t( , v) t(0.90,9) 1.833 (see Ref. 6)
ˆ 2 y,x (y ˆy )/ 0.0299/9 0.00333
2
i
i