Teaching the meaning of the correlation coefficient is difficult enough without having to use a confusing formula for it. This "introduces" a "new" equation whose very form helps explain the meaning of the correlation coefficient.
Jr imp, Maths IB Important, Mathematics IB, Mathematics, Jr. Maths, Mathematics AP board, Mathematics important, Maths AP Board, Inter Maths IB, Inter Maths IB Important.
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Jr imp, Maths IB Important, Mathematics IB, Mathematics, Jr. Maths, Mathematics AP board, Mathematics important, Maths AP Board, Inter Maths IB, Inter Maths IB Important.
Worried about timely submission of your econometrics projects. Hire expert econometrics assignment help from Statisticshelpdesk; we offer class academic help by top quality tutors quickly with free moderation and clarification online. Contact today.
Complete presentation On Regression Analysis.
Proved By Three methods, Least Square Method, Deviation method by assumed mean, Deviation method By Arithmetic mean.
Overviews non-parametric and parametric approaches to (bivariate) linear correlation. See also: http://en.wikiversity.org/wiki/Survey_research_and_design_in_psychology/Lectures/Correlation
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Solving the Pose Ambiguity via a Simple Concentric Circle ConstraintDr. Amarjeet Singh
Estimating the pose of objects with circle feature from images is a basic and important question in computer vision
community. This paper is focused on the ambiguity problem in pose estimation of circle feature, and a new method is proposed based
on the concentric circle constraint. The pose of a single circle feature, in general, can be determined from its projection in the image
plane with a pre-calibrated camera. However, there are generally two possible sets of pose parameters. By introducing the concentric
circle constraint, interference from the false solution can be excluded. On the basis of element at infinity in projective geometry and
the Euclidean distance invariant, cases that concentric circles are coplanar and non-coplanar are discussed respectively. Experiments
on these two cases are performed to validate the proposed method.
Complete presentation On Regression Analysis.
Proved By Three methods, Least Square Method, Deviation method by assumed mean, Deviation method By Arithmetic mean.
Overviews non-parametric and parametric approaches to (bivariate) linear correlation. See also: http://en.wikiversity.org/wiki/Survey_research_and_design_in_psychology/Lectures/Correlation
If you are looking for business statistics homework help, Statisticshelpdesk is your rightest destination. Our experts are capable of solving all grades of business statistics homework with best 100% accuracy and originality. We charge reasonable.
Solving the Pose Ambiguity via a Simple Concentric Circle ConstraintDr. Amarjeet Singh
Estimating the pose of objects with circle feature from images is a basic and important question in computer vision
community. This paper is focused on the ambiguity problem in pose estimation of circle feature, and a new method is proposed based
on the concentric circle constraint. The pose of a single circle feature, in general, can be determined from its projection in the image
plane with a pre-calibrated camera. However, there are generally two possible sets of pose parameters. By introducing the concentric
circle constraint, interference from the false solution can be excluded. On the basis of element at infinity in projective geometry and
the Euclidean distance invariant, cases that concentric circles are coplanar and non-coplanar are discussed respectively. Experiments
on these two cases are performed to validate the proposed method.
Solution of the Special Case "CLP" of the Problem of Apollonius via Vector Ro...James Smith
Using ideas developed in detail in http://www.slideshare.net/JamesSmith245/rotations-of-vectors-via-geometric-algebra-explanation-and-usage-in-solving-classic-geometric-construction-problems-version-of-11-february-2016, this document solves one of the special cases of the famous Problem of Apollonius. A new Appendix presents alternative solutions.
See also:
http://www.slideshare.net/JamesSmith245/solution-of-the-ccp-case-of-the-problem-of-apollonius-via-geometric-clifford-algebra
http://www.slideshare.net/JamesSmith245/rotations-of-vectors-via-geometric-algebra-explanation-and-usage-in-solving-classic-geometric-construction-problems-version-of-11-february-2016
http://www.slideshare.net/JamesSmith245/resoluciones-de-problemas-de-construccin-geomtricos-por-medio-de-la-geometra-clsica-y-el-lgebra-geomtrica-vectorial
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 10: Correlation and Regression
10.2: Regression
Hierarchical matrix approximation of large covariance matricesAlexander Litvinenko
We research class of Matern covariance matrices and their approximability in the H-matrix format. Further tasks are compute H-Cholesky factorization, trace, determinant, quadratic form, loglikelihood. Later H-matrices can be applied in kriging.
