1. Copyright © 2010 Pearson Education, Inc. 15-1 Chapter Fifteen Frequency Distribution, Cross-Tabulation, and Hypothesis Testing

2. Copyright © 2010 Pearson Education, Inc. 15-2 Chapter Outline Chapter 15a (Mid Sept.) 1) Frequency Distribution (slide 15-5) 2) Hypothesis Testing (slide 15-16) Chapter 15b (Late Sept.) 3) Cross-Tabulations (slide 15-27) Chapter 15c and d (October) 4) Testing for Differences; Parametric Tests (Slide 15-58) 5) Non-Parametric Tests (15-70)

3. Copyright © 2010 Pearson Education, Inc. 15-3 1) Internet Usage Data Respondent Sex Familiarity Internet Attitude Toward Usage of Internet Number Usage InternetTechnology Shopping Banking 1 1.00 7.00 14.007.00 6.00 1.001.00 2 2.00 2.00 2.003.00 3.00 2.002.00 3 2.00 3.00 3.004.00 3.00 1.002.00 4 2.00 3.00 3.007.00 5.00 1.002.00 5 1.00 7.00 13.007.00 7.00 1.001.00 6 2.00 4.00 6.005.00 4.00 1.002.00 7 2.00 2.00 2.004.00 5.00 2.002.00 8 2.00 3.00 6.005.00 4.00 2.002.00 9 2.00 3.00 6.006.00 4.00 1.002.00 10 1.00 9.00 15.007.00 6.00 1.002.00 11 2.00 4.00 3.004.00 3.00 2.002.00 12 2.00 5.00 4.006.00 4.00 2.002.00 13 1.00 6.00 9.006.00 5.00 2.001.00 14 1.00 6.00 8.003.00 2.00 2.002.00 15 1.00 6.00 5.005.00 4.00 1.002.00 16 2.00 4.00 3.004.00 3.00 2.002.00 17 1.00 6.00 9.005.00 3.00 1.001.00 18 1.00 4.00 4.005.00 4.00 1.002.00 19 1.00 7.00 14.006.00 6.00 1.001.00 20 2.00 6.00 6.006.00 4.00 2.002.00 21 1.00 6.00 9.004.00 2.00 2.002.00 22 1.00 5.00 5.005.00 4.00 2.001.00 23 2.00 3.00 2.004.00 2.00 2.002.00 24 1.00 7.00 15.006.00 6.00 1.001.00 25 2.00 6.00 6.005.00 3.00 1.002.00 26 1.00 6.00 13.006.00 6.00 1.001.00 27 2.00 5.00 4.005.00 5.00 1.001.00 28 2.00 4.00 2.003.00 2.00 2.002.00 29 1.00 4.00 4.005.00 3.00 1.002.00 30 1.00 3.00 3.007.00 5.00 1.002.00 Table 15.1

4. Copyright © 2010 Pearson Education, Inc. 15-4 1) Frequency Distribution In a frequency distribution, one variable is considered at a time. A frequency distribution for a variable produces a table of frequency counts, percentages, and cumulative percentages for all the values associated with that variable.

5. Copyright © 2010 Pearson Education, Inc. 15-5 1) Frequency of Familiarity with the Internet Table 15.2

6. Copyright © 2010 Pearson Education, Inc. 15-6 1) Frequency Histogram Fig. 15.1 2 3 4 5 6 7 0 7 4 3 2 1 6 5 Frequency Familiarity 8

7. Copyright © 2010 Pearson Education, Inc. 15-7 The mean, or average value, is the most commonly used measure of central tendency. The mean,,is given by Where, X i = Observed values of the variable X n = Number of observations (sample size) The mode is the value that occurs most frequently. The mode represents the highest peak of the distribution. The mode is a good measure of location when the variable has been grouped into categories (in other words, when there is no mean). 1) Statistics Associated with Frequency Distribution: Measures of Location X = X i / n X

8. Copyright © 2010 Pearson Education, Inc. 15-8 1) Statistics Associated with Frequency Distribution: Measures of Location The median of a sample is the middle value when the data are arranged in ascending or descending order. If the number of data points is even, the median is usually estimated as the midpoint between the two middle values – by adding the two middle values and dividing their sum by 2.

9. Copyright © 2010 Pearson Education, Inc. 15-9 1) Statistics Associated with Frequency Distribution: Measures of Variability The range measures the spread of the data. It is simply the difference between the largest and smallest values in the sample. Range = X largest – X smallest The interquartile range is the difference between the 75th and 25th percentile. In other words, the interquartile range encompasses the middle 50% of observations.

