1. Learning Objectives Copyright © 2002 South-Western/Thomson Learning Statistical Testing of Differences CHAPTER fifteen

2. Learning Objectives 1. To become aware of the nature of statistical significance. 2. To understand the concept of hypothesis development and how to test hypotheses. 3. To understand the differences between Type I and Type II errors. 4. To be familiar with several of the more common statistical tests of goodness of fit, hypotheses about one mean, hypothesis about two means, and hypotheses about proportions. 5.To learn about analysis of variance.

3. Learning Objectives Are certain measures different from one another? For example: Did top-of-mind awareness really increase? Did customer satisfaction really increase? To become aware of the nature of statistical differences. Differences and Changes

4. Learning Objectives Statistical Significance It is possible for numbers to be different in a mathematical sense but not statistically different in a statistical sense. Mathematical differences Statistical significance Managerially important differences To become aware of the nature of statistical differences.

5. Learning Objectives Hypothesis An assumption that a researcher makes about some characteristic of the population under study. Steps in Hypothesis Testing Step One: Stating the Hypothesis Null hypothesis: H o Alternative hypothesis: H a Step Two: Choosing the Appropriate Test Statistic To understand the concept of hypothesis development and how to test hypotheses. Hypothesis Testing

6. Learning Objectives Step Three: Developing a Decision Rule Step Four: Calculating the Value of the Test Statistic Use the appropriate formula Compare calculated value to the critical value. State the result in terms of: rejecting the null hypothesis failing to reject the null hypothesis Step Five: Stating the Conclusion To understand the concept of hypothesis development and how to test hypotheses. Hypothesis Testing

7. Learning Objectives Types of Errors in Hypothesis Testing Type I Error Rejection of the null hypothesis when, in fact, it is true. Type II Error Acceptance of the null hypothesis when, in fact, it is false. Accepting H o or Failing to Reject H o ? One-Tailed Test or Two-Tailed Test? To understand the differences between Type I and Type II errors. Other issues

8. Learning Objectives Table 15.2 Type I and Type II Errors Actual State of the Null Hypothesis Fail to Reject H o Reject H o H o is true H o is false Correct (1-  ) no error Type II error (  ) Type I error (  ) Correct (1-  ) no error Table 15.2

9. Learning Objectives Independent Versus Related Samples Independent samples Measurement of a variable in one population has no effect on the measurement of the other variable Related Samples Measurement of a variable in one population may influence the measurement of the other variable. Degrees of Freedom The number of observations minus the number of constraints. To understand the concept of hypothesis development and testing a hypothesis. Commonly Used Statistical Hypothesis Tests

10. Learning Objectives Chi-Square To determine whether an observed pattern of frequencies corresponds to an “expected” pattern. Chi-Square Test of a Single Sample 1. Specify the null and alternative hypotheses. 2. Determine the number of visitors that would be expected in each category if the null hypotheses were correct (E i ). To describe statistical tests of goodness of fit. Goodness of Fit

11. Learning Objectives 3. Calculate the X 2 value using the formula: To describe statistical tests of goodness of fit. O i = observed number in i th category E i = expected number in i th category k = number of categories X2X2  k I = 1 EiEi (O i - E i ) 2 = Goodness of Fit

12. Learning Objectives 4. Select the level of significance . 5. Compare the calculated X 2 value with the table value. State the result. To describe statistical tests of goodness of fit. Chi-Square Test of Two Independent Samples Association between two or more variables. 1. State the null and alternative hypotheses. Goodness of Fit

13. Learning Objectives To describe statistical tests of goodness of fit. 2. Place the observed (sample) frequencies in a k x r table using the k columns for the sample groups and the r rows for the conditions or treatments. Calculate the sum of each row and each column. Record marginal totals. Calculate total for entire table. 3. Determine the expected frequency for each cell in the contingency table by calculating the product of the two marginal totals common to that cell and dividing that value by N. Goodness of Fit

14. Learning Objectives To describe statistical tests of goodness of fit. 4. Calculate the value of X 2 using: X2X2  k j = 1 E ij (O ij - E ij ) 2 =  r I = 1 O ij = observed number in i th row of the jth column E ij = expected number in i th row of the jth column 5. Compare the calculated X 2 with the tabular X 2. State the result. Goodness of Fit

15. Learning Objectives Kolmogorov-Smirnov Test 1. Specify the null and alternative hypotheses. 2. Establish the cumulative frequency distribution expected under the null hypothesis. 3. Calculate the observed cumulative frequency distribution from the sample. 4. Select the level of significance . 5. Determine the K-S statistic. 6. State the result. To describe statistical tests of goodness of fit. Goodness of Fit

16. Learning Objectives Z - Test Hypothesis tests about a single mean if the sample mean if the sample is large enough and drawn from a normal population. 1. Specify the null and alternative hypotheses. 2. Specify the level of sampling error (  ) allowed. 3. Determine the sample standard deviation. 4. Calculate the estimated standard error of the mean: To describe statistical tests about one mean. About One Mean

17. Learning Objectives SxSx = S n√ where S x = the estimated standard error of the mean. To describe statistical tests about one mean. About One Mean

18. Learning Objectives About One Mean 5. Calculate the test statistic: (sample mean) population mean under the null hypothesis estimated standard error of the mean Z = To describe statistical tests about one mean. 6. State the result.

