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Instructions for the exam:
- You have 75 minutes to complete all problems. Show all of your work on these pages.
- The use of calculators is permitted.
- Many questions are multipart, please carefully read each question to avoid missing part of a
problem.
- To receive full credit write the formulas used and show all steps leading to the answers.
Possible
Points
Your
Score
Part 1. T/F 13
Part 2. Multiple Choice, Short
Answer
Part 3. Propagation of error and
uncertainty
Part 4. Interpreting a Spec Sheet 4
Part 5. Static Sensitivity 12
Total 70
I pledge that I have neither given nor received unauthorized assistance on this exam:
(Signature) ________________________________________________
ISAT 253: Foundations of Measurement and Instrumentation
Spring 2005
EXAM 2: Basic Principles of Instrumentation and Measurement
Part 1. True/False, Multiple Choice, Short Answer (1 point each)
A. Please indicate whether the statements below are true or false by checking the appropriate box.
False 1. Standard deviation is a useful indicator of instrument’s error and resolution.
True 2. The total derivative of a function of more than one variable is found using partial derivative.
False 3. You are measuring the voltage across a resistor by using Ohm’s Law, V=IR. In this case,
you can calculate the uncertainty of the voltage by simply multiplying the uncertainty of the
current measurement with the uncertainty of the resistor value.
True 4. The median is the value which divides an ascending series of data in half.
False 5. As technology continues to advance and instruments become more precise, eventually we
will be able to eliminate uncertainty and error in measurements.
True 6. Calibration implies that the measurements are referenced against a measurement standard.
False 7. The International Prototype Kilogram is a platinum-iridium cylinder kept at the International
Bureau of Weights and Measures in Washington, DC.
True 8. The repeatability of a measurement is the maximum variability of a succession of
measurements of the same value of input approached from the same direction.
False 9.^ The joule is one of the SI base units.
False 10.^ Zero-offset error is one of the most important sources of random error.
False 11. The coefficient of determination is always
2
0 r .
True 12.^ The scatter plot is merely an x – y plot of the data.
False 13.^ Systematic uncertainty is minimized by repeating measurement.
Part II. Multiple Choice (2 points each) and Very Short Answer (points as shown)
Please circle (or fill in) the correct answer. There is only one right answer per question.
- Three students are playing darts. The results of the first round are shown in the figure below, where the circle in
the center is the bull’s-eye. In terms of hitting the bull’s-eye, which player best demonstrates precision, but not
accuracy?
a. Player 1 (circles)
b. Player 2 (squares)
c. Player 3 (triangles)
d. None of the above
Solution: Player 3
The triangles represent precision because Player 3 threw the darts close
to each other.
- The following data were collected from a tire gauge by repeatedly measuring a known tire pressure of 35 psi.
Measurement 1 2 3 4 5
Pressure (psi) 37 35 33 36 39
Please answer the following questions and make sure to show your work for full credit.
a) (1 pt) The value of systematic error is:______
The value of the systematic error is 36 35 1
true
X X psi psi psi
b) (1 pt) The value of maximum random error is (value and units): _______
The value of the maximum random error is X^ ^ X^ furthestaway ^36 psi^ ^33 psi^ ^3 psi
c) (1 pt) The Average is = 36psi
d) (1 pt) The Mode is =
The mode is the most popular measurand. Subtle: if there is no most popular measurand the mode is
usually taken as the average.
e) (1 pt) The Standard Deviation is =
2
1
n
i
i
X X
s psi
n
- The voltage readings of a thermocouple are plotted as a function of change in temperature is shown in the
figure below. The ideal instrument behavior is also plotted. Label the following on the plot and provide the
associated values with their units. Show your work as necessary.
a) (1 pt) Sensitivity of Ideal instrument (label A) 0.9 – 0.1 V/C
b) (1 pt) Sensitivity of real instrument at 30ºC? (label B) ) ~ 0.08 V/C
c) (1 pt) Maximum Linearity error (Label C) approx. 3.3 V
d) (1 pt) Zero offset error (label D) _1.3-1.4 V
0
1
2
3
4
5
6
7
8
9
10
0 10 20 30 40 50 60 70 80 90 100
Input Range (C)
Output Range (V)
Real Instrument
Ideal Instrument
C
A
B
D
- Consider the following histogram of Pressure data:
This represents 100 samples of Pressure data (in psi). Looking at the histogram we would conclude that the
data are:
a) Data are skewed to the left
b) Data are skewed to the right
c) Data are probably from normal distribution.
d) Data are probably not from a normal distribution
e) Impossible to say with just 100 points.
Pressure (psi)
F
r
e
q
u
e
n
c
y
140 160 180 200 220 240
20
15
10
5
0
Histogram of Reactor Pressure (psi)
Worksheet: Worksheet 1 ; 2 / 28 / 2005 ; Dr. Ken Lewis
Part III. Propagation of Error and Uncertainty ( 14 points)
- Some car rental agencies use an onboard global positioning system (GPS) to track an auto. Assume that a
typical GPS’s precision uncertainty is 2% and its accuracy is 5%. Determine the total uncertainty in
measured position that the agency would have if (1) it uses the GPS system as is and (2) it recalibrates the
GPS to within an accuracy of 1%. (4 points)
Solution: (see also pp 268-269, Dunn textbook)
(1)
2 2
x x x
w B P
2 2 ( 0. 05 ) 0. 02
= 0.054 or 5.4% worth 2 points
(2)
2 2
x x x
w B P
2 2 ( 0. 01 ) 0. 02
= 0.022 or 2.2% worth 2 points
Partial credit (1 point) was awarded if answer to (2) correctly identified Bx = 0.01 or 1%.