10. Copyright © 2010 Pearson Education, Inc. 15-10 1) Statistics Associated with Frequency Distribution: Measures of Variability Deviation from the mean is the difference between the mean and an observed value. Mean = 4; observed value = 7 Deviation from the mean = ______________ The variance is the mean squared deviation from the mean. The variance can never be negative. Example: 3 point scale; 10 respondents = n Responses: 2 ppl. = 1; 6 ppl. = 2; 2 ppl. = 3 Mean = 2 =(1+1+2+2+2+2+2+2+3+3)/10 Variance = (2 * (1 – 2) 2 + 6 * (2 – 2) 2 + 2 * (3 – 2) 2 )/9 n-1 Frequency of response Actual response Mean

11. Copyright © 2010 Pearson Education, Inc. 15-11 1) Statistics Associated with Frequency Distribution: Measures of Variability Easy Method Difficult Method How do we calculate variance in Excel?

12. Copyright © 2010 Pearson Education, Inc. 15-12 1) Statistics Associated with Frequency Distribution: Measures of Variability The standard deviation is the square root of the variance. How do we calculate variance in Excel? Easy Method Difficult Method

13. Copyright © 2010 Pearson Education, Inc. 15-13 1) Statistics Associated with Frequency Distribution: Measures of Variability How to interpret standard deviation and variance: When the data points are scattered, the variance and standard deviation are large. When the data points are clustered around the mean, the variance and standard deviation are small.

14. Copyright © 2010 Pearson Education, Inc. 15-14 1) Statistics Associated with Frequency Distribution: Measures of Shape Skewness. The tendency of the deviations from the mean to be larger in one direction than in the other. It can be thought of as the tendency for one tail of the distribution to be heavier than the other. Kurtosis is a measure of the relative peakedness or flatness of the curve defined by the frequency distribution. If the kurtosis is positive, then the distribution is more peaked than a normal distribution. A negative value means that the distribution is flatter than a normal distribution.

15. Copyright © 2010 Pearson Education, Inc. 15-15 1) Skewness of a Distribution Fig. 15.2 Skewed Distribution Symmetric Distribution Mean Median Mode (a) Mean Median Mode (b)

16. Copyright © 2010 Pearson Education, Inc. 15-16 1.Formulate Hypotheses 2.Select Appropriate Test 3.Choose Level of Significance (usually.05) 4.Collect Data and Calculate Test Statistic 5.Compare with Level of Significance 6.Reject or Do Not Reject H 0 7.Draw Marketing Research Conclusion 2) Steps Involved in Hypothesis Testing

17. Copyright © 2010 Pearson Education, Inc. 15-17 2) A General Procedure for Hypothesis Testing Step 1: Formulate the Hypothesis A null hypothesis is a statement of the status quo, one of no difference or no effect. If the null hypothesis is not rejected, no changes will be made. An alternative hypothesis is one in which some difference or effect is expected. Accepting the alternative hypothesis will lead to changes in opinions or actions.

18. Copyright © 2010 Pearson Education, Inc. 15-18 2) A General Procedure for Hypothesis Testing Step 1: Formulate the Hypothesis A null hypothesis may be rejected, but it can never be accepted based on a single test. There is no way to determine whether the null hypothesis is true. In marketing research, the null hypothesis is formulated in such a way that its rejection leads to the acceptance of the desired conclusion. The alternative hypothesis represents the conclusion for which evidence is sought. Example: H 1 : Millennials shop more online than Baby Boomers. H 0 : Millennials shop the same amount online as Baby Boomers.

19. Copyright © 2010 Pearson Education, Inc. 15-19 The test statistic measures how close the sample has come to the null hypothesis. In our example, the paired samples t-test, is appropriate. The t-test is good for testing if there is a difference between groups. 2) A General Procedure for Hypothesis Testing Step 2: Select an Appropriate Test

20. Copyright © 2010 Pearson Education, Inc. 15-20 2) A General Procedure for Hypothesis Testing Step 3: Choose a Level of Significance Whenever we draw inferences about a population, we risk the following errors: Type I Error Type I error occurs when the sample results lead to the rejection of the null hypothesis when it is in fact true. The probability of type I error ( ) is also called the level of significance. generally =.05, but it is up to the researcher. Type II Error Type II error occurs when, based on the sample results, the null hypothesis is not rejected when it is in fact false.      

