This document contains excerpts from Chapter 3 and Chapter 12 of the 6th edition of the textbook "Business Statistics" by Ken Black. Chapter 3 discusses measures of shape such as skewness and the coefficient of skewness. Chapter 12 introduces regression analysis and correlation, covering topics like the Pearson correlation coefficient, least squares regression, and residual analysis. Examples are provided to demonstrate calculating the correlation coefficient and estimating the regression equation to predict costs from number of passengers for an airline.

1. Copyright 2010 John Wiley & Sons, Inc. 1
Copyright 2010 John Wiley & Sons, Inc.
Business Statistics, 6th ed.
by Ken Black
Chapter 3
Describing Data
Through Statistics

2. Copyright 2010 John Wiley & Sons, Inc. 2
Measures of Shape
Symmetrical – the right half is a mirror image of the
left half
Skewness – shows that the distribution lacks
symmetry; used to denote the data is sparse at one
end, and piled at the other end
Absence of symmetry
Extreme values in one side of a distribution

3. Copyright 2010 John Wiley & Sons, Inc. 3
Coefficient of Skewness
( )

 dM
Sk
−
=
3
Coefficient of Skewness (Sk) - compares the mean
and median in light of the magnitude to the standard
deviation; Md is the median; Sk is coefficient of
skewness; σ is the Std Dev

4. Copyright 2010 John Wiley & Sons, Inc. 4
Coefficient of Skewness
Summary measure for skewness
If Sk < 0, the distribution is negatively skewed
(skewed to the left).
If Sk = 0, the distribution is symmetric (not skewed).
If Sk > 0, the distribution is positively skewed (skewed
to the right).
( )

 d
k
M
S
−
=
3

5. Copyright 2010 John Wiley & Sons, Inc. 5
Copyright 2010 John Wiley & Sons, Inc.
Business Statistics, 6th ed.
by Ken Black
Chapter 12
Introduction to
Regression Analysis
and Correlation

6. Copyright 2010 John Wiley & Sons, Inc. 6
Learning Objectives
Compute the equation of a simple regression line from a
sample of data, and interpret the slope and intercept of
the equation.
Understand the usefulness of residual analysis in testing
the assumptions underlying regression analysis and in
examining the fit of the regression line to the data.
Compute a standard error of the estimate and interpret
its meaning.
Compute a coefficient of determination and interpret it.
Test hypotheses about the slope of the regression model
and interpret the results.
Estimate values of Y using the regression model.

7. Copyright 2010 John Wiley & Sons, Inc. 7
Regression and Correlation
Regression analysis is the process of constructing a
mathematical model or function that can be used to
predict or determine one variable by another variable.
Correlation is a measure of the degree of relatedness
of two variables.

8. Copyright 2010 John Wiley & Sons, Inc. 8
( )( )
( )( )
( ) ( )
( )( )
( ) ( )
r
SSXY
SSX SSY
X X Y Y
XY
X Y
n
n n
X X Y Y
X
X
Y
Y
=
=
− −
=
−
−








−









− −

 
 
2 2
2
2
2
2
−  1 1r
Pearson Product-Moment
Correlation Coefficient

9. Copyright 2010 John Wiley & Sons, Inc. 9
Degrees of Correlation
Correlation is a measure of the degree of relatedness
of variables
Coefficient of Correlation (r) - applicable only if both
variables being analyzed have at least an interval
level of data

10. Copyright 2010 John Wiley & Sons, Inc. 10
Degrees of Correlation
The term (r) is a measure of the linear correlation
of two variables
The number ranges from -1 to 0 to +1
Closer to +1, the higher the correlation between the
dependent and the independent variables
See the formula for Pearson Product Moment correlation
coefficient –
See slide 3-82 for the formula