19. Learning Objectives t - Test Hypothesis tests about a single mean if the sample mean if the sample is too small to use the Z - test 1. Specify the null and alternative hypotheses. 2. Specify the level of sampling error (  ) allowed. 3. Determine the sample standard deviation. To describe statistical tests about one mean. About One Mean

20. Learning Objectives S  n I = 1 n - 1 (X i - X) 2 = √ where X i = observed sales per week in the ith store X = average sales per week n = the number of stores To describe statistical tests about one mean. About One Mean

21. Learning Objectives 4. Calculate the estimated standard error of the mean using: To describe statistical tests about one mean. About One Mean SxSx = S n√

22. Learning Objectives 5. Calculate the t - statistic: (sample mean) population mean under the null hypothesis estimated standard error of the mean t = To describe statistical tests about one mean. About One Mean 6. State the result.

23. Learning Objectives Marketers are frequently interested in testing differences between groups. 1. The null and alternative hypotheses 2. Set the level of sampling error (  ) 3. Estimate the standard error of the differences between two means: About Two Means To describe statistical tests about two means.

24. Learning Objectives S x m-ƒ = S2mS2m S 2 ƒ nmnm n ƒ + √ S m = estimated standard deviation of population m (men) S f = estimated standard deviation of population f (women) n m = sample size for sample m n f = sample size for sample f To describe statistical tests about two means. About Two Means

25. Learning Objectives 4. Calculate the test statistic. Z is calculated as: difference between means of first sample and second sample difference between means under the null hypothesis standard error or the differences between the means Z = 5. Compare the calculated value of Z with the critical value. State the result. To describe statistical tests about two means. About Two Means

26. Learning Objectives Test of a Proportion, One Sample Phenomena expressed as percentages 1. Specify the null and alternative hypotheses 2. Set the level of sampling error (  ) 3. Estimate the standard error using the P-value specified in the null hypothesis: About Proportions To describe statistical tests about proportions.

27. Learning Objectives SpSp P(1- P) n - 1 = √ where: P = proportion specified in the null hypothesis n = sample size To describe statistical tests about proportions. About Proportions

28. Learning Objectives (observed proportion - proportion under null hypothesis) estimated standard error ( S p ) Compare the calculated Z-value with the critical Z-value. Z = To describe statistical tests about proportions. About Proportions 4. Calculate the test statistic as follows:

29. Learning Objectives Tests of Differences Between Two Proportions, Independent Samples Management may be interested in the difference between the proportions of people in two different groups that engage in a certain activity or have a certain characteristic. 1. Specify the null and alternative hypotheses 2. Set the level of sampling error (  ) 3. Estimate the standard error of the differences between the two proportions: To describe statistical tests about proportions. About Proportions

30. Learning Objectives To describe statistical tests about proportions. About Proportions where P = n m P m + n f P f n m + n f P m = proportion in sample m (men) P f = proportion in sample f (women) n m = sample of size m n f = sample of size f S pm-ƒ = 1 nmnm n ƒ + P(1- P) 1 √

31. Learning Objectives 4. Calculate the test statistic: difference between observed proportions difference between proportions under the null hypothesis estimated standard error or the differences between the two means Z = To describe statistical tests about proportions. About Proportions 5. Compare the calculated Z-value with the critical Z-value. State the result.

32. Learning Objectives Analysis of Variance Analysis of Variance (ANOVA) Test for the differences among the means of two or more variables. 1. Specify the null and alternative hypotheses 2. Calculate sum of squares among groups (SSA) SSA  j= 1 njnj (X i - X t ) 2 = To learn about analysis of variance.

33. Learning Objectives 3. Calculate the variation among the group means as measured by the mean sum of squares among groups (MSA): sum of squares among groups (SSA) degrees of freedom (d.f.) MSA = To learn about analysis of variance. Analysis of Variance where Degrees of freedom = number of groups (C) -1

34. Learning Objectives 4. Calculate the sum of squares within groups or sum of squared error (SSE): SSE  I = 1 = (X ij - X j ) 2  j = 1 c n j 5. Calculate the mean square error (MSE): sum of squares within groups (SSE) degrees of freedom (d.f.) MSE = To learn about analysis of variance. Analysis of Variance

35. Learning Objectives 6. Calculate the F - statistic as follows: F = MSA MSE To learn about analysis of variance. Analysis of Variance p - VALUES AND SIGNIFICANCE TESTING P- Value The exact probability of getting a computed test statistic that was largely due to chance. The smaller the p-value, the smaller the probability that the observed result occurred by chance. 7. State the result.

36. Learning Objectives Differences and Changes Statistical Significance Hypothesis Testing Other issues Hypothesis Tests About One Mean About Two Means About Proportions Analysis of Variance Goodness of Fit SUMMARY

37. Learning Objectives The End Copyright © 2002 South-Western/Thomson Learning