21. Copyright © 2010 Pearson Education, Inc. 15-21 2) Step 3: Potential Errors Outcome Null hypothesis (H 0 ) is true Null hypothesis (H 0 ) is false Reject null hypothesis Type I error False positive Correct outcome True positive Fail to reject null hypothesis Correct outcome True negative Type II error False negative

22. Copyright © 2010 Pearson Education, Inc. 15-22 The required data are collected and the value of the test statistic computed. 20 Millennials, 20 Baby Boomers 7 = high internet usage; 1 = low internet usage In SPSS: Analyze  Compare Means  Paired Samples T-test 2) A General Procedure for Hypothesis Testing Step 4: Collect Data and Calculate Test Statistic

23. Copyright © 2010 Pearson Education, Inc. 15-23 2) A General Procedure for Hypothesis Testing Step 4: Collect Data and Calculate Test Statistic The mean for Millennials is high than for Baby Boomers

24. Copyright © 2010 Pearson Education, Inc. 15-24 2) A General Procedure for Hypothesis Testing Step 5: Collect Data and Calculate Test Statistic Sig. is below.05

25. Copyright © 2010 Pearson Education, Inc. 15-25 2) A General Procedure for Hypothesis Testing Step 6: Making the Decision on H 0 If the probability associated with the calculated or observed value of the test statistic is less than the level of significance, the null hypothesis is rejected. The probability associated with the calculated or observed value of the test statistic is 0.000. This is less than the level of significance of 0.05. Hence, the null hypothesis is rejected.  H 0 : Millennials shop the same amount online as Baby Boomers.

26. Copyright © 2010 Pearson Education, Inc. 15-26 2) A General Procedure for Hypothesis Testing Step 7: Marketing Research Conclusion The conclusion reached by hypothesis testing must be expressed in terms of the marketing research problem. In our example, we conclude that there is evidence that Millennials shop online more than Baby Boomers. Hence, the recommendation to a retailer would be to target the online efforts towards primarily Millennials. NOT REJECTED: H 1 : Millennials shop more online than Baby Boomers.

27. Copyright © 2010 Pearson Education, Inc. 15-27 3) Cross-Tabulation A cross-tabulation is another method for presenting data. While a frequency distribution describes one variable at a time, a cross-tabulation describes two or more variables simultaneously. Cross-tabulation results in tables that reflect the joint distribution of two or more variables with a limited number of categories or distinct values, e.g., Table 15.3. Cross tabulation shows two types of results: some association or no association (see Fig. 15.7).

28. Copyright © 2010 Pearson Education, Inc. 15-28 3) Gender and Internet Usage Table 15.3 Gender Row Internet Usage MaleFemaleTotal Light (1) 5 10 15 Heavy (2) 10 5 15 Column Total 15 15

29. Copyright © 2010 Pearson Education, Inc. 15-29 3) Two Variables Cross-Tabulation Since two variables have been cross-classified, percentages could be computed either columnwise, based on column totals (Table 15.4), or rowwise, based on row totals (Table 15.5). The general rule is to compute the percentages in the direction of the independent variable, across the dependent variable. The correct way of calculating percentages is as shown in Table 15.4. Independent Variable (IV) is free to vary. Dependent Variable (DV) depends on the IV.

30. Copyright © 2010 Pearson Education, Inc. 15-30 3) Internet Usage by Gender Table 15.4: Columnwise Independent Variable (IV) = Gender Dependent Variable (DV) = Internet Usage What does this mean? Can we switch the IV and DV?

31. Copyright © 2010 Pearson Education, Inc. 15-31 3) Gender by Internet Usage Table 15.5: Rowwise Although table 15.5 is not wrong, table 15.4 is generally considered to be favorable.

32. Copyright © 2010 Pearson Education, Inc. 15-32 Introduction of a Third Variable in Cross- Tabulation Refined Association between the Two Variables No Association between the Two Variables No Change in the Initial Pattern Some Association between the Two Variables Fig. 15.7 Some Association between the Two Variables No Association between the Two Variables Introduce a Third Variable Original Two Variables The introduction of a third variable can result in four possibilities:

33. Copyright © 2010 Pearson Education, Inc. 15-33 Three Variables Cross-Tabulation: Refine an Initial Relationship Refine an Initial Relationship: As can be seen from Table 15.6: 52% of unmarried respondents fell in the high- purchase category 31% of married respondents fell in the high–purchase category Before concluding that unmarried respondents purchase more fashion clothing than those who are married, a third variable, the buyer's sex, was introduced into the analysis. Purchase of Fashion Current Marital Status Clothing Married Unmarried High31%52% Low69%48% Column100% Number of respondents 700300 Table 15.6

34. Copyright © 2010 Pearson Education, Inc. 15-34 Purchase of Fashion Clothing by Marital Status As shown in Table 15.7 (next slide): 60% of the unmarried females fall in the high-purchase category 25% of the married females fall in the high-purchase category However, the percentages are much closer for males: 40% of the unmarried males fall in the high-purchase category 35% of the married males fall in the high-purchase category Therefore, the introduction of sex (third variable) has refined the relationship between marital status and purchase of fashion clothing (original variables). Conclusion: Overall, unmarried respondents are more likely to fall in the high purchase category than married ones. However, this effect is much more pronounced for females than for males.

