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Types of errors in testing hypothesis.

There are two types of error in testing of hypothesis: Type I & Type II. Which error is more
dangerous? Discuss with examples. (Refer Chapter-14/ Module-12)
Embed course material concepts, principles, and theories (which require supporting citations), along
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posts as well.
Chapter 15
STAGE 4: MEASURES OF ASSOCIATION
Copyright © 2019 by The McGraw-Hill Companies, Inc. All rights reserved.
15-1
Learning Objectives
Understand . . .
ï‚— How correlation analysis may be applied to study relationships
between two or more variables.
ï‚— The uses, requirements, and interpretation of the product
moment correlation coefficient.
ï‚— How predictions are made with regression analysis using the
method of least squares to minimize errors in drawing a line of
best fit.
ï‚— How to test regression models for linearity and whether the
equation is effective in fitting the data.
ï‚— The nonparametric measures of association and the
alternatives they offer when key assumptions and
requirements for parametric techniques cannot be met.
Copyright © 2019 by The McGraw-Hill Companies, Inc. All rights reserved.
15-2
Measures of Association:
Interval/Ratio Data
Pearson correlation
coefficient
For continuous linearly related
variables
Correlation ratio (eta)
For nonlinear data or relating a
main effect to a continuous
dependent variable
Biserial
One continuous and one
dichotomous variable with an
underlying normal distribution
Partial correlation
Three variables; relating two with
the third’s effect taken out
Multiple correlation
Three variables; relating one
variable with two others
Bivariate linear regression
Predicting one variable from
another’s scores
Copyright © 2019 by The McGraw-Hill Companies, Inc. All rights reserved.
15-3
Measures of Association:
Ordinal Data
Gamma
Based on concordant-discordant
pairs; proportional reduction in
error (PRE) interpretation
Kendall’s tau b
P-Q based; adjustment for tied
ranks
Kendall’s tau c
P-Q based; adjustment for table
dimensions
Somers’s d
P-Q based; asymmetrical
extension of gamma
Spearman’s rho
Product moment correlation for
ranked data
Copyright © 2019 by The McGraw-Hill Companies, Inc. All rights reserved.
15-4
Measures of Association:
Nominal Data
Phi
Chi-square based for 2*2 tables
Cramer’s V
CS based; adjustment when one table
dimension >2
Contingency coefficient C
CS based; flexible data and distribution
assumptions
Lambda
PRE based interpretation
Goodman & Kruskal’s tau
PRE based with table marginals
emphasis
Uncertainty coefficient
Useful for multidimensional tables
Kappa
Agreement measure
Copyright © 2019 by The McGraw-Hill Companies, Inc. All rights reserved.
15-5
Researchers Search for Insights
Burke, one of the world’s
leading research companies,
claims researchers add the
most value to a project when
they look beyond the raw
numbers to the shades of
gray…discovering what the
data really mean.
Copyright © 2019 by The McGraw-Hill Companies, Inc. All rights reserved.
15-6
Pearson’s Product Moment Correlation r
Is there a relationship between X and Y?
What is the magnitude of the relationship?
What is the direction of the relationship?
Copyright © 2019 by The McGraw-Hill Companies, Inc. All rights reserved.
15-7
Connections and Disconnections
“To truly understand consumers’ motives
and actions, you must determine
relationships between what they think
and feel and what they actually do.”
David Singleton, vp of insights
Zyman Marketing Group
Copyright © 2019 by The McGraw-Hill Companies, Inc. All rights reserved.
15-8
Scatterplots of
Relationships
Copyright © 2019 by The McGraw-Hill Companies, Inc. All rights reserved.
15-9
Scatterplots
Copyright © 2019 by The McGraw-Hill Companies, Inc. All rights reserved.
15-10
Plot of Forbes 500 Net Profits with Cash
Flow
Copyright © 2019 by The McGraw-Hill Companies, Inc. All rights reserved.
15-11
Diagram
of
Common
Variance
Copyright © 2019 by The McGraw-Hill Companies, Inc. All rights reserved.
15-12
Interpretation of Correlations
X causes Y
Y causes X
X and Y are activated by
one or more other variables
X and Y influence each
other reciprocally
Copyright © 2019 by The McGraw-Hill Companies, Inc. All rights reserved.
15-13
Artifact
Correlations
Copyright © 2019 by The McGraw-Hill Companies, Inc. All rights reserved.
15-14
Interpretation of Coefficients

