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Essentials of Marketing Research (Second Edition)

Essentials of Marketing Research (Second Edition). Kumar Aaker & Day Instructor’s Presentation Slides. Chapter Thirteen. Hypothesis Testing: Basic Concepts and Tests of Association, Means and Proportion. Hypothesis Testing: Basic Concepts.

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Essentials of Marketing Research (Second Edition)

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  1. Essentials of Marketing Research (Second Edition) Kumar Aaker & Day Instructor’s Presentation Slides Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  2. Chapter Thirteen Hypothesis Testing: Basic Concepts and Tests of Association, Means and Proportion Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  3. Hypothesis Testing: Basic Concepts • Assumption (hypothesis) made about a population parameter (not sample parameter) • Purpose of Hypothesis Testing • To make a judgement about the difference between two sample statistics or the sample statistic and a hypothesized population parameter • Evidence has to be evaluated statistically before arriving at a conclusion regarding the hypothesis. Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  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. Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  5. The Logic of Hypothesis Testing • 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 Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  6. Problem Definition Clearly state the null and alternative hypotheses. Choose the relevant test and the appropriate probability distribution Determine the degrees of freedom Determine the significance level Choose the critical value Compare test statistic and critical value Compute relevant test statistic Decide if one-or two-tailed test Does the test statistic fall in the critical region? Do not reject null Reject null Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  7. Basic Concepts of Hypothesis Testing (Contd.) The Three Criteria Used Are • Significance Level • Degrees of Freedom • One or Two Tailed Test Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  8. 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 () Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  9. Significance Level (Contd.) • When choosing a level of significance, there is an inherent tradeoff between these two types of errors • Power of hypothesis test (1 - ) • A good test of hypothesis ought to reject a null hypothesis when it is false • 1 -  should be as high a value as possible Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  10. Degree 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 Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  11. 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 Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  12. 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 Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  13. DATA ANALYSIS OUTCOME In Population Accept Null Reject Null Hypothesis Hypothesis Null Hypothesis Correct Decision Type I Error True Null Hypothesis Type II Error Correct False Decision Hypothesis Testing Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  14. Hypothesis Testing Tests in this class Statistical Test • Frequency Distributions 2 • Means (one) z (if  is known) t (if  is unknown) • Means (two) t • Means (more than two) ANOVA Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  15. 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 Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  16. Chi-Square As a Test of Independence 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 Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  17. Chi-square As a Test of Independence (Contd.) 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 Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  18. Chi-square As a Test of Independence (Contd.) Degree 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 Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  19. - 2 ( O E ) n c = S 2 i i E = i 1 i Chi-square Statistic (2) • Measures of the difference between the actual numbers observed in cell i (Oi), and number expected (Ei) under independence if the null hypothesis were true With (r-1)*(c-1) degrees of freedom 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 and n is the total number of observations Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  20. Chi-square Step-by-Step 1) Formulate Hypotheses 2) Calculate row and column totals 3) Calculate row and column proportions 4) Calculate expected frequencies (Ei) 5) Calculate 2 statistic 6) Calculate degrees of freedom 7) Obtain Critical Value from table 8) Make decision regarding the Null-hypothesis Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  21. Example of Chi-square as a Test of Independence Class 1 2 A10 8 Grade B 20 16 C 45 18 D 16 6 E 9 2 This is a ‘Cell’ Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  22. Chi-square As a Test of Independence - Exercise OwnIncome Expensive Low Middle High Automobile Yes45 34 55 No 52 53 27 Task: Make a decision whether the two variables are independent! Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  23. F(x2) Critical value = 9.49 df = 4 5% of area under curve  = .05 The chi-square distribution • Probability distributions that are continuous, have one mode, and are skewed to the right. • Exact shape varies according to the number of degrees of freedom. • The critical value of a test statistic in a chi-square distribution is determined by specifying a significance level and the degrees of freedom. • Ex: Significance level = .05 Degrees of freedom = 4 CVx2 = 9.49 The decision rule when testing hypotheses by means of chi-square distribution is: If x2 is <=CVx2, accept H0 Thus, for 4 df and  = .05 If x2 is > CVx2, reject H0 If If x2 is <= 9.49, accept H0 x2 Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  24. Cross Tabulation Example • In a nationwide study of 1,402 adults a question was asked about institutions: “I am going to name some institutions in this country. As far as the people running these institutions are concerned, would you say have a great deal of confidence, only some confidence, or hardly any confidence at all in them?” One of the institutions was television. Answers to the question about television are cross-tabulated with three levels of income below. Annual Family Income Amount of confidence in television A great deal Only some Hardly any Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  25. Calculations for income-confidence data X2ts = 22.15 Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  26.  = .05 df = 4 [(r-1) (c-1)] n = 1402 X2cv = 9.5 X2ts = 22.15 F(x2) X2cv = 9.5 df = 4 5% of area under curve  = .05 22.15 Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  27. Strength of Association • Measured by contingency coefficient C = x2 o< c < 1  x2 + n • 0 - no association (i.e. Variables are statistically independent) • Maximum value depends on the size of table-compare only tables of same size Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  28. Limitations 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 Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  29. 