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

Basic Concepts and Tests of Association Chapter Seventeen Hypothesis Testing: Basic Concepts and Tests of Association

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

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.

Hypothesis Testing Process Problem Definition Clearly state the null and alternative hypotheses Choose the relevant test and the appropriate probability distribution Choose the critical value Compare test statistic & critical value Reject null Determine the significance level Compute relevant test statistic Determine the degrees of freedom Decide if one-or two-tailed test Do not reject null Does the test statistic fall in the critical region?

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

Significance Level Look at book page 473: explain Type I/II error 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 Look at book page 473: explain Type I/II error

Relationship between Type I & Type II Errors

Relationship between Type I & Type II Errors (Contd.)

Relationship between Type I & Type II Errors (Contd.)

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

Hypothesis Testing & Associated Statistical Tests

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

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

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

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

The Concept of Statistical Independence If n is equal to 200 and Ei is the number of outcomes expected in cell i,

Chi-Square As a Test of Independence

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.

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

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

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

Chi-square Step-by-Step Formulate Hypothesis Calculate row & column totals Calculate row & column proportions Calculate expected frequencies (Ei) Calculate χ2 statistic Calculate degrees of freedom Obtain critical value from table Make decision regarding Null-hypothesis

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

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

Measures of Association for Nominal Variables Measures based on Chi-Square Phi-squared Cramer’s V

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