1 College Algebra Final Examination---FHSU Math & C.S.docxjoyjonna282
1
College Algebra
Final Examination---FHSU Math & C.S. Department
Form G Name ____________________________
Multiple Choice:
Mark through (horizontally) the space on the answer card corresponding to the best answer for each
problem.
1. Solve the equation by factoring: 2 9 18 0x x
[A] 6, 3 [B] 6, 3 [C] 6, 3 [D] 6, 3
2. Write the product in the standard form a + bi: (2 6 )(2 9 )i i
[A] 58 6i [B] 50 30i [C] 58 6i [D]
2
54 6 4i i
3. Find the real solutions of the equation 14 21 2 .x x
[A] 5 [B] 3 [C] 4 [D] 5
4. Solve the inequality 3 7 2 4x x . Express your answer using interval notation.
[A] 11, [B] [3, ) [C] 3, [D] ( , 3]
5. Solve the equation 3 3 18.x
[A] 3, 9 [B] 3 [C] 9, 3 [D] no solution
6. Solve the inequality 5 1 5x . Express your answer using interval notation.
[A] 645 5, or , [B]
64
5 5
,
[C] 645 5, [D]
64
5 5
, or ,
7. A bank loaned $68,000, part of it at a rate of 15% per year and the rest at a rate of 5% per year. If the
interest received was $6600, how much was loaned at 15%?
[A] $36, 000 [B] $33, 000 [C] $32, 000 [D] $35, 000
8. Find the distance between the points 1 2(2, 2) and ( 10, 3).PP
[A] 14 [B] 26 [C] 169 [D] 13
9. List the intercepts for the graph of the equation
2
16 0.x y
[A] ( 4, 0), (0,16), (4, 0) [B] (4, 0), (0,16), (0, 16)
[C] (0, 4), (16, 0), (0, 4) [D] ( 4, 0), (0, 16), (4, 0)
2
10. Find the slope-intercept form of the equation of the line that is parallel with the given properties:
( 2, 3); and 9 20y x
[A] 9 15y x [B] 9 20y x [C]
1
20
9
y x [D]
1
20
9
y x
11. Write the standard from of the equation of the circle with radius 5 and center (1, 4) .
[A]
2 2
4 1 25x y [B]
2 2
1 4 5x y
[C]
2 2
4 1 25x y [D]
2 2
1 4 5x y
12. Write a general formula to describe the variation.
The illumination I produced on a surface by a source of light varies directly as the candlepower c of
the source and inversely as the square of the distance d between the source and the surface.
[A]
2
I kcd [B]
2
2
kc
I
d
[C]
2
kc
I
d
[D]
2
kd
I
c
13. For the given functions ( ) 6 3 and ( ) 7 9f x x g x x , determine ( )( ).f g x
[A]
2
( )( ) 4 27f g x x [B]
2
( )( ) 13 33 6f g x x x
[C]
2
( )( ) 42 33 27f g x x x [D]
2
( )( ) 42 12 27f g x x x
14. The graph of a function f is given. For what values of x is
( ) 0f x ?
[A] 15,17.5
[B] 15,17.5, 25
[C] 15
[D] 25, 15,17.5, 25
15. Determine whether the function
3 2
( ) 4 9f x x is even, odd, neither, or ...
JEE Mathematics/ Lakshmikanta Satapathy/3D Geometry theory part 9/ Equation of plane in intercept form and plane passing through the line of intersection of two planes
This article describes a new way to calculate binomial confidence limits, ones that define an interval that is narrower than the limits calculated by the Score, Wald, or Exact methods, and that is more defensible from a theoretical point of view since, unlike other binomial confidence limits, these Reasonable Limits are not statistically significantly different from the sample proportion on which they are based.
1. A Simpler Method for Teaching the Meaning of the Correlation Coefficient
Copyright 2012, by John N. Zorich Jr., Zorich Consulting & Training, www.johnzorich.com
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A SIMPLER METHOD FOR TEACHING THE MEANING
OF THE CORRELATION COEFFICIENT
by John N. Zorich, Jr.
For over 100 years, the concept of the correlation coefficient (CC) has been taught to
beginning students of statistical science but without serious reference to the equation on which it
is based. Instead, the meaning of the CC has been explained using wordy generalities and
textbook scatter plots ⎯ the CC being larger the less scattered the plot looks. Unfortunately,
such generalities result in most students internalizing subtle misconceptions. Figs. 1, 2, and 3
demonstrate some of the difficulties that cannot be explained using classic teaching methods.