11. Copyright 2010 John Wiley & Sons, Inc. 11
r < 0 r > 0
r = 0
Three Degrees of Correlation

12. Copyright 2010 John Wiley & Sons, Inc. 12
Day
Interest
X
Futures
Index
Y
1 7.43 221 55.205 48,841 1,642.03
2 7.48 222 55.950 49,284 1,660.56
3 8.00 226 64.000 51,076 1,808.00
4 7.75 225 60.063 50,625 1,743.75
5 7.60 224 57.760 50,176 1,702.40
6 7.63 223 58.217 49,729 1,701.49
7 7.68 223 58.982 49,729 1,712.64
8 7.67 226 58.829 51,076 1,733.42
9 7.59 226 57.608 51,076 1,715.34
10 8.07 235 65.125 55,225 1,896.45
11 8.03 233 64.481 54,289 1,870.99
12 8.00 241 64.000 58,081 1,928.00
Summations 92.93 2,725 720.220 619,207 21,115.07
X2 Y2 XY
Computation of r for
the Economics Example (Part 1)

13. Copyright 2010 John Wiley & Sons, Inc. 13
( )( )
( ) ( )
( )
( )( )
( )
( ) ( )
( )
r
X
X
Y
Y
XY
X Y
n
n n
=
−
−








−








=
−
−








−








=

 
 
2
2
2
2
2 2
21115 07
92 93 2725
12
720 22
12
619 207
12
92 93 2725
815
, .
.
. ,
.
.
Computation of r
Economics Example (Part 2)

14. Copyright 2010 John Wiley & Sons, Inc. 14
Computation of r
Economics Example (Part 2)
Means that 81.5% of the dependent variables
are explained by the independent variables.
Is 81.5% high or low?

15. Copyright 2010 John Wiley & Sons, Inc. 15
Bivariate (two variables) linear regression -- the most
elementary regression model
dependent variable, the variable to be predicted, usually
called Y
independent variable, the predictor or explanatory variable,
usually called X
Nonlinear relationships and regression models with
more than one independent variable can be explored
by using multiple regression models
Simple Regression Analysis

16. Copyright 2010 John Wiley & Sons, Inc. 16
Deterministic Regression Model
Y = 0 + 1X
Probabilistic Regression Model
Y = 0 + 1X + 
0 and 1 are population parameters
0 and 1 are estimated by sample statistics b0 and b1
Regression Models

17. Copyright 2010 John Wiley & Sons, Inc. 17
YY
where
XY
b
b
bb
ofvaluepredictedthe=ˆ
slopesamplethe=
interceptsamplethe=:
ˆ
1
0
10
+=
Equation of the Simple Regression Line

18. Copyright 2010 John Wiley & Sons, Inc. 18
Least Squares Analysis
Least squares analysis is a process whereby a
regression model is developed by producing the
minimum sum of the squared error values
The vertical distance from each point to the line is
the error of the prediction.
The least squares regression line is the regression line
that results in the smallest sum of errors squared.

19. Copyright 2010 John Wiley & Sons, Inc. 19
( )( )
( )
( )( )
1 2 2 2
2
2b
X X X X X X
X X Y Y XY nXY
n
XY
X Y
n
n
=
− −
=
−
−
=
−
−


−



 

0 1 1b b bY X
Y
n
X
n
= − = −
 
Least Squares Analysis

20. Copyright 2010 John Wiley & Sons, Inc. 20
( )( )
( )( )
( )
SS X X Y Y XY
X Y
n
SS
n
SS
SS
XY
XX
XY
XX
X X X X
b
= − − = −
= = −

=
 
 
− 
2 2
2
1
0 1 1b b bY X
Y
n
X
n
= − = −
 
Least Squares Analysis

21. Copyright 2010 John Wiley & Sons, Inc. 21
Number of
Passengers Cost ($1,000)
X Y X2
XY
61 4.28 3,721 261.08
63 4.08 3,969 257.04
67 4.42 4,489 296.14
69 4.17 4,761 287.73
70 4.48 4,900 313.60
74 4.30 5,476 318.20
76 4.82 5,776 366.32
81 4.70 6,561 380.70
86 5.11 7,396 439.46
91 5.13 8,281 466.83
95 5.64 9,025 535.80
97 5.56 9,409 539.32
X = 930 Y = 56.69  2
X = 73,764 XY = 4,462.22
Solving for b1 and b0 of the Regression
Line: Airline Cost Example (Part 1)

