Exam in Measurement Techniques MJ2440: Part I (Each Question 1 P)
Exam in Measurement Techniques MJ2440: Part I (Each Question 1 P)
Exam in Measurement Techniques MJ2440: Part I (Each Question 1 P)
2021-01-13, 14-18
Closed book exam. Calculator allowed. Max score 60 p. Required to pass is 3 p on each subsection of
Part I. To reach higher grades one must pass Part I and earn points on Part II. For a grade D (Part II ≥
10 p), for a grade C (Part II ≥ 20 p), for a grade B (Part II ≥ 30 p), for a grade A (Part II ≥ 35 p).
1. (experimental design)
a. Explain the benefit of Randomization in experimental design To reduce bias and to validate
statistical analyse.
b. Explain the benefit of Replication in experimental design To estimate dispersion of data and to
obtain s more precise estimate of a sample mean.
c. Explain the benefit of Blocking in experimental design To reduce variability from nuisance
factors to increase precision with which comparisons among the factors of interests are made.
d. Use an example to explain the concepts of factors, levels and responses. Baking a cake: factors
could be ingredients and temperature in oven, levels could be amount of ingredients and
temperature setting, response (or result) could be the taste of the cake.
e. What is a nuisance factor? A nuisance factor is a factor that probably has an effect on the
response, but we are not interested in that effect.
2. (sensors)
3. (measurement uncertainty)
a. What is the central limit theorem and why is it important in uncertainty analysis?
The theorem states that the sampling distribution of the mean becomes approximately
normal regardless of the distribution of the original variable as the sample size grows.
Moreover even the individual variable is often normally distributed if its value is dependent
of many other random variables. This validates the statistical analysis based on the normal
distribution.
b. What is type A and type B uncertainties. Explain the difference.
The type A uncertainty is based on repeated measurements. It is calculated as the experimental
standard deviation of the mean. This “domestic” uncertainty can be controlled by the
experimenter through the number of replications. The type B uncertainty is based on other
sources such as assumed distributions or calibration reports (which in turn may come from a
type A uncertainty but in an experiment done elsewhere). The type B uncertainty can be viewed
as an imported (fossilized) uncertainty.
c. Describe the Monte Carlo method in uncertainty analysis.
In the Monte Carlo method each input quantity is assigned with a probability distribution from
which random values are generated and used as input values to the measurement model,
together giving as a result one output quantity value. This is repeated many times to build up a
frequency distribution of the output quantity. From this distribution the interval having the
desired confidence level can be calculated
d. What is the difference between s(x) and σ ?
s(x) is the experimental standard deviation sometimes called the sample standard deviation. σ is
the population standard deviation
e. What are the two reasons (discussed in class) that the GUM technique fails and other techniques
(like the Monte Carlo) is used? Measurement model not linear. Convolved distributions not
normal.
4. (statistical techniques)
𝑛𝑛
1 1
𝑥𝑥𝑚𝑚 = � 𝑥𝑥𝑖𝑖 = (25,67) = 2,567
𝑛𝑛 10
𝑖𝑖=1
1� 1�
∑𝑛𝑛𝑖𝑖=1(𝑥𝑥𝑖𝑖 − 𝑥𝑥𝑚𝑚 )2 2 1 2
𝑠𝑠 = � � =� (0,159610)� = 0,13317
𝑛𝑛 − 1 10 − 1
2 2
𝑢𝑢𝑐𝑐 2 (𝑙𝑙) = ∑ 𝑢𝑢𝑖𝑖 2 (𝑙𝑙) => 𝑢𝑢𝑐𝑐 = �12 + � � = 1,5275 m
√3
𝑙𝑙 = (750,4 ± 3,2) 𝑚𝑚 where the number following the ± is the expanded uncertainty
𝑈𝑈 = 𝑘𝑘𝑢𝑢𝑐𝑐 with U determined from a combined standard uncertainty 𝑢𝑢𝑐𝑐 = 1,5275 m and a coverage
factor of k = 2,086 based on the t-distribution for 𝑣𝑣 = 20 degrees of freedom, and defines an
interval estimated to have a level of confidence of 95 %.
7. (15 p) Find out if the ambient temperature affects the golf result. The golf score from five
different persons are compared at two temperatures 20 and 10 °C. Use a significance level of 5
%.
person 20 10
1 101 106
2 87 86
3 125 135
4 66 76
5 55 60
𝐻𝐻0 : 𝜇𝜇1 = 𝜇𝜇2 𝐻𝐻1 : 𝜇𝜇2 ≠ 𝜇𝜇1 (Reject 𝐻𝐻0 if |𝑡𝑡0 | > 𝑡𝑡1−𝛼𝛼⁄2,𝜐𝜐 )
2 87 86 1 46.24
4 66 76 -10 17.64
5 55 60 -5 0.64
𝑑𝑑̅ −5.8
𝑡𝑡 = 𝑠𝑠 = = −2.85054
𝑑𝑑
� 4.549725�
√𝑛𝑛 √5
Step 5: Draw conclusions
We can reject the null hypothesis and conclude that the ambient temperature affects the results.
Type A evaluation of standard uncertainty
(based on repeated observations)
𝑛𝑛 n
1 1 2 𝑠𝑠(𝑞𝑞𝑘𝑘 )
𝑞𝑞� = � 𝑞𝑞𝑘𝑘 𝑠𝑠(𝑞𝑞𝑘𝑘 ) = � ��qj − q� � 𝑠𝑠(𝑞𝑞�) = 𝒖𝒖(𝒙𝒙𝒊𝒊 ) = s(q� )
𝑛𝑛 n−1 √𝑛𝑛
𝑘𝑘=1 j=1
Expanded uncertainty U
𝑈𝑈 = 𝑘𝑘 ∙ 𝑢𝑢𝑐𝑐 (𝑦𝑦) where k is chosen to be the t1−α/2,ν critical value from the t-table
with ν degrees of freedom. For large degrees of freedom, k=2 approximates 95 % coverage.
The t distribution is symmetric so that t1-α,ν = -tα,ν. The t table can be used for both
one-sided (lower and upper) and two-sided tests using the appropriate value of α.
ν 0.90 0.95 0.975 0.99 0.995 0.999
Non-Equal standard 𝑛𝑛 𝑠𝑠 2 𝑠𝑠 2
2
Paired t-test 2
1�
2
𝑑𝑑̅ ∑𝑛𝑛𝑖𝑖=1�𝑑𝑑𝑖𝑖 − 𝑑𝑑̅ �
𝑡𝑡 = 𝑠𝑠 𝑠𝑠𝑑𝑑 = � �
𝑑𝑑
𝑛𝑛 − 1 𝜈𝜈 = 𝑛𝑛 − 1
�
√𝑛𝑛
1
𝑑𝑑̅ = ∑𝑛𝑛𝑖𝑖=1 𝑑𝑑𝑖𝑖 𝑑𝑑𝑖𝑖 = 𝑦𝑦𝑖𝑖 − 𝑧𝑧𝑖𝑖
𝑛𝑛
Cumulative Distribution Function of the Standard Normal Distribution