MSE 210 Lab 3 Manual - Spring 2024
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Simon Fraser University *
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Course
210
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Date
Apr 3, 2024
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10
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MSE 210 Engineering Measurement and Data Analysis
1/10
Lab 3: Hypothesis testing & Empirical modeling Name:________________________ Student number:_____________ Name:________________________ Student number:_____________ Objectives This lab contains three parts. In the first part, you will conduct hypothesis testing on measured capacitance data. In the second part of this lab, you will employ the paired t
-test to investigate the effect of a bias voltage on the capacitance of ceramic capacitors. In the third part, you will collect data on the current-
voltage relationship for a thermistor
1
and build empirical models of the device based on your data. Based on the template, which is provided to you on Canvas, a complete lab report
(per group) is required. The report should include introduction (stating the motivation and goals/objectives), experimental plan (describing the different parts of the measurement system), results and discussion, and conclusion. The submission of an electronic version
on Canvas is due one week after each lab session. 1
A thermistor is a type of resistor whose resistance varies significantly with temperature.
MSE 210 Engineering Measurement and Data Analysis
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1. Inference on the mean 1.a. Data collection 10 nF capacitance data set (for problem 1.b) Copy the capacitance data (10 nF) measured in Lab 2. Otherwise obtain 25 capacitors (10 nF) from the TA, measure the capacitance using your digital multimeter, and record your observations in the table below. Wait for sufficient time for the reading to get stabilized. Find the mean and standard deviation of the measured capacitance. C
1
Capacitance (nF) C
1
Capacitance (nF) C
1
Capacitance (nF) C
1
Capacitance (nF) 1 8 15 22 2 9 16 23 3 10 17 24 4 11 18 25 5 12 19 6 13 20 7 14 21 𝑥𝑥̅
C1
= _____________ s
C1
= _____________
100 nF capacitance data set (for problem 1.c) Copy the capacitance data (100 nF) measured in Lab 2. Otherwise obtain 10 capacitors (100 nF) from the TA, and do the same measurements as in step 1 to find the mean and standard deviation of the measured capacitance. C
2
1 2 3 4 5 6 7 8 9 10 Capacitance 𝑥𝑥̅
C2
= _____________ s
C2
= _____________
MSE 210 Engineering Measurement and Data Analysis
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1.b. Large data set The manufacturer claims that C
1
has a mean value of 10 nF and a standard deviation of 5%. Use statistical hypothesis testing to confirm or reject this claim based on your collected data. 1. Parameter of interest: __________________________________ 2. H
0
: __________________________________ 3. H
1
: __________________________________ 4. Test statistic: __________________________________ 5. Reject H
0
if: the P
-value is less than 0.05. 6. Computations: 7. Conclusion: __________________________________ What is the 90% confidence interval for mean value of C
1
? ___________ ≤ μ
C
1
≤ ___________
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MSE 210 Engineering Measurement and Data Analysis
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1.c. Small data set Based on the datasheet, C
2 should have a mean value of 100 nF. Use statistical hypothesis testing on the data that you have collected for C
2
to confirm or reject this claim. 1. Parameter of interest: __________________________________ 2. H
0
: __________________________________ 3. H
1
: __________________________________ 4. Test statistic: __________________________________ 5. Reject H
0
if: the P
-value is less than 0.05. 6. Computations: 7. Conclusion: __________________________________ What is the 90% confidence interval for mean value of C
2
? ___________ ≤ μ
C2
≤ ___________
MSE 210 Engineering Measurement and Data Analysis
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2. Testing for the goodness of the fit Divide the data range into 6 intervals and build a stem-and-leaf diagram for measurements of C
1
. Stem Leaf Frequency Employ the procedure for testing for the goodness of a fit to decide whether you should use a normal or a lognormal distribution for this data. Which distribution better represents the data? _______________ Normal distribution: 𝑓𝑓
(
𝑥𝑥
) =
1
𝜎𝜎√2𝜋𝜋
𝑒𝑒
−
( 𝑥𝑥
−
𝜇𝜇
)
2
2
𝜎𝜎
2
𝑓𝑓𝑓𝑓𝑓𝑓
𝑥𝑥
> 0
Estimate for μ: ______________
Estimate for σ: ______________
Interval Observed frequency Expected frequency ( 𝐎𝐎
𝐢𝐢
−
𝐄𝐄
𝐢𝐢
)
𝟐𝟐
𝐄𝐄
𝐢𝐢
̶
̶
̶
̶
̶
̶
Test statistic: 𝑋𝑋
0
2
= ∑
( 𝑂𝑂
𝑖𝑖
−
𝐸𝐸
𝑖𝑖
)
2
𝐸𝐸
𝑖𝑖
𝐾𝐾
𝑖𝑖=1
= _____ Degrees of freedom: _____ P
-value = _____
MSE 210 Engineering Measurement and Data Analysis
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Lognormal distribution: 𝑓𝑓
(
𝑥𝑥
) =
1
𝑥𝑥𝑥𝑥√2𝜋𝜋
𝑒𝑒
−
(
ln
(
𝑥𝑥
) – 𝜃𝜃
)
2
2
𝜔𝜔
2
𝑓𝑓𝑓𝑓𝑓𝑓
𝑥𝑥
> 0
Estimate for θ: ______________
Estimate for ω: ______________
Interval Observed frequency Expected frequency ( 𝐎𝐎
𝐢𝐢
−
𝐄𝐄
𝐢𝐢
)
𝟐𝟐
𝐄𝐄
𝐢𝐢
̶
̶
̶
̶
̶
̶
Test statistic: 𝑋𝑋
0
2
= ∑
( 𝑂𝑂
𝑖𝑖
−
𝐸𝐸
𝑖𝑖
)
2
𝐸𝐸
𝑖𝑖
𝐾𝐾
𝑖𝑖=1
= _____ Degrees of freedom: _____ P
-value = _____ Note:
The Expected frequency for each bin can be obtained by calculating the probability for each interval according to the standardized normal distribution, and then multiply the probability by the sample size.