In Fig. 1, each of the four Data Sets (labeled A, B, C, and D) has a least squares linear
regression (LSLR) straight line drawn through the raw data points. Each Data Set has 2 different
Y values for each even X whole number from 2 through 18 (in Set D, the 2 different Y values at
each X value are so close together that they appear as a single dot). In spite of the obvious
differences between these data sets, they all yield the same high CC.
In Fig. 2, Data Sets B and C are, in effect, subsets of Set A. Set B is composed of the first six
data points from Set A after subtracting 3.0 from each Y value. Likewise, Set C is the first three
data points from Set A after subtracting 6.0. Notice that all three regression lines have the
identical slope and have data points that lie at exactly the same distance from their regression
line. It seems that a coefficient that purports to indicate correlation should indicate that these
three data sets are, correlatively speaking, the same. But, as indicated in the figure, the larger the
data set, the larger the CC.
In Fig. 3, we seem to have a contradiction to the conclusion reached regarding Fig. 2; that is,
in Fig. 3, the more data points, the lower the CC, despite the fact that all three regression lines
have the identical slope and have data points that lie at exactly the same distance from their
regression line (as in Fig. 2, Sets B and C are subsets of Set A, offset by a value of 3.0 or 6.0,
respectively).
The separation of CC meaning from the CC equation stems historically from the fact that the
equations that appeared most often in textbooks were difficult to teach or understand. The first
equations were developed in the late 1800s1; the most commonly cited one has been some
version of the following:
∑ ( X − X )( Y − Y )
CC = Equation 1 (the traditional equation)
∑ ( X − X ) ∑ (Y − Y )
2 2
The “Short Method” 2 (Equation 2) was popularized in the early 1900s, and became more
common after it was revised slightly and renamed the “computational form”3 for use with
sophisticated mechanical calculators and simple electronic ones:
Page 1 of 7
2. A Simpler Method for Teaching the Meaning of the Correlation Coefficient
Copyright 2012, by John N. Zorich Jr., Zorich Consulting & Training, www.johnzorich.com
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N ∑ XY − ∑ X ∑ Y
CC = Equation 2
⎛ ⎞⎛ ⎞
⎜ N ∑ X − (∑ X )2⎟⎜ N ∑ Y − (∑ Y )2⎟
2 2
⎝ ⎠⎝ ⎠
A simpler equation4 (Equation 3) was developed, but it didn’t have much pedagogic value. In
this equation, Sx and Sy are the standard deviation of the X and Y data, respectively, and “slope”
is the slope of the linear regression line calculated by the method of least squares:
(slope)Sx
CC = Equation 3
Sy
The ratio of slope to Sy in this equation is helpful in explaining the lack of dependence of the
CC on slope, since a large CC can result from either a large slope, a large Sx, and/or a small Sy.
This equation is unable to explain what the CC is, but is wonderful for explaining what the CC is
not.
About 1985, I thought I’d developed a new, more instructive equation. Alas, a few years later,
I found my “new” equation at the end of an appendix to a 1961 introductory statistics book
written by someone else.5
I discovered my “new” equation within the equation for the Coefficient of Determination
(CD). In the CD equation (see next), Ye represents the Y values calculated for each X value by
using the least squares linear regression analysis equation (Ye = a + bX), and Yi represents the
raw Y data. One Ye value is calculated for each Yi value. As always in least squares linear
regression, the mean of the Ye data is the same as that of the Yi, and so it is not subscripted in
the equation below:
= CC 2 =
(
∑ Ye − Y )2 Equation 4
CD
∑ (Yi − Y )
2
Dividing top and bottom of the fraction by N-1 (where N is the number of Yi data points), I
discovered an equation that is the ratio of two sample variances:
V a ria n c e ( Y e )
CC 2 = Equation 5
V a ria n c e ( Y i )
After taking the square root of both sides, I found an equation containing the absolute value of
the CC on one side, and the ratio of two sample standard deviations on the other:
S td D e v ia tio n ( Y e )
CC = Equation 6 (the “new” equation)
S td D e v ia tio n ( Y i )
Page 2 of 7
3. A Simpler Method for Teaching the Meaning of the Correlation Coefficient
Copyright 2012, by John N. Zorich Jr., Zorich Consulting & Training, www.johnzorich.com
-----------------------------------------------------------------------------------------------------------------------------------
Although this equation does not calculate the sign of the CC, this is not a limitation. In LSLR,
the sign of the CC is always the same as that of the regression coefficient (“b” in the LSLR
equation Y= a + bX) — that is, if the slope is negative, so is the CC, and vice versa, as easily
seen from Equation 3, above. Thus, the sign of the CC has no meaning independent of the
regression slope, and so the only unique aspect of the CC is its absolute value, which the “new”
equation calculates.
