1. McGraw-Hill/Irwin McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights reserved.

2. Chapter 15 Data Analysis: Testing for Significant Differences

3. 1. Explain measures of central tendency and dispersion. 2. Describe how to test hypotheses using univariate and bivariate statistics. 3. Apply and interpret analysis of variance (ANOVA). 4. Utilize perceptual mapping to present research findings. Learning Objectives 15-3

4. Value of Testing for Differences in Data Analysis of variance (ANOVA) Hypothesis testing t-distribution and associated confidence interval estimation Central tendency and dispersion Common to all marketing research projects Common to all marketing research projects Basic Statistics and Descriptive Analysis 15-4

5. Measures of Central Tendency MeanMeanMedianMedianModeMode 15-5

6. Mean – arithmetic average of the sample, all values of a distribution of responses are summed and divided by the number of valid responses. Mean – arithmetic average of the sample, all values of a distribution of responses are summed and divided by the number of valid responses. Mode – most common value is the set of responses to a question; i.e., the response most often given to a question. Mode – most common value is the set of responses to a question; i.e., the response most often given to a question. Median – middle value of a rank ordered distribution; half of the responses are above and half below the median value. Median – middle value of a rank ordered distribution; half of the responses are above and half below the median value. Measures of Central Tendency defined... 15-6

7. Measures of Central Tendency Types of Data Nominal = Mode Mode Ordinal = Median Median Interval & Ratio = Mean = Mean Interval & Ratio = Mean = Mean 15-7

8. Dialog Boxes for Calculating the Mean, Median and Mode 15-8

9. Output for Mean, Median and Mode for X25– Frequency of Eating at... 15-9

10. Measures of Dispersion VarianceVarianceRangeRangeStandardDeviationStandardDeviation... describe how close to the mean or other measure of central tendency the other values in the distribution fall. 15-10

11. Measures of Dispersion Standard deviation – the average distance of the dispersion of the values from the mean. Range – the distance between the smallest and largest values of the variable. Variance – the average squared deviation about the mean of a distribution of values. 15-11

12. Output – Measures of Dispersion 15-12

13. Purpose of inferential statistics – to make a determination about a population on the basis of a sample. Purpose of inferential statistics – to make a determination about a population on the basis of a sample. Sample – a subset of the population. Sample – a subset of the population. Sample statistics – measures obtained directly from sample data. Sample statistics – measures obtained directly from sample data. Population parameter – a measured characteristic of the population. Population parameter – a measured characteristic of the population. Actual population parameters are unknown since the cost to perform a census of the population is prohibitive. Actual population parameters are unknown since the cost to perform a census of the population is prohibitive. Frequency Distribution – used to display data calculated from the sample. Frequency Distribution – used to display data calculated from the sample. Analyzing Relationships of Sample Data 15-13

14. Hypothesis Testing Univariate statistical test = hypothesis tests one variable at a time. at a time. Univariate statistical test = hypothesis tests one variable at a time. at a time. Bivariate statistical test = hypothesis tests two variables. Bivariate statistical test = hypothesis tests two variables. 15-14

15. Hypothesis Testing... a preconceived notion that is empirically testable but unproven, and developed in order to explain phenomena. Hypothesis 15-15

16. Null Hypothesis (H 0 ) – a statement that asserts the status quo. Null Hypothesis (H 0 ) – a statement that asserts the status quo. Alternative Hypothesis (H 1 ) Alternative Hypothesis (H 1 ) a statement that is the opposite of the null hypothesis – that the difference in reality is not simply due to random error. a statement that is the opposite of the null hypothesis – that the difference in reality is not simply due to random error. Represents the condition desired. Represents the condition desired. Null hypothesis is accepted – there is no change in the status quo. Null hypothesis is accepted – there is no change in the status quo. Null hypothesis is rejected – the alternative hypothesis is accepted and the conclusion is that there has been a change in opinions or actions. Null hypothesis is rejected – the alternative hypothesis is accepted and the conclusion is that there has been a change in opinions or actions. Null hypothesis refers to a population parameter – not a sample statistic. Null hypothesis refers to a population parameter – not a sample statistic. Hypothesis Testing 15-16

17. Independent samples – two or more groups of respondents that are tested as though they may come from different populations (independent samples t-test). Independent samples – two or more groups of respondents that are tested as though they may come from different populations (independent samples t-test). Related samples – two or more groups of respondents that originated from the sample population (paired samples t-test). Related samples – two or more groups of respondents that originated from the sample population (paired samples t-test). Paired samples – questions are independent but respondents are the same. Paired samples – questions are independent but respondents are the same. Hypothesis Testing 15-17