35. Copyright © 2010 Pearson Education, Inc. 15-35 Purchase of Fashion Clothing by Marital Status Table 15.7 Purchase of Fashion Clothing Sex Male Female Married Not Married Not Married High35%40%25%60% Low65%60%75%40% Column totals100% Number of cases 400120300180 What are the IVs? What is the DV? Explain.

36. Copyright © 2010 Pearson Education, Inc. 15-36 Three Variables Cross-Tabulation: Initial Relationship was Spurious Initial Relationship was Spurious Table 15.8 (two variables): 32% of those with college degrees own an expensive car. 21% of those without college degrees own an expensive car. However, income may also be a factor… Table 15.9 (three variables): the percentages of those with and without college degrees who own expensive automobiles are the same for each of the income groups. When the data for the high income and low income groups are examined separately, the association between education and ownership of expensive automobiles disappears. Therefore, the initial relationship observed between these two variables was spurious. A spurious relationship exists when two events or variables have no direct causal connection, yet it may be wrongly inferred that they do, due to either coincidence or the presence of a certain third, unseen variable (referred to as a "confounding variable" or "lurking variable”).

37. Copyright © 2010 Pearson Education, Inc. 15-37 Ownership of Expensive Automobiles by Education Level Table 15.8: Education = IV; Own Expensive Car = DV Own Expensive Car Education College DegreeNo College Degree Yes No Column totals Number of cases 32% 68% 100% 250 21% 79% 100% 750

38. Copyright © 2010 Pearson Education, Inc. 15-38 Ownership of Expensive Automobiles by Education Level and Income Levels Table 15.9 What are the IVs? What is the DV? What do these results say? Also, note the number of respondents….

39. Copyright © 2010 Pearson Education, Inc. 15-39 Three Variables Cross-Tabulation: Reveal Suppressed Association Reveal Suppressed Association Table 15.10 shows no association between desire to travel abroad and age. Table 15.11: when is sex introduced as the third variable, a relationship between age and desire to travel abroad is revealed. Among men, 60% of those under 45 indicated a desire to travel abroad, as compared to 40% of those 45 or older. The pattern was reversed for women, where 35% of those under 45 indicated a desire to travel abroad as opposed to 65% of those 45 or older.

40. Copyright © 2010 Pearson Education, Inc. 15-40 Desire to Travel Abroad by Age Table 15.10 Since the association between desire to travel abroad and age runs in the opposite direction for males and females, the relationship between these two variables is masked when the data are aggregated across sex.

41. Copyright © 2010 Pearson Education, Inc. 15-41 Desire to Travel Abroad by Age and Gender Table 15.11 But when the effect of sex is controlled, as in Table 15.11, the suppressed association between desire to travel abroad and age is revealed for the separate categories of males and females.

42. Copyright © 2010 Pearson Education, Inc. 15-42 Three Variables Cross-Tabulations: No Change in Initial Relationship No Change in Initial Relationship Consider the cross-tabulation of family size and the tendency to eat out frequently in fast-food restaurants, as shown in Table 15.12. No association is observed. When income was introduced as a third variable in the analysis, Table 15.13 was obtained. Again, no association was observed.

43. Copyright © 2010 Pearson Education, Inc. 15-43 Eating Frequently in Fast-Food Restaurants by Family Size Table 15.12

44. Copyright © 2010 Pearson Education, Inc. 15-44 Eating Frequently in Fast-Food Restaurants by Family Size and Income Table 15.13 What do these findings mean?

45. Copyright © 2010 Pearson Education, Inc. 15-45 Statistics Associated with Cross-Tabulation: Chi-Square Pearson’s Chi-square statistic: Used to determine to test whether paired observations on two variables are independent of each other Used to determine the goodness of fit of an observed distribution to a theoretical distribution. An important characteristic of the chi-square statistic is the number of degrees of freedom (df) associated with it. That is, df = (r - 1) x (c -1). R=row; C=column The null hypothesis (H 0 ) will be rejected only when the calculated value of the test statistic is greater than the critical value of the chi-square distribution with the appropriate degrees of freedom (Table 3 in the Statistical Appendix). H 0 : there is no association between the two variables We say this is the H 0 because we believe that some association does exist!

