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Extra Homework Problems/Practice Problems
ME 236, Thorncroft
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Chapter 1
1. True or False:
a. A measurement with high precision (i.e., low precision error) has high accuracy.
b. Uncertainty is merely an estimate of the Error in a measurement.
c. In determining the weight of an object, Newton’s law, F = ma, is invoked, where a = g.
d. Precision Error is also called random or statistical error.
e. One pound-mass is equal to one pound-force on Earth.
2. Weighing yourself with clothes on (vs. weighing yourself without clothes on) introduces an
error to your measurement. This error is called:
a. reading error
b. bias error
c. precision error
d. instrument error
e. absolute error
3. We discussed in class that there are two general categories of error (bias and precision) but
three practical sources of error encountered in a measurement. Name them.
4. What are the definitions of (a) the newton in the SI system, (b) the pound-force in the “slug”
system, and (c) the pound-force in the “pound-mass” system?
Problem 1-A. The Reynolds number Re is a dimensionless number used in fluid mechanics and
is defined as
ρVD
Re =
,
µ
where ρ is the fluid density, V is the fluid velocity, D is some characteristic length of the body
immersed in the fluid, and µ is the fluid absolute viscosity.
a. Calculate the Reynolds number for the following properties: ρ = 1.16 kg/m3, V = 0.30
km/hr, D = 0.254 m, and µ = 1.85 × 10-5 N·s/m2. SHOW ALL CONVERSIONS
(CONVERSION FACTORS), and use only the most basic conversions (e.g., do not
convert kg/hr to m/s directly).
b. Convert the properties listed in part (a) to English Units (pound-mass system): lbm/ft3,
ft/s, ft, lbf·s/ft2 . Show all work.
c. Calculate the Reynolds number based on the English-unit properties calculated in part
(b).
d. What can you conclude about the (dimensionless) Reynolds number’s dependence on
unit system?
Extra Homework Problems/Practice Problems
Chapter 2
ME 236, Thorncroft
1. True or False:
a. The sum of all frequencies in a frequency distribution equals 1.
b. Relative frequency is the same as probability.
2. Which statement about the 3rd quartile is true?
a. The 3rd quartile is the range of values below the 75th percentile.
b. The 3rd quartile is the range of values above the 75th percentile.
c. The 3rd quartile is the range of values between the 50th and 75th percentile.
d. The 3rd quartile is the value such that 75 percent of the observations are smaller and 25
percent are larger.
3. Sixteen measurements of temperature range from 66.5 to 81.8 ºF. Select the most appropriate
bin assignment for this data from those below:
a.
b.
c.
d.
e.
65.0, 67.5, 70.0, 72.5, 75.0, 77.5, 80.0, 82.5
66.500, 68.845, 71.190, 73.535, 75.880, 78.225, 80.570, 82.915
66.5, 70.5, 74.5, 78.5, 82.5
66, 70, 74, 78, 82
None of the above
Problem 2-A. Consider the measurement of the temperature of hot gas flowing in a duct. The
relative frequency distribution (or at least part of it) is depicted below. Answer the following
questions:
a. What is the probability of
obtaining a measurement between
1090< T ≤ 1105 °C?
b. Three measurements fell within
the range 1110 <T ≤ 1115 ºC.
Are any measurements missing
from this graph, and if so, how
many?
Relative Frequency
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
1085
1090
1095
1100
1105
Tem perature (C)
1110
1115
1120
Extra Homework Problems/Practice Problems
Chapter 3
ME 236, Thorncroft
1. True or False:
a. All random measurements are normally distributed.
b. Using the z value in confidence intervals presumes that the population standard
deviation is known.
c. Approximately 92% of all data in a normal distribution lie within ±1.75 standard
deviations from the population mean.
2.
When predicting the mean of a population based on the mean of a sample (the population
standard deviation is known), what happens to the confidence interval when the sample
size approaches infinity? Circle all that apply:
a. the confidence interval approaches a finite value
b. the confidence interval approaches zero
c. the z value approaches 1.96
d. the confidence interval approaches infinity
2.11
3. Evaluate the integral: f ( x) =
∫
2
e 6−0.5 x dx . Hint: e a +b = e a e b . Do so without any integration
0.10
functions on your calculator.
Problem 3-A. A voltmeter is used to measure a known voltage of 100 V. Forty percent of the
readings are within 0.5 V of the mean value.
a. Assuming a normal distribution for the error, estimate the standard deviation for the
meter.
b. What is the probability that the mean of 10 readings will have an error greater than 0.75
V?
Extra Homework Problems/Practice Problems
Chapter 4
ME 236, Thorncroft
1. True or False:
a. Using the t distribution presumes that the population is normally distributed.
b. At the same level of confidence, the Student-t value is always less than or equal to the z
value.
c. The t table is used in place of the z table when the population standard deviation is
unknown.
d. The shape of the t distribution depends on the number of measurements in the sample.
