Copyright © 2014, 2011 Pearson Education, Inc. 1 Chapter 17 Comparison

Copyright © 2014, 2011 Pearson Education, Inc 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

Copyright © 2014, 2011 Pearson Education, Inc Data for Comparisons Comparison of Two Diets  Frame as a test of the difference between the proportions of two populations.  Let p A denote the population proportion on the Atkins diet who renew membership and p C denote the proportion on the conventional diet who renew membership.

Copyright © 2014, 2011 Pearson Education, Inc Data for Comparisons Comparison of Two Diets  The difference p A - p C measures the extra proportion who renew if on the Atkins diet.  To be profitable this difference must be more than 4%. H 0 : p A - p C ≤ 0.04 H A : p A - p C > 0.04

Copyright © 2014, 2011 Pearson Education, Inc Data for Comparisons Comparison of Two Diets Data used to compare two groups arise from: 1. Run an experiment that isolates a specific cause. 2. Obtain random samples from two populations. 3. Compare two sets of observations. Method 3 usually not reliable.

Copyright © 2014, 2011 Pearson Education, Inc 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 © 2014, 2011 Pearson Education, Inc 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 © 2014, 2011 Pearson Education, Inc Data for Comparisons Comparison of Two Diets  The factor in the comparison of diets is the diet offered.  It has two levels: Atkins and conventional.  The response is whether dieters renew membership in the fitness center.

Copyright © 2014, 2011 Pearson Education, Inc 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 © 2014, 2011 Pearson Education, Inc Two-Sample z-Test for Proportions Two-Sample z – Statistic using the estimated standard error of the difference between the sample proportions.

Copyright © 2014, 2011 Pearson Education, Inc Two-Sample z-Test for Proportions Two-Sample z – Test Summary

Copyright © 2014, 2011 Pearson Education, Inc Two-Sample z-Test for Proportions Two-Sample z – Test Checklist  No obvious lurking variables.  SRS condition.  Sample size condition. Observe at least 10 “successes” and 10 “failures in each sample: and

Copyright © 2014, 2011 Pearson Education, Inc Two-Sample z-Test for Proportions Comparison of Two Diets The p-value is Cannot reject H 0 at α = 0.05; can reject at α = 0.01.

Copyright © 2014, 2011 Pearson Education, Inc Two Sample Confidence Interval for Proportions Summary Statistics – Diet Comparison

Copyright © 2014, 2011 Pearson Education, Inc Two-Sample Confidence Interval for Proportions Summary Statistics – Diet Comparison Two 95% confidence intervals (one for each sample in the diet comparison) overlap indicating no significant difference.

Copyright © 2014, 2011 Pearson Education, Inc Two-Sample Confidence Interval for Proportions The two sample 100(1 – α)% confidence interval for p 1 - p 2 is. Checklist:No obvious lurking variables. SRS condition. Sample size condition (for proportion).

Copyright © 2014, 2011 Pearson Education, Inc Two-Sample Confidence Interval for Proportions Comparison of Two Diets The 95% confidence interval for the difference between the proportions who renew on the Atkins and conventional diets is between and 0.217; it does not contain zero.

Copyright © 2014, 2011 Pearson Education, Inc Two-Sample Confidence Interval for Proportions Interpreting the Confidence Interval  When the 95% confidence interval does not include zero, we say that the two proportions are statistically significantly different from each other.  Members on the Atkins diet renew memberships at a statistically significantly higher rate than those on the conventional diet.

Copyright © 2014, 2011 Pearson Education, Inc. 19 4M Example 17.1: 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 © 2014, 2011 Pearson Education, Inc. 20 4M Example 17.1: 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 © 2014, 2011 Pearson Education, Inc. 21 4M Example 17.1: COLOR PREFERENCES Mechanics

Copyright © 2014, 2011 Pearson Education, Inc. 22 4M Example 17.1: COLOR PREFERENCES Mechanics Based on the data, and the 95% confidence interval is ±1.96 ( ) [ to 0.308]

Copyright © 2014, 2011 Pearson Education, Inc. 23 4M Example 17.1: 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. Although the 95% confidence interval for the difference between proportions contains zero, before ordering the same lineup for both regions, consider gathering larger samples to narrow the confidence interval.

Copyright © 2014, 2011 Pearson Education, Inc Two-Sample t - Test 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 © 2014, 2011 Pearson Education, Inc Two-Sample t - Test Comparison of Two Diets  The null hypothesis specifies that the difference between population means is less than or equal to a predetermined constant D 0.  To compare diets D 0 = 5 pounds H 0 : µ A - µ C ≤ 5 H A : µ A - µ C > 5

Copyright © 2014, 2011 Pearson Education, Inc Two-Sample t - Test Two-Sample t – Statistic with approximate degrees of freedom calculated using software.