22. Copyright 2010 John Wiley & Sons, Inc. 22
Solving for b1 and b0 of the Regression
Line: Airline Cost Example (Part 2)
745.68
12
)69.56)(930(
22.462,4 =−=−=  
n
YX
XYSSXY
1689
12
)930(
764,73
)( 22
2
=−=−= 

n
X
XSSXX
0407.
1689
745.68
1 ===
XX
XY
SS
SS
b
57.1
12
930
)0407(.
12
69.56
10 =−=−=

n
X
b
n
Y
b
XY 0407.57.1ˆ +=

23. Copyright 2010 John Wiley & Sons, Inc. 23
SUMMARY OUTPUT
Regression Statistics
Multiple R 0.94820033
R Square 0.89908386
Adjusted R Square 0.88899225
Standard Error 0.17721746
Observations 12
ANOVA
df SS MS F Significance F
Regression 1 2.79803 2.79803 89.092179 2.7E-06
Residual 10 0.31406 0.03141
Total 11 3.11209
Coefficients Standard Error t Stat P-value
Intercept 1.56979278 0.33808 4.64322 0.0009175
Number of Passengers 0.0407016 0.00431 9.43887 2.692E-06
Airline Cost: Excel Summary Output

24. Copyright 2010 John Wiley & Sons, Inc. 24
Airline Cost: MINITAB Summary Output

25. Copyright 2010 John Wiley & Sons, Inc. 25
Residual Analysis: Airline Cost Example
Number of Predicted
Passengers Cost ($1,000) Value Residual
X Y Yˆ YY ˆ−
61 4.28 4.053 .227
63 4.08 4.134 -.054
67 4.42 4.297 .123
69 4.17 4.378 -.208
70 4.48 4.419 .061
74 4.30 4.582 -.282
76 4.82 4.663 .157
81 4.70 4.867 -.167
86 5.11 5.070 .040
91 5.13 5.274 -.144
95 5.64 5.436 .204
97 5.56 5.518 .042
 −=− 001.)ˆ( YY

26. Copyright 2010 John Wiley & Sons, Inc. 26
Compute the residuals for Demonstration Problem
12.1 in which a regression model was developed to
predict the number of full-time equivalent workers
(FTEs) by the number of beds in a hospital. Analyze
the residuals by using MINITAB graphic diagnostics.
Demonstration Problem 14.2

27. Copyright 2010 John Wiley & Sons, Inc. 27
Demonstration Problem 14.2 – MINITAB
Computations for Residuals

28. Copyright 2010 John Wiley & Sons, Inc. 28
Spearman’s Rank Correlation - Analyze the degree
of association of two variables
Applicable to ordinal level data (ranks)
2
2
6
1
( 1)
: = number of pairs being correlated
= the difference in the ranks of each pair
s
n n
where n
d
d
r = −
−

Spearman’s Rank Correlation

29. Copyright 2010 John Wiley & Sons, Inc. 29
Listed below are the average prices in dollars per 100
pounds for choice spring lambs and choice heifers
over a 10-year period. The data were published by
the National Agricultural Statistics Service of the U.S.
Department of Agriculture.
Suppose the researcher want to determine the
strength of association of the prices between these
two commodities by using Spearman’s rank
correlation.
Spearman’s Rank Correlation

30. Copyright 2010 John Wiley & Sons, Inc. 30
Spearman’s Rank Correlation for
Heifer and Lamb Prices

31. Copyright 2010 John Wiley & Sons, Inc. 31
345.0
)110(10
)108(6
1
)1(
6
1 22
2
=
−
−=
−
−= 
nn
d
sr
Spearman’s Rank Correlation for
Heifer and Lamb Prices

32. Copyright 2010 John Wiley & Sons, Inc. 32
The lamb prices are ranked and the heifer prices are
ranked.
The difference in ranks is computed for each year.
The differences are squared and summed, producing
∑d2 = 108.
The number of pairs, n, is 10.
The value of rs = 0.345 indicates that there is a very
modest if not poor positive correlation between lamb
and heifer prices.
Spearman’s Rank Correlation for
Heifer and Lamb Prices