2. Fill in the missing values. SKETCH the z or t distribution, AND graphically demonstrate the
probabilities and ranges.
a. P( -__________ ≤ z ≤ +__________ ) = 0.75 (both unknowns have same magnitude)
b. P(-2.35 ≤ z ≤ -0.88 ) = _____________
c. P( t ≤ __________ ) = 0.75 (50 degrees of freedom)
d. P( -0.695 ≤ t ≤ 3.055 ) = __________ (12 degrees of freedom)
Problem 4-A next page
Extra Homework Problems/Practice Problems
ME 236, Thorncroft
Problem 4-A. You are designing an automatic coin counter that stacks 50 pennies at a time to
be placed in coin wrappers. You plan on measuring the height of the stack to determine the
number of coins, but are concerned as to how accurate the method will be.
You measure the thickness of 30 pennies, with the data listed below.
No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Sums
Thickness,
x
(mm)
1.48
1.47
1.38
1.42
1.54
1.56
1.44
1.52
1.47
1.56
1.42
1.52
1.60
1.47
1.50
1.47
1.44
1.45
1.52
1.42
1.49
1.39
1.51
1.53
1.43
1.48
1.45
1.61
1.47
1.55
44.56
(x − x)
2
0.00003
0.00024
0.01110
0.00427
0.00299
0.00558
0.00206
0.00120
0.00024
0.00558
0.00427
0.00120
0.01315
0.00024
0.00022
0.00024
0.00206
0.00125
0.00120
0.00427
0.00002
0.00909
0.00061
0.00200
0.00306
0.00003
0.00125
0.01554
0.00024
0.00317
0.09735
a. Based on this data, estimate the height of a stack of 50 pennies.
How accurate is your estimate, at 95% confidence?
b. What percentage of pennies from the population have
thicknesses of at least 1.34 mm?
Extra Homework Problems/Practice Problems
Chapter 5
ME 236, Thorncroft
1. Using the t distribution in confidence intervals requires the population to be normally
distributed; however, it is approximately valid for non-normal populations as long as what
conditions are met?
2. Comment on the following statement: You can (and should) reject outliers from a set of data
as long as the statistical models identify them as outliers.
Extra Homework Problems/Practice Problems
Chapter 6
1. Consider the equation H =
ME 236, Thorncroft
ld 1 / 3
. If the uncertainty of each variable is the same fraction of
10k 1 / 5
their respective nominal value, what variable’s uncertainty has the largest effect on the
uncertainty in H?
a. l
b. d
c. k
d. H
2. Consider the function F = AB 0.5C 2 / 3 / D 4 / 7 , where A, B, C, and D are measured variables. If
all these variables have the same uncertainty (as a percentage of the nominal value), which
variable affects the uncertainty in the function F the most?
a. A
b. B
c. C
d. D
Problem 6-A. Earth’s gravity varies along its surface, and a large portion of this variation is
from centripetal acceleration due to the rotation of the earth. The centripetal acceleration at the
surface of the Earth can be calculated by
2
⎛ 2π ⎞
2
aR = ⎜
⎟ R E cos φ ,
⎝ T ⎠
where T = period of rotation of the Earth = 8.64x104 s (negligible uncertainty)
RE = radius of Earth = 6.37x106 m ± 1500 m
φ = Latitude on Earth (radians)
a. If our latitude on Earth is 35 degrees, 17 minutes ± 1 degree (35.3 ± 1.0°), determine our local
centripetal acceleration and its uncertainty. Use dimensional uncertainty propagation. (Hint:
and recall that 180° = π radians)
b. If the local gravity (without rotation) is 9.806 m/s2, what is the local effective gravity, and its
uncertainty? (In doing so, make an assumption about the uncertainty in the value of gravity
without rotation, g = 9.806 m/s2.)
Extra Homework Problems/Practice Problems
Chapter 7
ME 236, Thorncroft
1. True or false:
a. When choosing an appropriate model (equation) for a curve-fit, you should pick the
equation that yields an R2 value closest to 1.
b. The correlation coefficient, r, is a measure of the degree of linear correlation, but the
R2 value applies to any curve-fit function.
c. When choosing an appropriate model (equation) for a curve-fit, you should pick the
equation that minimizes the residuals.
2. In performing statistical analysis of curve fits, we assume that the data are scattered normally
about the curve-fit line. Why?
3. The following are plots of data and their predicted curve fits. Which values of the correlation
coefficient are most likely to be incorrect? (Choose all that apply)
a. r = 0.8
b. r = 0.9
c. r = - 0.7
d. r = 0.9
Extra Homework Problems/Practice Problems
Chapter 8
ME 236, Thorncroft
1. True or false:
a. A set of n x-y data pairs can be fit with a polynomial curve up to an order of n-1.
b. If a 6th order polynomial curve is fit to 5 data points, the resulting R2 value will be
equal to exactly 1.
2. Initially, what factor determines the choice of an appropriate curve fit model (sinusoidal,
linear, polynomial, etc.) to a set of experimental data?
3. What role does statistics play in refining a curve-fit model (for example, when choosing
between a 3rd-order and a 4th-order polynomial fit)?
4. Name two strategies for dealing with outliers in a curve-fitted set of data (aside from
discarding them).