A coefficient is not remarkable simply
because it is statistically significant!
It must be practically meaningful.
Copyright © 2019 by The McGraw-Hill Companies, Inc. All rights reserved.
15-15
Comparison of
Bivariate
Linear
Correlation
and
Regression
Copyright © 2019 by The McGraw-Hill Companies, Inc. All rights reserved.
15-16
Examples of Different Slopes
Copyright © 2019 by The McGraw-Hill Companies, Inc. All rights reserved.
15-17
Concept Application
X
Average Temperature (Celsius)
Y
Price per Case
(FF)
12
2,000
16
3,000
20
4,000
24
5,000
Mean =18
Mean = 3,500
Copyright © 2019 by The McGraw-Hill Companies, Inc. All rights reserved.
15-18
Plot of Wine Price by Average
Temperature
Copyright © 2019 by The McGraw-Hill Companies, Inc. All rights reserved.
15-19
Distribution of
Y for
Observation of
X
Copyright © 2019 by The McGraw-Hill Companies, Inc. All rights reserved.
15-20
Wine Price
Study
Example
Copyright © 2019 by The McGraw-Hill Companies, Inc. All rights reserved.
15-21
Least Squares
Line:
Wine Price
Study
Copyright © 2019 by The McGraw-Hill Companies, Inc. All rights reserved.
15-22
Plot of Standardized Residuals
Copyright © 2019 by The McGraw-Hill Companies, Inc. All rights reserved.
15-23
Prediction and Confidence Bands
Copyright © 2019 by The McGraw-Hill Companies, Inc. All rights reserved.
15-24
Testing Goodness of Fit
Y is completely unrelated to X
and no systematic pattern is evident
There are constant values of
Y for every value of X
The data are related but
represented by a nonlinear function
Copyright © 2019 by The McGraw-Hill Companies, Inc. All rights reserved.
15-25
Components of Variation
Copyright © 2019 by The McGraw-Hill Companies, Inc. All rights reserved.
15-26
F Ratio in Regression
Copyright © 2019 by The McGraw-Hill Companies, Inc. All rights reserved.
15-27
F Ratio in Regression
Copyright © 2019 by The McGraw-Hill Companies, Inc. All rights reserved.
15-28
Coefficient of Determination: r2
Total proportion of
variance in Y explained by X
Desired r2: 80% or more
Copyright © 2019 by The McGraw-Hill Companies, Inc. All rights reserved.
15-29
Chi-Square
Based
Measures
Copyright © 2019 by The McGraw-Hill Companies, Inc. All rights reserved.
15-30
Proportional
Reduction of
Error
Measures
Copyright © 2019 by The McGraw-Hill Companies, Inc. All rights reserved.
15-31
Statistical Alternatives for Ordinal
Measures
Copyright © 2019 by The McGraw-Hill Companies, Inc. All rights reserved.
15-32
Calculation of
Concordant (P),
Discordant (Q),
Tied (Tx,Ty), and
Total Paired
Observations:
KeyDesign
Example
Copyright © 2019 by The McGraw-Hill Companies, Inc. All rights reserved.
15-33
Calculation of
Concordant (P),
Discordant (Q),
Tied (Tx,Ty), and
Total Paired
Observations:
KeyDesign
Example
Copyright © 2019 by The McGraw-Hill Companies, Inc. All rights reserved.
15-34
Commonly Used
Measures of
Association
Copyright © 2019 by The McGraw-Hill Companies, Inc. All rights reserved.
15-35
KDL Data for Spearman’s Rho
_______ _____ Rank By_____ _____
_____
Applicant
Panel x
Psychologist y
d
d2
1
2
3
4
5
6
7
8
9
10
3.5
10.0
6.5
2.0
1.0
9.0
3.5
6.5
8.0
5.0
6.0
5.0
8.0
1.5
3.0
7.0
1.5
9.0
10.0
4.0
-2.5
5.0
-1.5
.05
-2
2.0
2.0
-2.5
-2
1.0
6.25
25.00
2.52
0.25
4.00
4.00
4.00
6.25
4.00
_1.00_
57.00
.
Copyright © 2019 by The McGraw-Hill Companies, Inc. All rights reserved.
15-36
Key Terms
1837
ï‚— Artifact correlations
ï‚— Concordant
ï‚— Bivariate correlation
ï‚— Correlation matrix
analysis
ï‚— Bivariate normal
distribution
ï‚— Chi-square-based measures
 Contingency coefficient C
 Cramer’s V
 Phi
ï‚— Coefficient of determination
(r2)
ï‚— Discordant
Copyright © 2019 by The McGraw-Hill Companies, Inc. All rights reserved.
ï‚— Error term
ï‚— Goodness of fit
ï‚— Lambda
ï‚— Linearity
ï‚— Method of least squares
15-37
Key Terms
1838
ï‚— Ordinal measures
ï‚— Regression coefficients
 Gamma
 Intercept
 Somers’s d
 Slope
 Spearman’s rho
 Residual
ï‚— Pearson correlation
ï‚— Scatterplot
coefficient
ï‚— Prediction and
confidence bands
ï‚— Proportional reduction in
error (PRE)
ï‚— Regression analysis
ï‚— Simple prediction
Copyright © 2019 by The McGraw-Hill Companies, Inc. All rights reserved.
ï‚— Tau
ï‚— Tau b
ï‚— Tau c
15-38

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