Hypothesis Testing: Basic Concepts and Tests of Association

Marketing ResearchAaker, Kumar, Leone and Day Twelfth EditionInstructor’s Presentation Slides1Chapter Seventeen2Hypothesis Testing: Basic Concepts and Tests of AssociationMarketing Research 12th Edition Hypothesis Testing: Basic ConceptsAssumption (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 parameterEvidence 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 observations3Marketing Research 12th Edition Hypothesis TestingThe 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.4Marketing Research 12th Edition 5Hypothesis Testing ProcessProblem DefinitionClearly state the null and alternative hypothesesChoose the relevant test and the appropriate probability distributionChoose the critical valueCompare test statistic & critical valueReject nullDetermine the significance levelCompute relevant test statisticDetermine the degrees of freedomDecide if one-or two-tailed testDo not reject nullDoes the test statistic fall in the critical region?Marketing Research 12th Edition YesNoBasic Concepts of Hypothesis TestingThree Criteria Used To Decide Critical Value (Whether To Accept or Reject Null Hypothesis):Significance LevelDegrees of FreedomOne or Two Tailed Test6Marketing Research 12th Edition Significance LevelIndicates 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 errorsA good test of hypothesis should reject a null hypothesis when it is false7Marketing Research 12th Edition Relationship between Type I & Type II Errors8Marketing Research 12th Edition Relationship between Type I & Type II Errors (Contd.)9Marketing Research 12th Edition Relationship between Type I & Type II Errors (Contd.)10Marketing Research 12th Edition Choosing The Critical ValuePower of hypothesis test (1 - ) should be as high as possibleDegrees of FreedomThe number or bits of "free" or unconstrained data used in calculating a sample statistic or test statisticA sample mean (X) has `n' degree of freedomA sample variance (s2) has (n-1) degrees of freedom11Marketing Research 12th Edition Hypothesis Testing & Associated Statistical Tests12Marketing Research 12th Edition Legend: σ = population standard deviation.One or Two-tail TestOne-tailed Hypothesis Test Determines whether a particular population parameter is larger or smaller than some predefined valueUses one critical value of test statisticTwo-tailed Hypothesis Test Determines the likelihood that a population parameter is within certain upper and lower boundsMay use one or two critical values13Marketing Research 12th Edition Basic Concepts of Hypothesis Testing (Contd.)Select the appropriate probability distribution based on two criteriaSize of the sampleWhether the population standard deviation is known or not14Marketing Research 12th Edition Hypothesis Testing15Data Analysis OutcomeAccept Null HypothesisReject Null HypothesisNull Hypothesis is TrueCorrect DecisionType I ErrorNull Hypothesis is FalseType II ErrorCorrect DecisionMarketing Research 12th Edition Cross-tabulation and Chi SquareIn Marketing Applications, Chi-square Statistic is used as:Test of IndependenceAre there associations between two or more variables in a study?Test of Goodness of FitIs there a significant difference between an observed frequency distribution and a theoretical frequency distribution?Statistical IndependenceTwo variables are statistically independent if a knowledge of one would offer no information as to the identity of the other16Marketing Research 12th Edition The Concept of Statistical Independence17If n is equal to 200 and Ei is the number of outcomes expected in cell i, Marketing Research 12th Edition Chi-Square As a Test of Independence18Marketing Research 12th Edition Chi-Square As a Test of Independence (Contd.)Null Hypothesis HoTwo (nominally scaled) variables are statistically independentAlternative Hypothesis HaThe two variables are not independentUse Chi-square distribution to test.19Marketing Research 12th Edition Chi-square DistributionA probability distributionTotal area under the curve is 1.0A different chi-square distribution is associated with different degrees of freedom20Cutoff points of the chi-square distribution functionMarketing Research 12th Edition Degrees of FreedomNumber of degrees of freedom, v = (r - 1) * (c - 1) r = number of rows in contingency table c = number of columnsMean of chi-squared distribution = Degree of freedom (v)Variance = 2vChi-square Distribution (Contd.)21Marketing Research 12th Edition 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 columnsExpected frequency in each cell, Ei = pc * pr * n Where pc and pr are proportions for independent variables n is the total number of observations22Marketing Research 12th Edition Chi-square Step-by-Step23Formulate HypothesisCalculate row & column totalsCalculate row & column proportionsCalculate expected frequencies (Ei)Calculate χ2 statisticCalculate degrees of freedomObtain critical value from tableMake decision regarding Null-hypothesisMarketing Research 12th Edition Strength of AssociationMeasured by contingency coefficient 0 = no association (i.e., Variables are statistically independent)Maximum value depends on the size of tableCompare only tables of same size24Marketing Research 12th Edition Limitations of Chi-square as an Association MeasureIt is basically proportional to sample sizeDifficult to interpret in absolute sense and compare cross-tabs of unequal sizeIt has no upper bound Difficult to obtain a feel for its value Does not indicate how two variables are related25Marketing Research 12th Edition Measures of Association for Nominal VariablesMeasures based on Chi-Square26Phi-squaredCramer’s VMarketing Research 12th Edition Chi-square Goodness of FitUsed to investigate how well the observed pattern fits the expected patternResearcher may determine whether population distribution corresponds to either a normal, Poisson or binomial distribution27To determine degrees of freedom:Employ (k-1) ruleSubtract an additional degree of freedom for each population parameter that has to be estimated from the sample dataMarketing Research 12th Edition 28End of Chapter SeventeenMarketing Research 12th Edition