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Marketing Research
Aaker, Kumar, Day
Ninth Edition
Instructor’s Presentation Slides
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Chapter Seventeen
Hypothesis Testing:
Basic Concepts and Tests of
Association
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Hypothesis Testing: Basic Concepts
• Assumption (hypothesis) made about a population
parameter (not sample parameter)
• Purpose of Hypothesis Testing
▫ To make a judgment about the difference between two sample
statistics or between sample statistic and a hypothesized
population parameter
• Evidence has to be evaluated statistically before arriving
at a conclusion regarding the hypothesis.
▫ Depends on whether information generated from the sample is
with fewer or larger observations
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Hypothesis Testing
• The null hypothesis (Ho) is tested against the
alternative hypothesis (Ha).
• At least the null hypothesis is stated.
• Decide upon the criteria to be used in making
the decision whether to “reject” or "not reject"
the null hypothesis.
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Hypothesis Testing Process
Problem Definition
Clearly state the null and alternative
hypotheses
Choose the relevant test and the
appropriate probability distribution
Determine the
significance level
Compute relevant
test statistic
Choose the critical value
Determine the
degrees of freedom
Decide if one-or twotailed test
Compare test statistic & critical value
Does
the test statistic
fall in the critical
region?
Do not reject null
Reject null
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Basic Concepts of Hypothesis Testing
Three Criteria Used To Decide Critical Value
(Whether To Accept or Reject Null Hypothesis):
• Significance Level
• Degrees of Freedom
• One or Two Tailed Test
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Significance Level
• Indicates the percentage of sample means that is outside the cut-off limits
(critical value)
• The higher the significance level () used for testing a hypothesis, the higher
the probability of rejecting a null hypothesis when it is true (Type I error)
• Accepting a null hypothesis when it is false is called a Type II error and its
probability is ()
• When choosing a level of significance, there is an inherent tradeoff between
these two types of errors
• A good test of hypothesis should reject a null hypothesis when it is false
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Relationship between Type I & Type II Errors
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Relationship between Type I &
Type II Errors (Contd.)
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Relationship between Type I &
Type II Errors (Contd.)
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Choosing The Critical Value
• Power of hypothesis test
▫ (1 - ) should be as high as possible
• Degrees of Freedom
▫ The number or bits of "free" or unconstrained data used
in calculating a sample statistic or test statistic
▫ A sample mean (X) has `n' degree of freedom
▫ A sample variance (s2) has (n-1) degrees of freedom
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Hypothesis Testing &
Associated Statistical Tests
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One or Two-tail Test
• One-tailed Hypothesis Test
▫ Determines whether a particular population parameter
is larger or smaller than some predefined value
▫ Uses one critical value of test statistic
• Two-tailed Hypothesis Test
▫ Determines the likelihood that a population parameter
is within certain upper and lower bounds
▫ May use one or two critical values
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Basic Concepts of Hypothesis Testing (Contd.)
• Select the appropriate probability distribution
based on two criteria
▫ Size of the sample
▫ Whether the population standard deviation is known or
not
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Hypothesis Testing
Data Analysis Outcome
Accept Null Hypothesis Reject Null Hypothesis
Null Hypothesis is True
Correct Decision
Type I Error
Null Hypothesis is False
Type II Error
Correct Decision
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Cross-tabulation and Chi Square
In Marketing Applications, Chi-square Statistic is
used as:
• Test of Independence
▫ Are there associations between two or more variables in a study?
• Test of Goodness of Fit
▫ Is there a significant difference between an observed frequency
distribution and a theoretical frequency distribution?
• Statistical Independence
▫ Two variables are statistically independent if a knowledge of one
would offer no information as to the identity of the other
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The Concept of Statistical Independence
If n is equal to 200 and Ei is the number of outcomes expected in cell i,
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Chi-Square As a Test of Independence
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Chi-Square As a Test of
Independence (Contd.)
Null Hypothesis Ho
• Two (nominally scaled) variables are statistically
independent
Alternative Hypothesis Ha
• The two variables are not independent
Use Chi-square distribution to test.
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Chi-square Distribution
• A probability distribution
• Total area under the curve is 1.0
• A different chi-square distribution is associated with different degrees
of freedom
Cutoff points of the chi-square distribution function
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Chi-square Distribution (Contd.)
Degrees of Freedom
• Number of degrees of freedom, v = (r - 1) * (c - 1)
r = number of rows in contingency table
c = number of columns
• Mean of chi-squared distribution = Degree of freedom (v)
• Variance = 2v
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Chi-square Statistic (2)
• Measures of the difference between the actual numbers observed in cell i (Oi), and
number expected (Ei) under assumption of statistical independence if the null
hypothesis were true
(Oi  Ei ) 2
 
i 1
Ei
2
n
With (r-1)*(c-1) degrees of freedom
Oi = observed number in cell i
Ei = number in cell i expected under independence
r = number of rows
c = number of columns
•
Expected frequency in each cell, Ei = pc * pr * n
Where
pc and pr are proportions for independent variables
n is the total number of observations
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Chi-square Step-by-Step
Formulate
Hypothesis
Calculate row &
column totals
Calculate row &
column
proportions
Calculate
expected
frequencies (Ei)
Make decision
regarding Nullhypothesis
Obtain critical
value from
table
Calculate
degrees of
freedom
Calculate χ2
statistic
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Strength of Association
• Measured by contingency coefficient
• 0 = no association (i.e., Variables are statistically
independent)
• Maximum value depends on the size of table
• Compare only tables of same size
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Limitations of Chi-square as an
Association Measure
• It is basically proportional to sample size
 Difficult to interpret in absolute sense and compare
cross-tabs of unequal size
• It has no upper bound
 Difficult to obtain a feel for its value
 Does not indicate how two variables are related
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Measures of Association for Nominal Variables
• Measures based on Chi-Square
Phi-squared
Cramer’s V
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Chi-square Goodness of Fit
• Used to investigate how well the observed pattern fits the
expected pattern
• Researcher may determine whether population
distribution corresponds to either a normal, Poisson or
binomial distribution
To determine degrees of freedom:
• Employ (k-1) rule
• Subtract an additional degree of freedom for each population
parameter that has to be estimated from the sample data
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Marketing Research 10th Edition