46. Copyright © 2010 Pearson Education, Inc. 15-46 Statistics Associated with Cross-Tabulation Chi-Square The chi-square statistic ( ) is used to test the statistical significance of the observed association in a cross-tabulation. We need to determine expected frequencies to determine the chi-square stat. The expected frequency for each cell can be calculated by using a simple formula:   f e = n r n c n wheren r = total number in the row n c = total number in the column n= total sample size

47. Copyright © 2010 Pearson Education, Inc. 15-47 An Old Example: Gender and Internet Usage Table 15.3 Gender Row Internet Usage MaleFemaleTotal Light (1) 5 10 15 Heavy (2) 10 5 15 Column Total 15 15 Statistically, is there an association between gender and internet usage? H 0 : there is no association between gender and internet usage

48. Copyright © 2010 Pearson Education, Inc. 15-48 Statistics Associated with Cross-Tabulation Chi-Square For the data in Table 15.3, the expected frequencies for the cells going from left to right and from top to bottom, are: Then the value of is calculated as follows: 15X 30 =7.50 15X 30 =7.50 15X 30 =7.50 15X 30 =7.50  2 = (f o -f e ) 2 f e    o=observation e=expected

49. Copyright © 2010 Pearson Education, Inc. 15-49 Statistics Associated with Cross-Tabulation Chi-Square For the data in Table 15.3, the value of is calculated as: = (5 -7.5) 2 + (10 - 7.5) 2 + (10 - 7.5) 2 + (5 - 7.5) 2 7.5 7.5 7.5 7.5 =0.833 + 0.833 + 0.833+ 0.833 = 3.333  

50. Copyright © 2010 Pearson Education, Inc. 15-50 Statistics Associated with Cross-Tabulation Chi-Square For the cross-tabulation given in Table 15.3, there are (2-1) x (2-1) = 1 degree of freedom. Table 3 in the Statistical Appendix contains upper-tail areas of the chi-square distribution for different degrees of freedom. For 1 degree of freedom, the probability of exceeding a chi- square value of 3.841 is 0.05. The calculated chi-square statistic had a value of 3.333. Since this is less than the critical value of 3.841, the null hypothesis of no association can not be rejected indicating that the association is not statistically significant at the 0.05 level. Another example: http://www.youtube.com/watch?v=Ahs8jS5mJKk http://www.youtube.com/watch?v=Ahs8jS5mJKk

51. Copyright © 2010 Pearson Education, Inc. 15-51 Statistics Associated with Cross-Tabulation Phi Coefficient The phi coefficient ( ) is used as a measure of the strength of association in the special case of a table with two rows and two columns (a 2 x 2 table). The phi coefficient is proportional to the square root of the chi-square statistic It takes the value of 0 when there is no association, which would be indicated by a chi-square value of 0 as well. When the variables are perfectly associated, phi assumes the value of 1 and all the observations fall just on the main or minor diagonal. =    2 n

52. Copyright © 2010 Pearson Education, Inc. 15-52 Statistics Associated with Cross-Tabulation Contingency Coefficient While the phi coefficient is specific to a 2 x 2 table, the contingency coefficient (C) can be used to assess the strength of association in a table of any size. The contingency coefficient varies between 0 and 1. The maximum value of the contingency coefficient depends on the size of the table (number of rows and number of columns). For this reason, it should be used only to compare tables of the same size. C =  2  2 + n

53. Copyright © 2010 Pearson Education, Inc. 15-53 Statistics Associated with Cross-Tabulation Cramer’s V Cramer's V is a modified version of the phi correlation coefficient,, and is used in tables larger than 2 x 2. or  V =  2 min (r-1), (c-1) V =  2 /n min (r-1), (c-1)

54. Copyright © 2010 Pearson Education, Inc. 15-54 Cross-Tabulation in Practice While conducting cross-tabulation analysis in practice, it is useful to proceed along the following steps. 1.Test the null hypothesis that there is no association between the variables using the chi-square statistic. If you fail to reject the null hypothesis, then there is no relationship. 2.If H 0 is rejected, then determine the strength of the association using an appropriate statistic (phi-coefficient, contingency coefficient, Cramer's V, lambda coefficient, or other statistics), as discussed earlier. 3.If H 0 is rejected, interpret the pattern of the relationship by computing the percentages in the direction of the independent variable, across the dependent variable. 4.If the variables are treated as ordinal rather than nominal, use tau b, tau c, or Gamma as the test statistic. If H 0 is rejected, then determine the strength of the association using the magnitude, and the direction of the relationship using the sign of the test statistic.