Calculation of the CC using the new equation is shown by example in Table 1.
Not only can this new equation be used to easily explain the surprising CC results shown in Figs.
1, 2, and 3, but it can also be use to explain other interesting facts, such as:
1. The correlation coefficient can never equal exactly 1.000, unless all the Yi’s form a
perfectly straight line ⎯ which is the only case in which the standard deviations of Ye and
Yi are identical.
2. The CC can never equal exactly 0.000, unless the standard deviation of Ye is also zero ⎯
which would occur only if the calculated linear regression line were perfectly horizontal.
3. The CC represents the fraction of the total variation in Yi, as measured in units of standard
deviation, that can be explained by a linear relationship between Yi and X. The larger the
CC, the larger the fraction of the Yi variation which can be explained this way. The
remaining variation can’t be explained, at least not by the CC.
Page 3 of 7
4. A Simpler Method for Teaching the Meaning of the Correlation Coefficient
Copyright 2012, by John N. Zorich Jr., Zorich Consulting & Training, www.johnzorich.com
-----------------------------------------------------------------------------------------------------------------------------------
Fig. 1
Linear Regression by Least Squares
Correlation Coefficient = 0.955 for Each Data Set
14
A
12
10
B
8
6
C
4
2
D
0
0 2 4 6 8 10 12 14 16 18 20
Page 4 of 7
5. A Simpler Method for Teaching the Meaning of the Correlation Coefficient
Copyright 2012, by John N. Zorich Jr., Zorich Consulting & Training, www.johnzorich.com
-----------------------------------------------------------------------------------------------------------------------------------
Fig. 2
14
Linear Regression by Least Squares
12
A
10
8
B
6
4
Correlation Coefficients
Data Set A = 0.955
Data Set B = 0.906
C
Data Set C = 0.714
2
0
0 2 4 6 8 10 12 14 16 18 20
Page 5 of 7
6. A Simpler Method for Teaching the Meaning of the Correlation Coefficient
Copyright 2012, by John N. Zorich Jr., Zorich Consulting & Training, www.johnzorich.com
-----------------------------------------------------------------------------------------------------------------------------------
Fig. 3
Linear Regression by Least Squares
14
Correlation Coefficients
Data Set A = 0.955
Data Set B = 0.962
12
Data Set C = 0.971 A
10
B
8
6 C
4
2
0
0 2 4 6 8 10 12 14 16 18 20
Page 6 of 7
7. A Simpler Method for Teaching the Meaning of the Correlation Coefficient
Copyright 2012, by John N. Zorich Jr., Zorich Consulting & Training, www.johnzorich.com
-----------------------------------------------------------------------------------------------------------------------------------
Table 1.
Calculation of the Correlation Coefficient using the “New Equation”
(Ye is calculated using the regression equation at the bottom of this table)
X raw data Yi raw data Ye, calculated
6 10 9.7609
7 11 10.2065
7 10 11.5435
8 11 11.5435
9 12 11.5435
10 12 11.9891
10 12 12.4348
12 13 12.8804
13 14 13.3261
15 14 13.7717
Standard Deviation = Standard Deviation =
1.4491 1.4023
Least Squares Linear Regression Equation is Ye = 7.2221 + 0.4823X
Correlation Coefficient (CC) using the “Traditional Equation” = 0.9677
CC using the “New Equation,” (StdDev Ye) / (StdDev Yi) = 0.9677
1
S. M. Stigler, The History of Statistics, 1986 (Belknap Press, Cambridge MA), chapter 9.
2
J. G. Smith, Elementary Statistics, 1934 (Henry Holt & Co., New York) p. 374.
3
H. L. Alder, E. B. Roessler, Introduction to Probability and Statistics, 6th ed., 1977 (W. H. Freeman & Co., San
Francisco) p. 230.
4
Ibid, p. 231. Alder & Roessler use different symbols than are used here.
5
W. J. Reichmann, Use and Abuse of Statistics, 1961 (Oxford University Press, New York) p. 306.
Page 7 of 7