2. Copyright © 2011 Pearson Education, Inc. Comparison Chapter 18

3. 18.1 Data for Comparisons A fitness chain is considering licensing a proprietary diet at a cost of $200,000. Is it more effective than the conventional free government recommended food pyramid?  Use inferential statistics to test for differences between two populations  Test for the difference between two means Copyright © 2011 Pearson Education, Inc. 3 of 46

4. 18.1 Data for Comparisons Comparison of Two Diets  Frame as a test of the difference between the means of two populations (mean number of pounds lost on Atkins versus conventional diets)  Let µ A denote the mean weight loss in the population if members go on the Atkins diet and µ C denote the mean weight loss in the population if members go on the conventional diet. Copyright © 2011 Pearson Education, Inc. 4 of 46

5. 18.1 Data for Comparisons Comparison of Two Diets  In order to be profitable for the fitness chain, the Atkins diet has to win by more than 2 pounds, on average.  State the hypotheses as: H 0 : µ A - µ C ≤ 2 H A : µ A - µ C > 2 Copyright © 2011 Pearson Education, Inc. 5 of 46

6. 18.1 Data for Comparisons Comparison of Two Diets  Data used to compare two groups typically arise in one of three ways: 1. Run an experiment that isolates a specific cause. 2. Obtain random samples from two populations. 3. Compare two sets of observations. Copyright © 2011 Pearson Education, Inc. 6 of 46

7. 18.1 Data for Comparisons Experiments  Experiment: procedure that uses randomization to produce data that reveal causation.  Factor: a variable manipulated to discover its effect on a second variable, the response.  Treatment: a level of a factor. Copyright © 2011 Pearson Education, Inc. 7 of 46

8. 18.1 Data for Comparisons Experiments  In the ideal experiment, the experimenter 1. Selects a random sample from a population. 2. Assigns subjects at random to treatments defined by the factor. 3. Compares the response of subjects between treatments. Copyright © 2011 Pearson Education, Inc. 8 of 46

9. 18.1 Data for Comparisons Comparison of Two Diets  The factor in the comparison of diets is the diet offered.  There are two treatments: Atkins and conventional.  The response is the amount of weight lost (measured in pounds). Copyright © 2011 Pearson Education, Inc. 9 of 46

10. 18.1 Data for Comparisons Confounding  Confounding: mixing the effects of two or more factors when comparing treatments.  Randomization eliminates confounding.  If it is not possible to randomize, then sample independently from two populations. Copyright © 2011 Pearson Education, Inc. 10 of 46

11. 18.2 Two-Sample t - Test Two-Sample t – Statistic with approximate degrees of freedom calculated using software. Copyright © 2011 Pearson Education, Inc. 11 of 46

12. 18.2 Two-Sample t - Test Two-Sample t – Test Summary Copyright © 2011 Pearson Education, Inc. 12 of 46

13. 18.2 Two-Sample t - Test Two-Sample t – Test Checklist  No obvious lurking variables.  SRS condition.  Similar variances. While the test allows the variances to be different, should notice if they are similar.  Sample size condition. Each sample must satisfy this condition. Copyright © 2011 Pearson Education, Inc. 13 of 46

14. 4M Example 18.1: COMPARING TWO DIETS Motivation Scientists at U Penn selected 63 subjects from the local population of obese adults. They randomly assigned 33 to the Atkins diet and 30 to the conventional diet. Do the results show that the Atkins diet is worth licensing? Copyright © 2011 Pearson Education, Inc. 14 of 46

15. 4M Example 18.1: COMPARING TWO DIETS Method Use the two-sample t-test with α = 0.05. The hypotheses are H 0 : µ A - µ C ≤ 2 H A : µ A - µ C > 2. Copyright © 2011 Pearson Education, Inc. 15 of 46

16. 4M Example 18.1: COMPARING TWO DIETS Method – Check Conditions Since the interquartile ranges of the boxplots appear similar, we can assume similar variances. Copyright © 2011 Pearson Education, Inc. 16 of 46