18. First Step – to develop the hypotheses that are to be tested... First Step – to develop the hypotheses that are to be tested... Developed prior to the collection of data. Developed prior to the collection of data. Developed as part of a research plan. Developed as part of a research plan. Make comparisons between two groups of respondents to determine if there are important differences between the groups. Make comparisons between two groups of respondents to determine if there are important differences between the groups. Important considerations in hypothesis testing are: Important considerations in hypothesis testing are: Magnitude of the difference between the means. Magnitude of the difference between the means. Size of the sample used to calculate the means. Size of the sample used to calculate the means. Hypothesis Testing 15-18

19. Statistical Significance Statistical Significance Inference regarding the population Inference regarding the population Type I Error – made by rejecting the null hypothesis when it is true – the probability of alpha (α) Type I Error – made by rejecting the null hypothesis when it is true – the probability of alpha (α) Level of Significance –.10,.05, or.01 Level of Significance –.10,.05, or.01 Hypothesis Testing 15-19

20. Type II Error – failing to reject the null hypothesis when the alternative hypothesis is true – the probability of beta (β). Type II Error – failing to reject the null hypothesis when the alternative hypothesis is true – the probability of beta (β). Unlike alpha (α), which is specified by the researcher, beta (β) depends on the actual population parameter. Unlike alpha (α), which is specified by the researcher, beta (β) depends on the actual population parameter. Type I and Type II errors – sample size can help control these errors. Type I and Type II errors – sample size can help control these errors. Can select an alpha (α) and the sample size in order to increase the power of the test and beta (β). Can select an alpha (α) and the sample size in order to increase the power of the test and beta (β). Hypothesis Testing 15-20

21. Analyzing Relationships of Sample Data Univariate Tests of Significance t-testt-test z-testz-test... involve hypothesis testing using one variable at a time.... if sample size <30 and the standard deviation is unknown, assumption of a normal distribution is not valid, use t-test.... if sample size >30 and the standard deviation is unknown, use z-test use z-test.... if sample size >30 and the standard deviation is unknown, use z-test use z-test. 15-21

22. Analyzing Relationships of Sample Data UnivariateandBivariatet-testsUnivariateandBivariatet-testsBivariatet-testBivariatet-test... require interval or ratio data.... assumption is the samples are drawn from populations with normal distributions and the variances of the populations are equal. 15-22

23. Univariate Hypothesis Test Using X16–Reasonable Prices 15-23

24. Analyzing Relationships of Sample Data BivariateHypothesisBivariateHypothesisNullHypothesisNullHypothesis... more than one group is involved.... there is no difference between the group means. the group means. µ1 = µ2 or that µ1 - µ2 = 0 µ1 = µ2 or that µ1 - µ2 = 0... there is no difference between the group means. the group means. µ1 = µ2 or that µ1 - µ2 = 0 µ1 = µ2 or that µ1 - µ2 = 0 15-24

25. Analyzing Relationships of Sample Data The formula for calculating the t value is... _ _ _ _ Z = x 1 – x 2 Z = x 1 – x 2 Sx 1 – x 2 15-25

26. Bivariate Statistical Tests Cross-tabulation – is useful for examining relationships and reporting the findings for two variables. The purpose of cross-tabulation is to determine if differences exist between subgroups of the total sample. 15-26

27. Dialog Boxes for Crosstab 15-27

28. Example of a Cross-Tabulation: Gender by Ad Recall 15-28

29. Chi-Square (X 2 ) Analysis... test for significance between the frequency distributions of two or more nominally scaled variables in a cross-tabulation table to determine if there is any association. 15-29

30. Assesses how closely the observed frequencies fit the pattern of the expected frequencies and is referred to as a ”goodness-of-fit” test. Assesses how closely the observed frequencies fit the pattern of the expected frequencies and is referred to as a ”goodness-of-fit” test. Used to analyze nominal data which cannot be analyzed with other types of statistical analysis, such as ANOVA or t-tests. Used to analyze nominal data which cannot be analyzed with other types of statistical analysis, such as ANOVA or t-tests. Results will be distorted if more than 20 percent of the cells have an expected count of less than 5. Results will be distorted if more than 20 percent of the cells have an expected count of less than 5. Chi-Square (X 2 ) Analysis 15-30

31. Chi-Square Analysis Do college students and high school students differ in their preference for Coke versus Pepsi? Do part-time and full-time workers differ in terms of how often they are absent from work (seldom, occasionally, frequently)? Does frequency of eating out (infrequent, moderately frequent, and very frequent) differ between males and females? Is usage of the Internet (low, moderate and high) related to gender? Examples of Research Questions 15-31

32. SPSS Chi-Square Crosstab Example 15-32

33. Analyzing Relationships of Sample Data IndependentSamplesIndependentSamplesRelatedSamplesRelatedSamples Example... interviews with male and female coffee drinkers. Example... interviews of only female students and comparing number of Cokes consumed versus number of cups of coffee. 15-33