55. Copyright © 2010 Pearson Education, Inc. 15-55 A Classification of Hypothesis Testing Procedures for Examining Differences Independent Samples Paired Samples Independent Samples Paired Samples * Two-Group t test * Z test * Paired t test * Chi-Square * Mann-Whitney * Median * K-S * Sign * Wilcoxon * McNemar * Chi-Square Fig. 15.9 Hypothesis Tests One Sample Two or More Samples One Sample Two or More Samples * t test * Z test * Chi-Square * K-S * Runs * Binomial Parametric Tests (Metric Tests) Non-parametric Tests (Nonmetric Tests)

56. Copyright © 2010 Pearson Education, Inc. 15-56 Hypothesis Testing Related to Differences Parametric tests assume that the variables of interest are measured on at least an interval scale (chapter 11). Nonparametric tests assume that the variables are measured on a nominal or ordinal scale (chapter 11). These tests can be further classified based on whether one or two or more samples are involved. The samples are independent if they are drawn randomly from different populations. For the purpose of analysis, data pertaining to different groups of respondents, e.g., males and females, are generally treated as independent samples. The samples are paired when the data for the two samples relate to the same group of respondents.

57. Copyright © 2010 Pearson Education, Inc. 15-57 Parametric v. Non-parametric Scales Non-parametric level scales: Nominal level Scale: Any numbers used are mere labels: they express no mathematical properties. Examples are SKU inventory codes and UPC bar codes. Ordinal level scale: Numbers indicate the relative position of items, but not the magnitude of difference. An example is a preference ranking. Parametric level scales: Interval level scale: Numbers indicate the magnitude of difference between items, but there is no absolute zero point. Examples are attitude (Likert) scales and opinion scales. Ratio level scale: Numbers indicate magnitude of difference and there is a fixed zero point. Ratios can be calculated. Examples include: age, income, price, and costs.

58. Copyright © 2010 Pearson Education, Inc. 15-58 Parametric Tests Parametric tests provide inferences for making statements about the means of parent populations. A t-test is a common procedure used with parametric data. It can be used for: One-sample test: e.g. Does the market share for a given product exceed 15 percent? Two-sample test: e.g. Do the users and nonusers of a brand differ in terms of their perception of a brand? Paired-sample test The t statistic assumes that the variable is: Normally distributed The mean of the population is known And the population variance is estimated from the sample

59. Copyright © 2010 Pearson Education, Inc. 15-59 Hypothesis Testing Using the t Statistic 1.Formulate the null (H 0 ) and the alternative (H 1 ) hypotheses. H0: the market share for a given product does not exceed 15 percent. H1: the market share for a given product exceeds 15 percent. 2.Select the appropriate formula for the t statistic (as seen on p. 472). 3.Select a significance level, , for testing H 0. Typically, the 0.05 level is selected. 4.Take one or two samples and compute the mean and standard deviation for each sample. 5.Calculate the t statistic assuming H 0 is true.

60. Copyright © 2010 Pearson Education, Inc. 15-60 Hypothesis Testing Using the t Statistic 6.Calculate the degrees of freedom and estimate the probability of getting a more extreme value of the statistic from Table 4. 7.If the probability computed in step 5 is smaller than the significance level selected in step 2, reject H 0. If the probability is larger, do not reject H 0. Failure to reject H 0 does not necessarily imply that H 0 is true. It only means that the true state is not significantly different than that assumed by H 0. 8.Express the conclusion reached by the t test in terms of the marketing research problem.

61. Copyright © 2010 Pearson Education, Inc. 15-61 SPSS Windows: One Sample t Test 1. Select ANALYZE from the SPSS menu bar. 2. Click COMPARE MEANS and then ONE SAMPLE T TEST. 3. Choose your TEST VARIABLE(S). E.g. “Market share” 4. Choose your TEST VALUE box. E.g. “15 percent” 5. Click OK.

62. Copyright © 2010 Pearson Education, Inc. 15-62 One Sample : Z Test Note that if the population standard deviation was assumed to be known, rather than estimated from the sample, a z test would be appropriate. Generally, the population standard deviation is unknown, so a t-test is more common.

63. Copyright © 2010 Pearson Education, Inc. 15-63 Two Independent Samples Means In the case of means for two independent samples, the hypotheses take the following form. H0: Users and nonusers of a brand are the same in terms of their perception of a brand. H1: Users and nonusers of a brand differ in terms of their perception of a brand. The two populations are sampled and the means and variances computed based on samples of sizes n1 and n2. If both populations are found to have the same variance, a pooled variance estimate is computed from the two sample variances as follows: 2 (( 21 11 2 2 2 2 1 1 2 12 ))       nn X X X X s nn ii ii or s 2 = (n 1 -1)s 1 2 +(n 2 -1)s 2 2 n 1 +n 2 -2

64. Copyright © 2010 Pearson Education, Inc. 15-64 Two Independent Samples Means The standard deviation of the test statistic can be estimated as: The appropriate value of t can be calculated as: The degrees of freedom in this case are (n 1 + n 2 -2).