17. 4M Example 18.1: COMPARING TWO DIETS Method – Check Conditions  No obvious lurking variables because of randomization.  SRS condition satisfied.  Both samples meet the sample size condition. Copyright © 2011 Pearson Education, Inc. 17 of 46

18. 4M Example 18.1: COMPARING TWO DIETS Mechanics with 60.8255 df and p-value = 0.0308; reject H 0 Copyright © 2011 Pearson Education, Inc. 18 of 46

19. 4M Example 18.1: COMPARING TWO DIETS Message The experiment shows that the average weight loss of obese adults on the Atkins diet does exceed the average weight loss of obese adults on the conventional diet. Unless the fitness chain’s membership resembles this population (obese adults), these results may not apply. Copyright © 2011 Pearson Education, Inc. 19 of 46

20. 18.3 Confidence Interval for the Difference 95% Confidence Intervals for µ A and µ C Copyright © 2011 Pearson Education, Inc. 20 of 46

21. 18.3 Confidence Interval for the Difference 95% Confidence Intervals for µ A and µ C The confidence intervals overlap. If they were nonoverlapping, we could conclude a significant difference. However, this result is inconclusive. Copyright © 2011 Pearson Education, Inc. 21 of 46

22. 18.3 Confidence Interval for the Difference 95% Confidence Interval for µ 1 - µ 2 The 100(1 – α)% two-sample confidence interval for the difference in means is Copyright © 2011 Pearson Education, Inc. 22 of 46

23. 18.3 Confidence Interval for the Difference 95% Confidence Interval for µ A - µ c Since the 95% confidence interval for µ A - µ B does not include zero, the means are statistically significantly different (those on the Atkins diet lose on average between 1.7 and 15.2 pounds more than those on a conventional diet). Copyright © 2011 Pearson Education, Inc. 23 of 46

24. 4M Example 18.2: EVALUATING A PROMOTION Motivation To evaluate the effectiveness of a promotional offer, an overnight service pulled records for a random sample of 50 offices that received the promotion and a random sample of 75 that did not. Copyright © 2011 Pearson Education, Inc. 24 of 46

25. 4M Example 18.2: EVALUATING A PROMOTION Method Use the two-sample t –interval. Let µ yes denote the mean number of packages shipped by offices that received the promotion and µ no denote the mean number of packages shipped by offices that did not. Copyright © 2011 Pearson Education, Inc. 25 of 46

26. 4M Example 18.2: EVALUATING A PROMOTION Method – Check Conditions All conditions are satisfied with the exception of no obvious lurking variables. Since we don’t know how the overnight delivery service distributed the promotional offer, confounding is possible. For example, it could be the case that only larger offices received the promotion. Copyright © 2011 Pearson Education, Inc. 26 of 46

27. 4M Example 18.2: EVALUATING A PROMOTION Mechanics Copyright © 2011 Pearson Education, Inc. 27 of 46

28. 4M Example 18.2: EVALUATING A PROMOTION Message The difference is statistically significant. Offices that received the promotion used the overnight service to ship from 4 to 21 more packages on average than those offices that did not receive the promotion. There is the possibility of a confounding effect. Copyright © 2011 Pearson Education, Inc. 28 of 46

29. 18.4 Other Comparisons Comparisons Using Confidence Intervals  Other possible comparisons include comparing two proportions or comparing two means from paired data.  95% confidence intervals generally have the form Estimated Difference ± 2 Estimated Standard Error of the Difference Copyright © 2011 Pearson Education, Inc. 29 of 46

30. 18.4 Other Comparisons Comparing Proportions The 100(1 – α)% confidence z-interval for p 1 - p 2 is. Checklist:No obvious lurking variables. SRS condition. Sample size condition (for proportion). Copyright © 2011 Pearson Education, Inc. 30 of 46

31. 4M Example 18.3: COLOR PREFERENCES Motivation A department store sampled customers from the east and west and each was shown designs for the coming fall season (one featuring red and the other violet). If customers in the two regions differ in their preferences, the buyer will have to do a special order for each district. Copyright © 2011 Pearson Education, Inc. 31 of 46