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MSE 210 Engineering Measurement and Data Analysis
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3. Paired t-test You will be provided with 7 capacitors with a nominal value of 100 nF (we refer to these capacitors as C
1
). Build the circuit shown in Figure 1 on your breadboard. Apply a 1 V sine wave signal with a frequency of 1 kHz to the input of the circuit. Measure the magnitude of the AC voltages with the digital multimeter (make sure that you are using the AC voltage measurement function). Measure the input voltage at the beginning and do not change the voltage from the function generator for the rest of the experiment. v
in
= ___________ V Pick one of the capacitors and connect it between the ground and the output node of the circuit. Measure the output voltage using the multimeter and record your observation in the table below using 4 significant digits. Do not change the capacitor but switch its bottom electrode to a fixed 20 V DC voltage from the power supply. Measure the AC voltage at the output again and report it with 4 significant digits in the table. Repeat these steps for the other 6 capacitors and record your observations. For each reading wait for at least 2 to 3 minutes until it gets stabilized. Figure 1: Circuit for part 1 of the lab; instead of the illustrated switch, you can use a piece of wire and manually switch between ground and 20 V on your breadboard. Use the paired t
-test to decide whether applying a DC bias voltage has any effect on the magnitude of the output voltage. Assume that a confidence level of better than 90% is desired. 1 2 3 4 5 6 7 V
bias = 0V V
bias = 20V μ
D
= _____________ S
D
= _____________
MSE 210 Engineering Measurement and Data Analysis
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1. Parameter of interest: __________________________________ 2. H
0
: __________________________________ 3. H
1
: __________________________________ 4. Test statistic: __________________________________ 5. Reject H
0
if: __________________________________ 6. Computations: 7. Conclusion: __________________________________ The magnitude of the output voltage can be calculated from v
out
= v
in
1
√1+R
2
C
2
ω
2
where ω is the radial frequency of the signal. What happens to the capacitance value under a bias voltage? Answer: Can you think of the reason for this phenomenon? Answer:
MSE 210 Engineering Measurement and Data Analysis
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4. Building empirical models Measure and record the resistance of your thermistor with the multimeter (report 4 significant digits). We refer to this value as R
0
. R
0 = _______________
Ω
4a. Simple linear regression Use the DC power supply to set the voltage across the thermistor and measure the current through it with your multimeter (see Figure 2). Apply 8 DC voltages in the 0.5 to 10 V range and record the resulting currents with 4 significant digits in the table below. 1 2 3 4 5 6 7 8 V
DC
(V) 0.5 10 I
in
(mA) R =V
out
/I
in
Δ
R=R-R
0 Note 1:
The current through the thermistor or the voltage across it must NEVER exceed 10 mA or 30 V. Note 2:
Do not touch the thermistor during the experiments (use the breadboard). Note 3:
The thermistor can become very hot for currents larger than a few mA. Note 4:
After changing the voltage, wait for at least one full minute
before you take your measurement. What is the correlation coefficient? r
= _____________________ Figure 2: Circuit to measure the resistance of the thermistor as a function of the voltage across it.
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MSE 210 Engineering Measurement and Data Analysis
10/10
Construct a scatter-plot of V
vs I
in
in suitable software of your choice (e.g., MATLAB, Excel). What are the intercept and slope of the linear least square regression model for this data? β
0
= ________________ β
1
= ________________ Plot the regression line on the scatter-plot. Calculate the R
2
coefficient and test for the significance of regression (using a hypothesis test on β
1
=0). Describe your results qualitatively based on the expected function of the thermistor. 4b. Nonlinear regression Apply 8 DC voltages in the 12 to 30 V range and record the resulting currents with 4 significant digits in the table below. Record your observations in the table below (
follow the notes in section 2a
). 9 10 11 12 13 14 15 16 V
DC
(V) 12 30 I
in
(mA) R =V
out
/I
in
Δ
R=R-R
0 Construct a scatterplot of Δ
R
vs I
in
using the data from all 16 measurements. We want to use a regression model of the form Δ
R
=
β
0
+β
1 I
in
+
β
2 I
in
2
+
β
3 I
in
3
for the electrical behavior of the thermistor. Use the regress
command in MATLAB to calculate the regression coefficients based on your data. Plot the data and the regression polynomial on one graph using MATLAB. Explain qualitatively the performance of the thermistor based on your measured data and regression analysis and explain the reason for the nonlinear behavior. β
0
=____________ β
1
=____________ β
2
=____________ β
3
=____________