2340 MGMT Section W01
Business Statistics I
Instructor: E. Mark Leany
contact via Blackboard
online.uen.org
alternately:
[e-mail]
Linear Regression and Correlation
Chapter 13
444
GOALS
1.
2.
3.
4.
5.
Understand and interpret the terms dependent and
independent variable.
Calculate and interpret the coefficient of correlation, the
coefficient of determination, and the standard error of
estimate.
Conduct a test of hypothesis to determine whether the
coefficient of correlation in the population is zero.
Calculate the least squares regression line.
Construct and interpret confidence and prediction intervals
for the dependent variable.
444
Regression Analysis - Introduction
Chapter 4 - idea of showing the relationship between
two variables with a scatter diagram was introduced.
Example: As the age of the buyer increased, the amount
spent for the vehicle also increased.
Correlation: Numerical measures to express the
strength of relationship between two variables are
developed.
Regression: An equation is used to express the
relationship between variables, allowing us to estimate
one variable on the basis of another.
456
Regression Analysis - Examples
1.
Is there a relationship between the amount Healthtex
spends per month on advertising and its sales in the
month?
2.
Can we base an estimate of the cost to heat a home in
January on the number of square feet in the home?
3.
Is there a relationship between the miles per gallon
achieved by large pickup trucks and the size of the
engine?
4.
Is there a relationship between the number of hours
that students studied for an exam and the score
earned?
456
Correlation Analysis
Correlation Analysis is the study of the relationship between
variables. It is also defined as group of techniques to measure
the association between two variables.
Scatter Diagram is a chart that portrays the relationship
between the two variables. It is the usual first step in
correlations analysis
The Dependent Variable is the variable being predicted or
estimated - scaled on the Y axis
The Independent Variable provides the basis for estimation. It is
the predictor variable - scaled on the X axis
456
Scatter Diagram Example
The sales manager of Copier Sales of America, which has a large sales force
throughout the United States and Canada, wants to determine whether
there is a relationship between the number of sales calls made in a month
and the number of copiers sold that month. The manager selects a random
sample of 10 representatives and determines the number of sales calls
each representative made last month and the number of copiers sold.
456
The Coefficient of Correlation, r
The Coefficient of Correlation (r) is a measure of the strength
of the relationship between two variables.
r is used to estimate (rho)
It shows the direction and strength of the linear relationship
between two interval or ratio-scale variables
It can range from -1.00 to +1.00.
Values of -1.00 or +1.00 indicate perfect and strong
correlation.
Values close to 0.0 indicate weak correlation.
Negative values indicate an inverse relationship and
positive values indicate a direct relationship.
457
Correlation Coefficient - Interpretation
459
Scatter Plot and the Related
Coefficient of Correlation (Ex. 1)
5
4
r = 1.00
3
2
1
0
0
1
2
3
4
5
Scatter Plot and the Related
Coefficient of Correlation (Ex. 2)
5
4
r = -1.00
3
2
1
0
0
1
2
3
4
5
Scatter Plot and the Related
Coefficient of Correlation (Ex. 3)
5
4
r = 0.00
3
2
1
0
0
1
2
3
4
5
Scatter Plot and the Related
Coefficient of Correlation (Ex. 4)
5
4
r = 0.80
3
2
1
0
0
1
2
3
4
5
Correlation Coefficient - Example
Using the Copier Sales of America data which
a scatterplot is shown below, compute the
correlation coefficient and coefficient of
determination.
Using the formula:
460
Correlation Coefficient - Example
(The Full Calculation)
Sales Representative
Tom Keller
Jeff Hall
Brian Virost
Greg Fish
Susan Welch
Carlos Ramirez
Rich Niles
Mike Kiel
Mark Reynolds
Soni Jones
AVERAGE =
STANDARD DEV. =
r
Calls, X
20
40
20
30
10
10
20
20
20
30
22
9.1894
X X )(Y Y )
(n 1) s x s y
Sales, Y
30
60
40
60
30
40
40
50
30
70
45
14.3372
X - Xbar
Y - Ybar
-2
-15
18
15
-2
-5
8
15
-12
-15
-12
-5
-2
-5
-2
5
-2
-15
8
25
(X - Xbar)(Y - Ybar) =
r=
(X - Xbar)(Y - Ybar)
30
270
10
120
180
60
10
-10
30
200
900
0.759014109
900
(10 1)(9.189)(14,337)
460
Correlation Coefficient - Example
(The Shortcut)
Sales Representative
Tom Keller
Jeff Hall
Brian Virost
Greg Fish
Susan Welch
Carlos Ramirez
Rich Niles
Mike Kiel
Mark Reynolds
Soni Jones
Calls, X
20
40
20
30
10
10
20
20
20
30
=CORREL(B2:B11,C2:C11)
0.759014109
Sales, Y
30
60
40
60
30
40
40
50
30
70
0.759
Correlation Coefficient - Example
How do we interpret a correlation of 0.759?