32. 4M Example 18.3: COLOR PREFERENCES Method Data were collected on a random sample of 60 customers from the east and 72 from the west. Construct a 95% confidence interval for p E - p W. SRS and sample size conditions are satisfied. However, can’t rule out a lurking variable (e.g., customers may be younger in the west compared to the east). Copyright © 2011 Pearson Education, Inc. 32 of 46

33. 4M Example 18.3: COLOR PREFERENCES Mechanics Copyright © 2011 Pearson Education, Inc. 33 of 46

34. 4M Example 18.3: COLOR PREFERENCES Mechanics Based on the data, and the 95% confidence interval is 0.1389 ±1.96 (0.08645) [-0.031 to 0.308] Copyright © 2011 Pearson Education, Inc. 34 of 46

35. 4M Example 18.3: COLOR PREFERENCES Message There is no statistically significant difference between customers from the east and those from the west in their preferences for the two designs. The 95% confidence interval for the difference between proportions contains zero. Copyright © 2011 Pearson Education, Inc. 35 of 46

36. 18.4 Other Comparisons Paired Comparisons  Paired comparison: a comparison of two treatments using dependent samples designed to be similar (e.g., the same individuals taste test Coke and Pepsi).  Pairing isolates the treatment effect by reducing random variation that can hide a difference. Copyright © 2011 Pearson Education, Inc. 36 of 46

37. 18.4 Other Comparisons Paired Comparisons  Given paired data, we begin the analysis by forming the difference within each pair (i.e., d i = x i – y i ).  A two-sample analysis becomes a one-sample analysis. Let denote the mean of the differences and s d their standard deviation. Copyright © 2011 Pearson Education, Inc. 37 of 46

38. 18.4 Other Comparisons Paired Comparisons The 100(1 - α)% confidence paired t- interval is with n-1 df Checklist:No obvious lurking variables. SRS condition. Sample size condition. Copyright © 2011 Pearson Education, Inc. 38 of 46

39. 4M Example 18.4: SALES FORCE COMPARISON Motivation The merger of two pharmaceutical companies (A and B) allows senior management to eliminate one of the sales forces. Which one should the merged company eliminate? Copyright © 2011 Pearson Education, Inc. 39 of 46

40. 4M Example 18.4: SALES FORCE COMPARISON Method Both sales forces market similar products and were organized into 20 comparable geographical districts. Use the differences obtained from subtracting sales for Division B from sales for Division A in each district to obtain a 95% confidence t-interval for µ A - µ B. Copyright © 2011 Pearson Education, Inc. 40 of 46

41. 4M Example 18.4: SALES FORCE COMPARISON Method – Check Conditions Inspect histogram of differences: All conditions are satisfied. Copyright © 2011 Pearson Education, Inc. 41 of 46

42. 4M Example 18.4: SALES FORCE COMPARISON Mechanics The 95% t-interval for the mean differences does not include zero. There is a statistically significant difference. Copyright © 2011 Pearson Education, Inc. 42 of 46

43. 4M Example 18.4: SALES FORCE COMPARISON Message On average, sales force B sells more per day than sales force A. The high correlation (r = 0.97) of sales between Sales Force A and Sales Force B in these districts confirms the benefit of a paired comparison. Copyright © 2011 Pearson Education, Inc. 43 of 46

44. Best Practices  Use experiments to discover causal relationships.  Plot your data.  Use a break-even analysis to formulate the null hypothesis. Copyright © 2011 Pearson Education, Inc. 44 of 46

45. Best Practices (Continued)  Use one confidence interval for comparisons.  Compare the variances in the two samples.  Take advantage of paired comparisons. Copyright © 2011 Pearson Education, Inc. 45 of 46

46. Pitfalls  Don’t forget confounding.  Do not assume that a confidence interval that includes zero means that the difference is zero.  Don’t confuse a two-sample comparison with a paired comparison.  Don’t think that equal sample sizes imply paired data. Copyright © 2011 Pearson Education, Inc. 46 of 46