7
Consider again the voltage/flow rate vortex flowmeter
calibration data given above. Suppose now that in the laboratory,
you measured three different flow rates with the vortex flowmeter,
such that the voltage readings were:
1.50 V, 3.45 V and 4.61 V. What would be the expected flow rate,
together with its +/- error term for each voltage?
Suppose we want a 95% confidence interval; then α = 0.05. From
the calibration data above, n = 8, Σx
i
= 21.34, Σx
i
2
= 69.3154,
Σ(y
i
-ax
i
-b)
2
= 0.000831191, S = 0.011769957 liter/sec (Note that S
has the units of y), and t
0.025, 6
= 2.447, so the three predicted
values of flow rates, together with their estimated uncertainties due
to the calibration curve fitting are:
x* (V) y* (liter/s) +/- y* (liter/s) Relative
error in y (%)
1.50 0.3391 0.013962 3.7 %
3.45 1.710138 0.012436 0.73 %
4.61 2.5256644 0.018876 0.75 %
You can also use the t-equation above to find uncertainties in
extrapolated values of y, provided you know that the linear
relationship holds in the extrapolated regime (e.g., you couldn't
extrapolate friction factor vs. Reynolds number data taken for
1<Re<2000 out to Re=10,000, since data were taken in the laminar
range and the extrapolation goes to the turbulent range). In fact, we
cannot reliably extrapolate the flowmeter data above, because if we
go to x* lower than the experimental range, the flow would be
predicted to be negative, which is not physically realistic, while if we
go to higher voltages, we exceed the 5-Volt limit of the instrument.
If you care to do so, you can use the t-equation to reconstruct
the equation above for error in the intercept of an extrapolated line.
The error prediction is one standard deviation, which corresponds
to a 68% confidence interval, and effectively assumes an infinite
number of data points; in this case t
0.16,
= 1.
Reference for this section:
Probability and Statistics for Engineering and the Sciences, Second
Edition, by Jay L. Devore, Brooks/Cole Publishing Company, p. 478
Monterey, CA 1987 (ISBN 0-534-06828-6)