First, it is positive, so we see there is a direct relationship
between the number of sales calls and the number of copiers
sold. The value of 0.759 is fairly close to 1.00, so we
conclude that the association is strong.
However, does this mean that more sales calls cause more
sales?
No, we have not demonstrated cause and effect here, only
that the two variablessales calls and copiers soldare
related.
460
LYING with Statistics
At certain times of the day, there are
more police officers on I-15
During those times, they give out more
speeding ticket
The correlation is significant
Conclusion: Police officers cause
speeding
Coefficient of Determination
The coefficient of determination (r2) is the proportion of
the total variation in the dependent variable (Y) that is
explained or accounted for by the variation in the
independent variable (X). It is the square of the coefficient
of correlation.
It ranges from 0 to 1.
It does not give any information on the direction of
the relationship between the variables.
462
Coefficient of Determination (r2) Copier Sales
Example
In the copier example, r = 0.576
The coefficient of determination, r2 ,is 0.576,
found by (0.759)2
This is a proportion or a percent; we can say that
57.6 percent of the variation in the number of
copiers sold is explained, or accounted for, by the
variation in the number of sales calls.
462
Testing the Significance (IS IT) of
the Correlation Coefficient
H0:
H1:
H0:
H1:
465
= 0 (the correlation in the population is 0)
0 (the correlation in the population is not 0)
Reject H0 if:
t > t /2,n-2 or t < -t /2,n-2
> 0 (the correlation in the population is positive)
<= 0 (the correlation in the population is not positive)
Reject H0 if:
t > t ,n-2
Testing the Significance of
the Correlation Coefficient - Question
At the 0.05 level, is there a significant correlation between
the number of sales calls made and the number of
copiers sold?
(Remember that n = 10)
465
Testing the Significance of
the Correlation Coefficient - 5 Steps
(1) H 0 :
( 2)
(3) t
0 vs. H A :
0.05
rn2
1 r2
(4) Reject H 0 if t
(5) t
0
2.306 OR t - 2.306
0.759 10 2
1 0.759
2
3.2973
2.306 so REJECT H 0
465
Testing the of
the Significance Correlation Coefficient Copier Sales Example
H0:
H1:
465
= 0 (the correlation in the population is 0)
0 (the correlation in the population is not 0)
Reject H0 if:
t > t /2,n-2 or t < -t /2,n-2
t > t0.025,8 or t < -t0.025,8
t > 2.306 or t < -2.306
Testing the Significance of
the Correlation Coefficient Copier Sales Example
Computing t, we get
The computed t (3.297) is within the rejection region, therefore, we will reject H0. This
means the correlation in the population is not zero. From a practical standpoint, it
indicates to the sales manager that there is correlation with respect to the number of
sales calls made and the number of copiers sold in the population of salespeople.
465
Linear Regression Model - Words
467
Linear Regression Model - Graphics
y = dependent
a
s
alue
v
cted
i
pred
b
1
x = independent
Regression Analysis
In regression analysis we use the independent variable
(X) to estimate the dependent variable (Y).
The relationship between the variables is linear.
Both variables must be at least interval scale.
The least squares criterion is used to determine the
equation.
REGRESSION EQUATION An equation that expresses the linear
relationship between two variables.
LEAST SQUARES PRINCIPLE Determining a regression equation
by minimizing the sum of the squares of the vertical distances
between the actual Y values and the predicted values of Y.
467
Regression Analysis Least Squares Principle
The least squares principle is used to obtain a and b.
468
Regression Analysis Least Squares Principle
This calculates which line would give the smallest sum of the
SQUARED distances from the trendline
Note that, since it is not just smallest distances, the "outliers" have
more of an effect on the line
50
40
Actual Line
30
"Eyeball" Line
20
10
0
0
5
10
Computing the Slope of the Line and the Yintercept
b
a
469
n(
( X )( Y )
XY)
2
( X )2
n( X )
Y
X
b
n
n
Regression Equation - Example
Recall the example involving Copier Sales of America. The sales manager
gathered information on the number of sales calls made and the number
of copiers sold for a random sample of 10 sales representatives. Use
the least squares method to determine a linear equation to express the
relationship between the two variables.