34. Analyzing Relationships of Sample Data ot-test for differences between group means – is the difference between the means divided by the variability of random means. t-value – ratio of the difference between two sample means and the standard error. t-value – ratio of the difference between two sample means and the standard error. t-test – provides a rational way of determining if the difference between the two sample means occurred by chance. t-test – provides a rational way of determining if the difference between the two sample means occurred by chance. ot-test for differences between group means – is the difference between the means divided by the variability of random means. t-value – ratio of the difference between two sample means and the standard error. t-value – ratio of the difference between two sample means and the standard error. t-test – provides a rational way of determining if the difference between the two sample means occurred by chance. t-test – provides a rational way of determining if the difference between the two sample means occurred by chance. 15-34

35. Comparing Two Means with the Independent-Samples t-Test 15-35

36. Paired Samples t-Test 15-36

37. Analyzing Relationships of Sample Data ANOVA (analysis of variance) ANOVA... determines if three or more means are statistically different from each other (single dependent variable)... same as ANOVA but multiple dependent variables can be analyzed together. MANOVA (multivariate analysis of variance) MANOVA 15-37

38. Requirements for ANOVA Requirements for ANOVA dependent variable can be either interval or ratio scaled. dependent variable can be either interval or ratio scaled. independent variable is categorical. independent variable is categorical. Null hypothesis for ANOVA – states there is no difference between the groups – the null hypothesis is... Null hypothesis for ANOVA – states there is no difference between the groups – the null hypothesis is... µ1 = µ2 = µ3 µ1 = µ2 = µ3 ANOVA – focuses on the behavior of the variance within a set of data. ANOVA – focuses on the behavior of the variance within a set of data. ANOVA – if the calculated variance between the groups is compared to the variance within the groups, a determination can be made as to whether the means are significantly different. ANOVA – if the calculated variance between the groups is compared to the variance within the groups, a determination can be made as to whether the means are significantly different. Analyzing Relationships of Sample Data 15-38

39. ANOVA Total variance – separated into between-group and within-group variance. F-test – used to statistically evaluate the differences between the group means. DeterminingStatisticalSignificanceDeterminingStatisticalSignificance 15-39

40. Analyzing Relationships of Sample Data Total Variance WithinGroupsWithinGroupsBetweenGroupsBetweenGroups 15-40

41. ANOVA – Testing Statistical Significance The larger the F ratio...... the larger the difference in the variance between groups... the larger the difference in the variance between groups.... the more likely the null hypothesis will be rejected. Based on the F- distribution...... Examines the ratio of two components of total variance and is calculated as shown below... F ratio = Variance between groups Variance within groups Variance within groups F ratio = Variance between groups Variance within groups Variance within groups... implies significant differences between the groups. 15-41

42. ANOVA – cannot identify which pairs of means are significantly different from each other... ANOVA – cannot identify which pairs of means are significantly different from each other... Follow-up Tests – identify the means that are statistically different from each other. Follow-up Tests – identify the means that are statistically different from each other. Sheffé Sheffé Tukey, Duncan and Dunn Tukey, Duncan and Dunn Analyzing Relationships of Sample Data 15-42

43. In a one-way ANOVA – only one independent variable is possible. In a one-way ANOVA – only one independent variable is possible. For several independent variables – an n- way ANOVA would be used. For several independent variables – an n- way ANOVA would be used. Can be used with experimental designs – exposes different groups in a sample to several different information (treatments) to see if their responses change. Can be used with experimental designs – exposes different groups in a sample to several different information (treatments) to see if their responses change. N-way ANOVA 15-43

44. ANOVA Comparing Two Restaurants on Selected Variables 15-44

45. 45 MANOVA CharacteristicsCharacteristics Statistical calculations for MANOVA – similar to n-way ANOVA and are in statistical software packages such as SPSS and SAS.... designed to examine multiple dependent variables across single or multiple independent variables. 15-45

46. Perceptual Mapping... process used to develop maps showing the perceptions of respondents. The maps are visual representations of respondents’ perceptions of a company, product, service, brand, or any other object in two dimensions. 15-46

47. Perceptual Maps... have a vertical and a horizontal axis that are labeled with descriptive adjectives. To develop perceptual maps – can use rankings, mean ratings, and multivariate methods. 15-47

48. Ratings of Six Fast-Food Restaurants 15-48

49. Perceptual Map of Six Fast-Food Restaurants 15-49

50. Importance Ratings for Restaurant Selection Factors 15-50

51. One-Way ANOVA for Three Restaurant Competitors 15-51

52. One-Way ANOVA of Differences in Restaurant Perceptions Variables 15-52

53. Summary of ANOVA Findings from Exhibits 15.14–15.16 15-53

54. Importance-Performance Chart for Remington’s Steak House 15-54

55. Perceptual Mapping DistributionDistribution AdvertisingAdvertising Image development New product development Applications in Marketing ResearchApplications Research 15-55