65. Copyright © 2010 Pearson Education, Inc. 15-65 SPSS Windows: Two Independent Samples t Test 1. Select ANALYZE from the SPSS menu bar. 2. Click COMPARE MEANS and then INDEPENDENT SAMPLES T TEST. 3. Chose your TEST VARIABLE(S). E.g. perception of brand. 4. Chose your GROUPING VARIABLE. E.g. usage rate. 5. Click DEFINE GROUPS. 6. Type “1” in GROUP 1 box and “2” in GROUP 2 box. 7. Click CONTINUE. 8. Click OK.

66. Copyright © 2010 Pearson Education, Inc. 15-66 Two Independent Samples F Test An F test of sample variance may be performed if it is not known whether the two populations have equal variance. We will discuss F tests in more detail when we discuss ANOVA (analysis of variance) in chapter 16.

67. Copyright © 2010 Pearson Education, Inc. 15-67 Paired Samples Perhaps we want to see differences in variables from the same respondents. The difference in these cases is examined by a paired samples t test. To compute t for paired samples, the paired difference variable, denoted by D, is formed and its mean and variance calculated. Then the t statistic is computed. The degrees of freedom are n - 1, where n is the number of pairs. The relevant formulas are: There is no difference in perception of brand between users and nonusers of a brand.

68. Copyright © 2010 Pearson Education, Inc. 15-68 SPSS Windows: Paired Samples t Test 1. Select ANALYZE from the SPSS menu bar. 2. Click COMPARE MEANS and then PAIRED SAMPLES T TEST. 3. Select two potentially paired variables and move these variables in to the PAIRED VARIABLE(S) box. E.g. ‘perception of brand’ and ‘usage rate’. 4. Click OK.

69. Copyright © 2010 Pearson Education, Inc. 15-69 Paired-Samples t Test Based on mean values and p-values (less than.05), we can determine that a difference exists between respondents’ attitude towards internet and their attitude towards technology.

70. Copyright © 2010 Pearson Education, Inc. 15-70 Nonparametric Tests Nonparametric tests are used when the independent variables are nonmetric (e.g. nominal or ordinal). Like parametric tests, nonparametric tests are available for testing variables from one sample, two independent samples, or two related samples.

71. Copyright © 2010 Pearson Education, Inc. 15-71 Nonparametric Tests One Sample Nonparametric tests: used when the IVs are nonmetric. Like parametirc tests, nonparametric tests are available for testing variables from one sample, two independent samples, or two related samples. As we know, the chi-square test can be performed on a single variable from one sample. In this context, the chi- square serves as a goodness-of-fit test. It tests whether a significant difference exists between our observed cases and some expected outcome. For example, do the responses from our sample of internet users fit a normal distribution? The chi-square test could help answer this. The Kolmogorov-Smirnov (K-S) one-sample test, runs test and binomial test can also test one sample. PLEASE NOTE: Nonparametric data is considered inferior and therefore rarely gets published.

72. Copyright © 2010 Pearson Education, Inc. 15-72 A Summary of Hypothesis Tests Related to Differences Table 15.19 SampleApplicationLevel of ScalingTest/Comments One Sample DistributionsNonmetric K-S and chi-square for goodness of fit Runs test for randomness Binomial test for goodness of fit for dichotomous variables One SampleMeansMetricttest, if variance is unknown z test, if variance is known Proportion Metric Z test

73. Copyright © 2010 Pearson Education, Inc. 15-73 A Summary of Hypothesis Tests Related to Differences Table 15.19, cont.

74. Copyright © 2010 Pearson Education, Inc. 15-74 A Summary of Hypothesis Tests Related to Differences Table 15.19, cont.

75. Copyright © 2010 Pearson Education, Inc. 15-75 Questions?? Thank you!

76. Copyright © 2010 Pearson Education, Inc. 15-76 Appendix: SPSS Windows The main program in SPSS is FREQUENCIES. It produces a table of frequency counts, percentages, and cumulative percentages for the values of each variable. It gives all of the associated statistics. If the data are interval scaled and only the summary statistics are desired, the DESCRIPTIVES procedure can be used. The EXPLORE procedure produces summary statistics and graphical displays, either for all of the cases or separately for groups of cases. Mean, median, variance, standard deviation, minimum, maximum, and range are some of the statistics that can be calculated.

