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Regression
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Transforming Numerical Methods Education for the STEM undergraduate
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Applications
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Mousetrap Car
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Torsional Stiffness of a Mousetrap Spring
Torque (N-m)
0.4
0.3
T  k 0  k1 θ
0.2
0.1
0.5
1
1.5
2
θ (radians)
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Stress vs Strain in a Composite Material
Stress, σ (Pa)
3.0E+09
2.0E+09
1.0E+09
0.0E+00
0
0.005
0.01
0.015
0.02
Strain, ε (m/m)
  E
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A Bone Scan
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Radiation intensity from Technitium-99m
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Trunnion-Hub Assembly
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Thermal Expansion Coefficient Changes
with Temperature?
6.00E-06
5.00E-06
(in/in/o F)
Thermal expansion coefficient, α
7.00E-06
4.00E-06
3.00E-06
2.00E-06
-400
-300
-200
1.00E-06
-100
0
100
200
Temperature, o F
α  a0  a1T  a2T
2
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THE END
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Pre-Requisite Knowledge
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This rapper’s name is
A.
B.
C.
D.
E.
Da Brat
Shawntae Harris
Ke$ha
Ashley Tisdale
Rebecca Black
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Close to half of the scores in a test given to
a class are above the
A.
B.
C.
D.
average score
median score
standard deviation
mean score
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Given y1, y2,……….. yn, the standard
deviation is defined as
1.
2
.   yi  y 
n
/n
i 1
2.
.
 y
i 1
3.
2
n
i
 y / n
2
. yi  y 
n
/(n  1)
i 1
4.
.
2
n
 y
i 1
i
 y  /(n  1)
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THE END
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Linear Regression
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Given (x1,y1), (x2,y2),……….. (xn,yn), best fitting
data to y=f (x) by least squares requires
minimization of
 y  f x 
i
i 1
n
 y  f x 
n
 y
i
i 1
D. )
2
 f xi 
n
 y  y  , y 
2
n
i 1
i
y
i 1
n
i
0%
0%
0%
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0%
)
C. )
i
)
i 1
i
)
B. )
i
)
A. )
n
The following data
x
y
1
20
30
40
1
400
800
1300
is regressed with least squares regression to a
straight line to give y=-116+32.6x. The
observed value of y at x=20 is
1. -136
2. 400
3. 536
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6
0%
53
0
0%
40
-1
3
6
0%
The following data
x
y
1
20
30
40
1
400
800
1300
is regressed with least squares regression to a
straight line to give y=-116+32.6x. The
predicted value of y at x=20 is
1. -136
2. 400
3. 536
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Transforming Numerical Methods Education for the STEM Undergraduate
6
0%
53
0
0%
40
-1
3
6
0%
The following data
x
y
1
20
30
40
1
400
800
1300
is regressed with least squares regression to a
straight line to give y=-116+32.6x. The
residual of y at x=20 is
1. -136
2. 400
3. 536
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6
0%
53
0
0%
40
-1
3
6
0%
THE END
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Nonlinear Regression
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When transforming the data to find the constants of the
regression model y=aebx to best fit (x1,y1), (x2,y2),………..
(xn,yn), the sum of the square of the residuals that is
minimized is
1.
2.
3.
4.
 y ae 
n
i 1
bxi 2
i
n
 ln(y )  ln a  bx 
2
i
i 1
i
n
  y  ln a  bx 
i 1
2
i
i
n
 ln(y )  ln a  b ln(x )
i 1
2
i
i
0%
0%
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0%
0%
  k1ek 
for concrete in compression, where  is the stress and 
When transforming the data for stress-strain curve
2
is the strain, the model is rewritten as
A. ) ln   ln k1  ln   k2

B. ) ln  ln k1  k 2

0%
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0%
)
0%
)
0%
)
D. ) ln   ln(k1 )  k2
)

 ln k1  k 2
C. ) ln

Adequacy of Linear
Regression Models
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The case where the coefficient of determination
for regression of n data pairs to a straight line
is one if
33%
33%
33%
A. none of data points fall
exactly on the straight
line
B. the slope of the straight
line is zero
C. all the data points fall on
the straight line
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A.
B.
Transforming Numerical Methods Education for the STEM Undergraduate
C.
The case where the coefficient of determination
for regression of n data pairs to a general
straight line is zero if the straight line model
25%
25%
25% 25%
A. has zero intercept
B. has zero slope
C. has negative slope
D. has equal value for
intercept and the slope
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A.
B.
C.
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D.
The coefficient of determination varies between
A. -1 and 1
B. 0 and 1
C. -2 and 2
an
-2
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d
1
an
d
0
d
an
-1
0%
2
0%
1
0%
The correlation coefficient varies between
A. -1 and 1
B. 0 and 1
C. -2 and 2
an
-2
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d
1
an
d
0
d
an
-1
0%
2
0%
1
0%
If the coefficient of determination is 0.25, and
the straight line regression model is y=2-0.81x,
the correlation coefficient is
20%
A.
B.
C.
D.
E.
20%
20%
20%
20%
-0.25
-0.50
0.00
0.25
0.50
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A.
B.
C.
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D.
E.
If the coefficient of determination is 0.25, and
the straight line regression model is y=2-0.81x,
the strength of the correlation is
20%
A.
B.
C.
D.
E.
20%
20%
20%
20%
Very strong
Strong
Moderate
Weak
Very Weak
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A.
B.
C.
Transforming Numerical Methods Education for the STEM Undergraduate
D.
E.
If the coefficient of determination for a
regression line is 0.81, then the percentage
amount of the original uncertainty in the data
explained by the regression model is
A. 9
B. 19
C. 81
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0%
81
0%
19
9
0%
The percentage of scaled residuals expected
to be in the domain [-2,2] for an adequate
regression model is
85
90
95
100
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0
0%
10
0%
95
0%
90
0%
85
A.
B.
C.
D.
THE END
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The average of the following numbers is
0%
0%
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0%
.
0%
10
4.0
7.0
7.5
10.0
14
7.
5
1.
2.
3.
4.
10
7.
4
4.
2
The following data
x
y
1
20
30
40
1
400
800
1300
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0%
40
.
0%
.6
2
.9
5
6
0%
28
27
.4
8
0%
32
1. 27.480
2. 28.956
3. 32.625
4. 40.000
5
is regressed with least squares regression to
y=a1x. The value of a1 most nearly is
A scientist finds that regressing y vs x data given below to
straight-line y=a0+a1x results in the coefficient of
determination, r2 for the straight-line model to be zero.
x
y
1
2
3
6
11
22
17
?
The missing value for y at x=17 most nearly is
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0%
34
.
0%
6.
88
2.
0%
-2
.
44
4
0%
9
1. -2.444
2. 2.000
3. 6.889
4. 34.00
A scientist finds that regressing y vs x data given below to
straight-line y=a0+a1x results in the coefficient of
determination, r2 for the straight-line model to be one.
x
y
1
2
3
6
11
22
17
?
The missing value for y at x=17 most nearly is
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0%
34
.
0%
6.
88
2.
0%
-2
.
44
4
0%
9
1. -2.444
2. 2.000
3. 6.889
4. 34.00
The average of 7 numbers is given 12.6. If 6
of the numbers are 5, 7, 9, 12, 17 and 10,
the remaining number is
1. -47.9
2. -47.4
3. 15.6
4. 28.2
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