Copyright © 2014, 2011 Pearson Education, Inc Two-Sample t - Test Two-Sample t – Test Summary

Copyright © 2014, 2011 Pearson Education, Inc 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 © 2014, 2011 Pearson Education, Inc. 29 4M Example 17.2: 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 at α = 0.05 that the Atkins diet is worth the extra effort and produces 5 more pounds of weight loss?

Copyright © 2014, 2011 Pearson Education, Inc. 30 4M Example 17.2: COMPARING TWO DIETS Method Use the two-sample t-test with α = The hypotheses are H 0 : µ A - µ C ≤ 5 H A : µ A - µ C > 5

Copyright © 2014, 2011 Pearson Education, Inc. 31 4M Example 17.2: COMPARING TWO DIETS Method – Check Conditions Since the interquartile ranges of the boxplots appear similar, we can assume similar variances.

Copyright © 2014, 2011 Pearson Education, Inc. 32 4M Example 17.2: COMPARING TWO DIETS Method – Check Conditions  No obvious lurking variables because of randomization.  SRS condition satisfied.  Both samples meet the sample size condition.

Copyright © 2014, 2011 Pearson Education, Inc. 33 4M Example 17.2: COMPARING TWO DIETS Mechanics with df, p-value = 1572; cannot reject H 0

Copyright © 2014, 2011 Pearson Education, Inc. 34 4M Example 17.2: COMPARING TWO DIETS Message The experiment shows that the average weight loss of obese adults on the Atkins diet exceeds the average weight loss of obese adults on the conventional diet by 5 pounds. The difference is not statistically significant. Unless the fitness chain’s membership resembles this population (obese adults), these results may not apply.

Copyright © 2014, 2011 Pearson Education, Inc Confidence Interval for the Difference between Means Summary Statistics – Diet Comparison

Copyright © 2014, 2011 Pearson Education, Inc Confidence Interval for the Difference between Means Summary Statistics – Diet Comparison The confidence intervals overlap. If they were nonoverlapping, we could conclude a significant difference. However, this result is inconclusive.

Copyright © 2014, 2011 Pearson Education, Inc Confidence Interval for the Difference between Means The 100(1 – α)% confidence t-interval for µ 1 - µ 2 is. Checklist:No obvious lurking variables. SRS condition. Similar variances. Sample size condition.

Copyright © 2014, 2011 Pearson Education, Inc Confidence Interval for the Difference between Means 95% Confidence Interval for µ A - µ c Since the 95% confidence interval for µ A - µ C 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 © 2014, 2011 Pearson Education, Inc. 39 4M Example 17.3: 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 © 2014, 2011 Pearson Education, Inc. 40 4M Example 17.3: 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 © 2014, 2011 Pearson Education, Inc. 41 4M Example 17.3: EVALUATING A PROMOTION Method – Check Conditions

Copyright © 2014, 2011 Pearson Education, Inc. 42 4M Example 17.3: 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 © 2014, 2011 Pearson Education, Inc. 43 4M Example 17.3: EVALUATING A PROMOTION Mechanics

Copyright © 2014, 2011 Pearson Education, Inc. 44 4M Example 17.3: 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 © 2014, 2011 Pearson Education, Inc 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.  Randomization remains relevant.

Copyright © 2014, 2011 Pearson Education, Inc Paired 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 © 2014, 2011 Pearson Education, Inc 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 © 2014, 2011 Pearson Education, Inc. 48 4M Example 17.4: 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 © 2014, 2011 Pearson Education, Inc. 49 4M Example 17.4: 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 © 2014, 2011 Pearson Education, Inc. 50 4M Example 17.4: SALES FORCE COMPARISON Method – Check Conditions Inspect histogram of differences: All conditions are satisfied.

Copyright © 2014, 2011 Pearson Education, Inc. 51 4M Example 17.4: SALES FORCE COMPARISON Mechanics The 95% t-interval for the mean differences does not include zero. There is a statistically significant difference.

Copyright © 2014, 2011 Pearson Education, Inc. 52 4M Example 17.4: SALES FORCE COMPARISON Mechanics The benefit of paired comparison; sales in these districts are highly correlated (r = 0.97).

Copyright © 2014, 2011 Pearson Education, Inc. 53 4M Example 17.4: SALES FORCE COMPARISON Message On average, sales force B sells more per day than sales force A. By comparing sales per representative, head to head in each district, a statistically significant difference in performance is detected.

Copyright © 2014, 2011 Pearson Education, Inc. 54 Best Practices  Use experiments to discover causal relationships.  Plot your data.  Use a break-even analysis to formulate the null hypothesis.

Copyright © 2014, 2011 Pearson Education, Inc. 55 Best Practices (Continued)  Use one confidence interval for comparisons.  Compare the variances in the two samples.  Take advantage of paired comparisons.

Copyright © 2014, 2011 Pearson Education, Inc. 56 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.