77. Copyright © 2010 Pearson Education, Inc. 15-77 SPSS Windows To select these procedures click: Analyze>Descriptive Statistics>Frequencies Analyze>Descriptive Statistics>Descriptives Analyze>Descriptive Statistics>Explore The major cross-tabulation program is CROSSTABS. This program will display the cross-classification tables and provide cell counts, row and column percentages, the chi-square test for significance, and all the measures of the strength of the association that have been discussed. To select these procedures, click: Analyze>Descriptive Statistics>Crosstabs

78. Copyright © 2010 Pearson Education, Inc. 15-78 SPSS Windows The major program for conducting parametric tests in SPSS is COMPARE MEANS. This program can be used to conduct t tests on one sample or independent or paired samples. To select these procedures using SPSS for Windows, click: Analyze>Compare Means>Means … Analyze>Compare Means>One-Sample T Test … Analyze>Compare Means>Independent-Samples T Test … Analyze>Compare Means>Paired-Samples T Test …

79. Copyright © 2010 Pearson Education, Inc. 15-79 SPSS Windows The nonparametric tests discussed in this chapter can be conducted using NONPARAMETRIC TESTS. To select these procedures using SPSS for Windows, click: Analyze>Nonparametric Tests>Chi-Square … Analyze>Nonparametric Tests>Binomial … Analyze>Nonparametric Tests>Runs … Analyze>Nonparametric Tests>1-Sample K-S … Analyze>Nonparametric Tests>2 Independent Samples … Analyze>Nonparametric Tests>2 Related Samples …

80. Copyright © 2010 Pearson Education, Inc. 15-80 SPSS Windows: Frequencies 1. Select ANALYZE on the SPSS menu bar. 2. Click DESCRIPTIVE STATISTICS and select FREQUENCIES. 3. Move the variable “Familiarity [familiar]” to the VARIABLE(s) box. 4. Click STATISTICS. 5. Select MEAN, MEDIAN, MODE, STD. DEVIATION, VARIANCE, and RANGE.

81. Copyright © 2010 Pearson Education, Inc. 15-81 SPSS Windows: Frequencies 6. Click CONTINUE. 7. Click CHARTS. 8. Click HISTOGRAMS, then click CONTINUE. 9. Click OK.

82. Copyright © 2010 Pearson Education, Inc. 15-82 SPSS Windows: Cross-tabulations 1. Select ANALYZE on the SPSS menu bar. 2. Click on DESCRIPTIVE STATISTICS and select CROSSTABS. 3. Move the variable “Internet Usage Group [iusagegr]” to the ROW(S) box. 4. Move the variable “Sex[sex]” to the COLUMN(S) box. 5. Click on CELLS. 6. Select OBSERVED under COUNTS and COLUMN under PERCENTAGES.

83. Copyright © 2010 Pearson Education, Inc. 15-83 SPSS Windows: Cross-tabulations 7. Click CONTINUE. 8. Click STATISTICS. 9. Click on CHI-SQUARE, PHI AND CRAMER’S V. 10. Click CONTINUE. 11. Click OK.

84. Copyright © 2010 Pearson Education, Inc. 15-84 SPSS Windows: One Sample t Test 1. Select ANALYZE from the SPSS menu bar. 2. Click COMPARE MEANS and then ONE SAMPLE T TEST. 3. Move “Familiarity [familiar]” in to the TEST VARIABLE(S) box. 4. Type “4” in the TEST VALUE box. 5. Click OK.

85. Copyright © 2010 Pearson Education, Inc. 15-85 SPSS Windows: Two Independent Samples t Test 1. Select ANALYZE from the SPSS menu bar. 2. Click COMPARE MEANS and then INDEPENDENT SAMPLES T TEST. 3. Move “Internet Usage Hrs/Week [iusage]” in to the TEST VARIABLE(S) box. 4. Move “Sex[sex]” to GROUPING VARIABLE box. 5. Click DEFINE GROUPS. 6. Type “1” in GROUP 1 box and “2” in GROUP 2 box. 7. Click CONTINUE. 8. Click OK.

86. Copyright © 2010 Pearson Education, Inc. 15-86 SPSS Windows: Paired Samples t Test 1. Select ANALYZE from the SPSS menu bar. 2. Click COMPARE MEANS and then PAIRED SAMPLES T TEST. 3. Select “Attitude toward Internet [iattitude]” and then select “Attitude toward technology [tattitude].” Move these variables in to the PAIRED VARIABLE(S) box. 4. Click OK.