What is the expected number of copiers sold by a representative who
made 20 calls?
Average sales
of 20 calls =
38
(NOT the answer)
469
Finding the Regression Equation - Example
Step 1 Find the slope (b) of the line
Step 2 Find the y-intercept (a)
The regression equation is :
^
Y
^
Y
^
Y
^
470
Y
a bX
18.9476 1.1842 X
18.9476 1.1842(20)
42.6316
NOT 38 from earlier
Plotting the Estimated and the Actual Ys
470
Computing the Estimates of Y
Using the regression equation, substitute the value of
each X to solve for the estimated sales
Tom Keller
^
Y
^
Y
^
Y
Soni Jones
^
18.9476 1.1842 X
Y
^
18.9476 1.1842(20)
Y
42.6316
Y
^
18.9476 1.1842 X
18.9476 1.1842(30)
54.4736
471
The Standard Error of Estimate
The standard error of estimate measures the
scatter, or dispersion, of the observed values
around the line of regression
Formulas used to compute the standard error:
^
s y. x
s y. x
475
(Y Y ) 2
n2
Y 2 a Y b XY
n2
Standard Error of the Estimate - Example
Recall the example involving
Copier Sales of America. The
sales manager determined
the least squares regression
equation is given below.
Determine the standard error of
estimate as a measure of
how well the values fit the
regression line.
^
Y
^
18 . 9476
1 . 1842 X
s y.x
(Y Y ) 2
n2
784 .211
10 2
9.901
476
Standard Error of the Estimate - Excel
477
Assumptions Underlying Linear
Regression
For each value of X, there is a group of Y values, and these
Y values are normally distributed. The means of these normal
distributions of Y values all lie on the straight line of regression.
The standard deviations of these normal distributions are equal.
The Y values are statistically independent. This means that in
the selection of a sample, the Y values chosen for a particular X
value do not depend on the Y values for any other X values.
477
Confidence Interval and Prediction
Interval Estimates of Y
A confidence interval reports the mean value of Y
for a given X.
A prediction interval reports the range of values
of Y for a particular value of X.
479
Confidence Interval Estimate - Example
We return to the Copier Sales of America
illustration. Determine a 95 percent confidence
interval for all sales representatives who make
25 calls.
480
Confidence Interval Estimate - Example
Step 1 Compute the point estimate of Y
In other words, determine the number of copiers we expect a sales
representative to sell if he or she makes 25 calls.
The regression equation is :
^
Y
^
Y
^
481
Y
18.9476 1.1842 X
18.9476 1.1842(25)
48.5526
Confidence Interval Estimate - Example
Step 2 Find the value of t
To find the t value, we need to first know the number
of degrees of freedom. In this case the degrees of
freedom is n - 2 = 10 2 = 8.
We set the confidence level at 95 percent. To find
the value of t, move down the left-hand column of
Appendix B.2 to 8 degrees of freedom, then move
across to the column with the 95 percent level of
confidence.
The value of t is 2.306.
481
Confidence Interval Estimate - Example
2
Step 3 Compute X X and
X
480
X
2
X
X
2
Confidence Interval Estimate - Example
Step 4 Use the formula above by substituting the numbers computed
in previous slides
481
Thus, the 95 percent confidence interval for the average sales of all
sales representatives who make 25 calls is from 40.9170 up to
56.1882 copiers.
Prediction Interval Estimate - Example
We return to the Copier Sales of America
illustration. Determine a 95 percent
prediction interval for Sheila Baker, a West
Coast sales representative who made 25
calls.
480
Prediction Interval Estimate - Example
Step 1 Compute the point estimate of Y
In other words, determine the number of copiers we
expect a sales representative to sell if he or she
makes 25 calls.
The regression equation is :
^
Y
^
Y
^
480
Y
18.9476 1.1842 X
18.9476 1.1842(25)
48.5526
Prediction Interval Estimate - Example
Step 2 Using the information computed
earlier in the confidence interval estimation
example, use the formula above.
481
If Sheila Baker makes 25 sales calls, the number of copiers she
will sell will be between about 24 and 73 copiers.
Confidence and Prediction Intervals
Minitab Illustration
482
GOALS
1.
2.
3.
4.
5.
444
Understand and interpret the terms dependent and
independent variable.
Calculate and interpret the coefficient of correlation, the
coefficient of determination, and the standard error of
estimate.
Conduct a test of hypothesis to determine whether the
coefficient of correlation in the population is zero.
Calculate the least squares regression line.
Construct and interpret confidence and prediction intervals
for the dependent variable.