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1 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 3 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 4 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 5 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 6 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 http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 7 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 8 Relationship between Type I & Type II Errors http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 9 Relationship between Type I & Type II Errors (Contd.) http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 10 Relationship between Type I & Type II Errors (Contd.) http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 11 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 12 Hypothesis Testing & Associated Statistical Tests http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 13 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 http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 14 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 15 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 16 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 http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 17 The Concept of Statistical Independence If n is equal to 200 and Ei is the number of outcomes expected in cell i, http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 18 Chi-Square As a Test of Independence http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 19 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. http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 20 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 21 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 http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 22 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 23 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 http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 24 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 http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 25 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 http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 26 Measures of Association for Nominal Variables • Measures based on Chi-Square Phi-squared Cramer’s V http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 27 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