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Marketing Research
Aaker, Kumar, Day
Ninth Edition
Instructor’s Presentation Slides
                                                                       2



        Chapter Seventeen

                          Hypothesis Testing:
                      Basic Concepts and Tests of
                             Association




http://www.drvkumar.com/mr10/                 Marketing Research 10th Edition
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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
http://www.drvkumar.com/mr10/                                     Marketing Research 10th Edition
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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.


http://www.drvkumar.com/mr10/                          Marketing Research 10th Edition
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       Hypothesis Testing Process
                                         Problem Definition

                                Clearly state the null and alternative
                                             hypotheses
                                                                            Determine the
                                  Choose the relevant test and the        degrees of freedom
                                 appropriate probability distribution
                                                                          Decide if one-or two-
    Determine the
                                                                               tailed test
  significance level                  Choose the critical value

  Compute relevant              Compare test statistic & critical value
    test statistic
                                                   Does
                                           the test statistic
                                           fall in the critical              Do not reject null
                                                 region?



                                             Reject null
http://www.drvkumar.com/mr10/                                                  Marketing Research 10th Edition
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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

http://www.drvkumar.com/mr10/                                               Marketing Research 10th Edition
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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



http://www.drvkumar.com/mr10/                             Marketing Research 10th Edition
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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




http://www.drvkumar.com/mr10/                             Marketing Research 10th Edition
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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




http://www.drvkumar.com/mr10/                                            Marketing Research 10th Edition
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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




http://www.drvkumar.com/mr10/                                                           Marketing Research 10th Edition
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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
                                                    n
                                             
                                              2
                                               i 1     Ei
                  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



http://www.drvkumar.com/mr10/                                                         Marketing Research 10th Edition
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        Chi-square Step-by-Step

                                                  Calculate row &          Calculate
             Formulate          Calculate row &
                                                      column                expected
             Hypothesis          column totals
                                                   proportions          frequencies (Ei)




           Make decision        Obtain critical     Calculate
                                                                           Calculate χ2
          regarding Null-        value from         degrees of
                                                                            statistic
            hypothesis              table            freedom




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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




http://www.drvkumar.com/mr10/                                            Marketing Research 10th Edition


				
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