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 Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  30. Chi-square 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 Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  31. Goodness-of-Fit Test Suppose a researcher is investigating preferences for four possible names of a new lightweight brand of sandals: Camfo, Kenilay, Nemlads, and Dics. Since the names are generated from random combinations of syllables, thre researcher expects preferences will be equally distributed across the four names (that is, each name will receive 25 percent of the available preferences). After sampling 300 people at reandom and asking them which one of the four names was most preferred, the following distribution resulted (each expected value is 300 * .25 = 75). Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  32. Goodness-of-Fit Test (cont.) • There are (d – 1) or three degrees of freedom in this instance. If  is specified as 0.01, the critical value is 11.325 from Statistical Appendix Table 3.18 Given this information, the hypothesis to be tested can be stated as: H0: preferences are equal for the names Ha: preferences are not equal for the names And the decision rule is If x2 is <= 11.325, accept H0. If x2 is > 11.325, reject H0. The test statistic is calculated as x2 = (30-75)2 / 75 + (80-75)2 / 75 + (120-75)2 / 75 + (70-75)2 / 75 = 27.00 + .33 + 27.00 + .33 = 54.66 Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  33. Hypothesis Testing For Differences Between Means • Commonly used in experimental research • Statistical technique used is analysis Of variance (ANOVA) Hypothesis Testing Criteria Depends on • Whether the samples are obtained from different or related populations • Whether the population is known on not known • If the population standard deviation is not known, whether they can be assumed to be equal or not Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  34. The Probability Values (P-value) Approach to Hypothesis Testing • P-value provides researcher with alternative method of testing hypothesis without pre-specifying  • Largest level of significance at which we would not reject ho Difference Between Using  and p-value • Hypothesis testing with a pre-specified  • Researcher is trying to determine, "is the probability of what has been observed less than ?" • Reject or fail to reject ho accordingly Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  35. The Probability Values (P-value) Approach to Hypothesis Testing (Contd.) Using the p-Value • Researcher can determine "how unlikely is the result that has been observed?" • Decide whether to reject or fail to reject ho without being bound by a pre-specified significance level • In general, the smaller the p-value, the greater is the researcher's confidence in sample findings • P-value is generally sensitive to sample size • A large sample should yield a low p-value • P-value can report the impact of the sample size on the reliability of the results Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  36. Hypothesis Testing About a Single Mean - Step-by-Step 1) Formulate Hypotheses 2) Select appropriate formula 3) Select significance level 4) Calculate z or t statistic 5) Calculate degrees of freedom (for t-test) 6) Obtain critical value from table 7) Make decision regarding the Null-hypothesis Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  37. Hypothesis Testing About a Single Mean - Example 1 • Ho:  = 5000 (hypothesized value of population) • Ha:   5000 (alternative hypothesis) • n = 100 • X = 4960 •  = 250 •  = 0.05 Rejection rule: if |zcalc| > z/2 then reject Ho. Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  38. Hypothesis Testing About a Single Mean - Example 2 • Ho:  = 1000 (hypothesized value of population) • Ha:   1000 (alternative hypothesis) • n = 12 • X = 1087.1 • s = 191.6 •  = 0.01 Rejection rule: if |tcalc| > tdf, /2 then reject Ho. Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  39. Hypothesis Testing About a Single Mean - Example 3 • Ho:   1000 (hypothesized value of population) • Ha:  > 1000 (alternative hypothesis) • n = 12 • X = 1087.1 • s = 191.6 •  = 0.05 Rejection rule: if tcalc > tdf,  then reject Ho. Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  40. - m ( X ) ± = = X ts t x s x Confidence Intervals • Hypothesis testing and Confidence Intervals are two sides of the same coin. interval  estimate of  Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  41. s ± X Z n s s - £ £ + = P ( X Z u X z ) . 95 n n 15 15 - £ £ + = P ( 290 2 . 58 ( u 290 2 . 58 ( )) . 99 s 15 75 75 = n 75 £ £ = P ( 285 . 54 u 294 . 46 ) 0 . 99 Confidence Interval Estimation If  = .95 then, Problem: n = 75  = .01 Since CI is for both sides, z-value is got for /2 = .005 Z /2 = 2.58 Test the hypothesis that the true mean weight of the Hawkeyes football team is greater than or equal to 300 pounds with  = .05 Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  42. H0: uW  300 H1: uW < 300 At  = 0.05, CVZ = -1.645 (for a one-tailed test) Since Zts falls in the critical region We ______________________ the null hypothesis Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  43. Test the hypothesis that the true mean weight of the Hawkeyes football team is equal to 286 pounds with  = 0.01 H0: uW = 286 uW 286 AT  = .01 CVZ = 2.58 Since Zts < CvZ we __________________ the null hypothesis Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  44. H0: PA = PB HA: PA not equal to PB Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  45. df = n1+n2-2 • (n1-1) + (n2-1) • = .05 df = 113 And = weighted average of sample proportions Computation of tts would proceed as follows: Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  46. + + 40 (. 45 ) 75 (. 40 ) 18 30 = = = ˆ p . 42 + 40 75 115 Since then and -1.96 +1.96 .025 .025 - + Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  47. Descriptive Statistics for two samples of students, liberal arts majors (n = 317) and engineering majors (n = 592) include The smaller the mean, the more students agree with the statement. The formula for a t-test of mean differences for independent samples is With being the standard error of the mean difference Where Is a weighted average of sample standard deviations. In this situation the hypothesis: Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  48. Pooled Std. dev = 1.07 Tts= 2.59-2.29 / .07 = .30 / .07 = 4.29 Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  49. Statistical techniques Analysis of Variance (ANOVA) Correlation Analysis Regression Analysis Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

  50. Analysis of Variance • ANOVA mainly used for analysis of • experimental data • Ratio of “between-treatment” variance • and “within- treatment” variance Essentials of Marketing Research ,Second Edition Kumar , Aaker & Day

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