Marketing Research

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Marketing Research

• Aaker, Kumar, Day
• Ninth Edition
• Instructors 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
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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 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
• 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
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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
Cramers 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