- A jar full of US coins contains 50 pennies, 50 nickels, 50 dimes, and 50 quarters. All of the coins have been
weighed and the resulting means and standard deviations are shown in the Table below. (10 points)
Pennies Nickels Dimes Quarters
Mean (grams per coin) 2.500 5.000 2.268 5.
Standard deviation (grams) 0.004 0.007 0.002 0.
For a confidence level of 95%:
A) What’s the random uncertainty in the mean for the dimes? (1 pt)
grams
n
S
P t
x
x^ v
7 x 10
071
Note that the t-value is for 49 degrees of freedom, a value that doesn’t appear in the t-table. However,
the values for 40 and 50 are shown as 2.021 and 2.010, respectively. Since 49 is closest to 50, 2.01 is a
better choice, however, as long as the t-value is between 2.01 and 2.02, full credit was given.
If wrong units or no units were included, ½ point was deducted. If wrong t-value was used, ¼ point was
deducted.
In cases where students applied a formula for the uncertainty in the mean that was consistent with the
equation sheet they were given, full credit was awarded.
B) What’s the random uncertainty in the mean for the quarters? (1 pt)
grams
n
S
P t
x
x^ v
3 x 10
071
E) If you took a nickel out of your pocket and weighed it, what’s the random uncertainty in this single
measurement? (1 pt)
Px tvs x
Since v=49 degrees of freedom, we’ll use t=2.01 (see earlier comment in part a)
Therefore, Px =(2.01)(0.007 grams) = 0.014 grams
(Notice that the uncertainty in a single measurement is larger than the uncertainty in the mean.)
A common mistaken answer identified the t-value as 12.706 based on 1 degree of freedom. Since the
standard deviation (0.007) is based on a sample size of 50, this is NOT correct.
F) If you were going to design a vending machine to accept QUARTERS ONLY based on weight, what is the
minimum resolution you would want in your measurement? Explain your answer. (2 pts)
95% of the quarters that are put into our vending machine will be 5.67 +/- 0.030 grams. Therefore, we need to
measure their weight to a hundredth of a gram in order to be sure that they are quarters.
However, if we simply want to distinguish quarters from nickels (5.00 +/- 0.014 grams), we only need to resolve
the weight to a tenth of a gram. In other words, if the coin weighs 5.6 – 5.8 grams, it’s a quarter and cannot be a
nickel.
Both answers (0.01 gram and 0.1 gram) were accepted for full credit if they were explained properly.
A common mistaken answer identified the resolution as 5.6xx grams. This type of answer demonstrated a lack of
understanding of the concept of resolution. An instrument with a resolution of 5.6 grams wouldn’t be able to
distinguish two coins that differed in weight by 5.6 grams!
Another common mistaken answer expressed the resolution as some number of “significant figures”. These
concepts are NOT related. For example, an instrument may have a resolution of 0.01 grams or 0.001 grams. In
both of these examples, there is only 1 significant figure. The number of significant figures has NO bearing on
the resolution.
G) What’s the total uncertainty in part c if a scale with an accuracy of + 0.02 gram was used in the measurement.
(2 pts)
2 2 wx Bx P x Bx = 0.02, worth 1 point
2 2 ( 0. 02 ) 0. 0. 017
= 0.026 grams worth 1 point
The accuracy is the same as the bias uncertainty, Bx. The precision uncertainty, Px, was estimated in part
d. If the answer from part d (even if it was wrong)was used for Px, full credit was awarded.
Part IV. Interpreting A Spec Sheet (4 pts)
All instruments come with “spec sheets,” a document that indicates the precision, accuracy, and resolution of the
device, instructions for its calibration, and any other special operating conditions. Below is part of the spec sheet
for the Kestrel 1000 anemometer, a device that measures wind speed.
a) Assuming that the data display in the photo is in m/s, what is the resolution of this device? 0.1 m/s (1 pt)
Deducted ½ point for not showing units.
Gave partial credit (½ point) if the answer was 3 significant digits.
⅓ of the class got this answer completely wrong and clearly do not understand the concept of
resolution.
b) Can you use the Kestrel 1000 if the wind speed is 20 m/s? How do you know? Yes, because the operating
range for this instrument is 0.3 to 40 m/s. (1 pt)
Deducted ½ point if the answer did not specifically state the operating range of 0.3 to 40 m/s
c) Supposed you take a wind speed reading of 17.3 m/s. Which of the two accuracy ratings should you use? Why?
± 3% of reading, because it provides the largest uncertainty estimate. **Explanation: Remember that these are
error and uncertainty estimates; therefore we should go with the accuracy rating that gives us the larger
uncertainty. ± 0.1 m/s would be used for very low wind speeds. (2 pts)
The two most common incorrect answers were:
(a) ± 0.1 m/s because this rating was expressed in the units that matched the wind speed reading. The
units in which the accuracy rating is reported is not relevant here.
(b) ) ± 0.1 m/s because it gives us the most precise estimate. Remember that this is an estimate of
accuracy, not precision.
