Statistics for business decision making and analysis robert stine and foster chapter 06

Chapter 6

Association between
Quantitative Variables

Copyright © 2011 Pearson Education, Inc.


6.1 Scatterplots
Is household natural gas consumption
associated with climate?


Annual household natural gas consumption
measured in thousands of cubic feet (MCF)



Climate as measured by the National
Weather Service using heating degree days
(HDD)
3 of 30
Copyright © 2011 Pearson Education, Inc.


6.1 Scatterplots
Association between Numerical Variables


A graph displaying pairs of values as points

on a two-dimensional grid



The explanatory variable is placed on the xaxis



The response variable is placed on the y-axis
4 of 30
Copyright © 2011 Pearson Education, Inc.


6.1 Scatterplots
Scatterplot of Natural Gas Consumption (y)
versus Heating Degree-Days (x)

5 of 30
Copyright © 2011 Pearson Education, Inc.


6.2 Association in Scatterplots
Visual Test for Association


Compare the original scatterplot to others
that randomly match the coordinates



If you can pick the original out as having a
pattern, then there is an association

6 of 30
Copyright © 2011 Pearson Education, Inc.


6.2 Association in Scatterplots
Describing Association
1.
2.
3.

4.

Direction. Does it trend up or down?
Curvature. Is the pattern linear or curved?
Variation. Are the points tightly clustered
around the trend?
Outliers. Is there something unexpected?

7 of 30
Copyright © 2011 Pearson Education, Inc.


6.2 Association in Scatterplots
Gas Consumption vs. Heating Degree Days
1.
2.
3.

4.

Direction: Positive.
Curvature: Linear.
Variation: Considerable scatter.
Outliers: None apparent.

8 of 30
Copyright © 2011 Pearson Education, Inc.


6.3 Measuring Association
Covariance
x1 − x ) ( y1 − y ) + ( x2 − x ) ( y 2 − y ) + L ( x n − x ) ( y n − y )
(
cov( x, y ) =
n −1





A measure that quantifies the linear
association
Depends on units of measurement and is
therefore difficult to interpret
9 of 30
Copyright © 2011 Pearson Education, Inc.

6.3 Measuring Association
Correlation (r)
cov( x, y )
corr( x, y ) =
S x Sy





Standardized measure of the strength of
the linear association (has no units)
Always between -1 and +1
Easy to interpret
10 of 30
Copyright © 2011 Pearson Education, Inc.


6.3 Measuring Association
Gas Consumption and Heating Degree Days
Cov (HDD, Gas) = 63,357 HDD X MCF
Corr (HDD, Gas) = 0.55
The association is positive and moderate.

11 of 30
Copyright © 2011 Pearson Education, Inc.


6.3 Measuring Association
Scatterplot for r = 1

12 of 30
Copyright © 2011 Pearson Education, Inc.


6.3 Measuring Association
Scatterplot for r = -0.95

13 of 30
Copyright © 2011 Pearson Education, Inc.


6.3 Measuring Association
Scatterplot for r = 0.75

14 of 30
Copyright © 2011 Pearson Education, Inc.


6.3 Measuring Association
Scatterplot for r = -0.50

15 of 30
Copyright © 2011 Pearson Education, Inc.


6.3 Measuring Association
Scatterplot for r = 0

16 of 30

Copyright © 2011 Pearson Education, Inc.


6.3 Measuring Association
Correlation Matrix
A table showing all of the correlations among
a set of numerical variables.

17 of 30
Copyright © 2011 Pearson Education, Inc.


6.4 Summarizing Association with a Line
Expressed using z-scores
zˆ = rzx

Slope-Intercept Form
yˆ = a + bx
a = y − bx and b = rsy / s x

18 of 30
Copyright © 2011 Pearson Education, Inc.


6.4 Summarizing Association with a Line
Line Relating Gas Consumption (y) to Heating
Degree Days (x)

yˆ = 42.6 + 0.0126 x

19 of 30
Copyright © 2011 Pearson Education, Inc.


6.4 Summarizing Association with a Line
Lines and Prediction
 Use the correlation line to customize an ad for estimated savings from insulation
based on climate.
 For a home in a cold climate (HDD = 8,800), the predicted gas consumption is 154
MCF.
 At $10 / MCF, the predicted cost is $1,540.
 Assuming that insulation saves 30% on gas bill, estimated savings is $462.

20 of 30
Copyright © 2011 Pearson Education, Inc.


6.5 Spurious Correlation
Lurking Variables


Scatterplots and correlation reveal association, not causation



Spurious correlations result from underlying lurking variables

21 of 30
Copyright © 2011 Pearson Education, Inc.

6.5 Spurious Correlation
Checklist: Covariance and Correlation





Numerical variables
No obvious lurking variables
Linear
Outliers

22 of 30
Copyright © 2011 Pearson Education, Inc.


4M Example 6.1:
LOCATING A NEW STORE
Motivation
Is it better to locate a new retail outlet far
from competing stores?

23 of 30
Copyright © 2011 Pearson Education, Inc.


4M Example 6.1:
LOCATING A NEW STORE
Method

Is there an association between sales at the
retail outlets and distance to nearest
competitor? For 55 stores in the chain,
data are gathered for total sales in the
prior year and distance in miles from the
nearest competitor.
24 of 30
Copyright © 2011 Pearson Education, Inc.


4M Example 6.1:
LOCATING A NEW STORE
Mechanics

25 of 30
Copyright © 2011 Pearson Education, Inc.