Chapter 18 Comparison Copyright © 2011 Pearson Education, Inc 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 of 46 Copyright © 2011 Pearson Education, Inc 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 µA 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 of 46 Copyright © 2011 Pearson Education, Inc 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 pounds, on average State the hypotheses as: H0: µA - µC ≤ HA: µA - µC > of 46 Copyright © 2011 Pearson Education, Inc 18.1 Data for Comparisons Comparison of Two Diets Data used to compare two groups typically arise in one of three ways: Run an experiment that isolates a specific cause Obtain random samples from two populations Compare two sets of observations of 46 Copyright © 2011 Pearson Education, Inc 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 of 46 Copyright © 2011 Pearson Education, Inc 18.1 Data for Comparisons Experiments In the ideal experiment, the experimenter Selects a random sample from a population Assigns subjects at random to treatments defined by the factor Compares the response of subjects between treatments of 46 Copyright © 2011 Pearson Education, Inc 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) of 46 Copyright © 2011 Pearson Education, Inc 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 10 of 46 Copyright © 2011 Pearson Education, Inc 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 pE - 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) 32 of 46 Copyright © 2011 Pearson Education, Inc 4M Example 18.3: COLOR PREFERENCES Mechanics 33 of 46 Copyright © 2011 Pearson Education, Inc 4M Example 18.3: COLOR PREFERENCES Mechanics Based on the data, pˆ E pˆ W 0.5833 0.4444 0.1389 and the 95% confidence interval is 0.1389 ±1.96 (0.08645) [-0.031 to 0.308] 34 of 46 Copyright © 2011 Pearson Education, Inc 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 35 of 46 Copyright © 2011 Pearson Education, Inc 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 36 of 46 Copyright © 2011 Pearson Education, Inc 18.4 Other Comparisons Paired Comparisons Given paired data, we begin the analysis by forming the difference within each pair (i.e., di = xi – yi ) A two-sample analysis becomes a one-sample analysis Let d denote the mean of the differences and sd their standard deviation 37 of 46 Copyright © 2011 Pearson Education, Inc 18.4 Other Comparisons Paired Comparisons The 100(1 - α)% confidence paired t- interval is d t / ; n sd n with n-1 df Checklist: No obvious lurking variables SRS condition Sample size condition 38 of 46 Copyright © 2011 Pearson Education, Inc 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? 39 of 46 Copyright © 2011 Pearson Education, Inc 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 µA - àB 40 of 46 Copyright â 2011 Pearson Education, Inc 4M Example 18.4: SALES FORCE COMPARISON Method – Check Conditions Inspect histogram of differences: All conditions are satisfied 41 of 46 Copyright © 2011 Pearson Education, Inc 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 42 of 46 Copyright © 2011 Pearson Education, Inc 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 43 of 46 Copyright © 2011 Pearson Education, Inc Best Practices Use experiments to discover causal relationships Plot your data Use a break-even analysis to formulate the null hypothesis 44 of 46 Copyright © 2011 Pearson Education, Inc Best Practices (Continued) Use one confidence interval for comparisons Compare the variances in the two samples Take advantage of paired comparisons 45 of 46 Copyright © 2011 Pearson Education, Inc 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 46 of 46 Copyright © 2011 Pearson Education, Inc ... Education, Inc 18. 3 Confidence Interval for the Difference 95% Confidence Intervals for µA and µC 20 of 46 Copyright © 2011 Pearson Education, Inc 18. 3 Confidence Interval for the Difference... Use inferential statistics to test for differences between two populations Test for the difference between two means of 46 Copyright © 2011 Pearson Education, Inc 18. 1 Data for Comparisons Comparison... pulled records for a random sample of 50 offices that received the promotion and a random sample of 75 that did not 24 of 46 Copyright © 2011 Pearson Education, Inc 4M Example 18. 2: EVALUATING