1. Example 10.4 Measuring the Effects of Traditional and New Styles of Soft-Drink Cans Hypothesis Test for Other Parameters

2. 10.110.1 | 10.1a | 10.2 | 10.3 | 10.5 | 10.6 | 10.7 | 10.8 | 10.910.1a10.210.310.510.610.710.810.9 Objective To use paired-sample t tests for differences between means to see whether consumers rate the attractiveness, and their likelihood to purchase, higher for a new-style can than for the traditional-style can.

3. 10.110.1 | 10.1a | 10.2 | 10.3 | 10.5 | 10.6 | 10.7 | 10.8 | 10.910.1a10.210.310.510.610.710.810.9 Background Information n Beer and soft-drink companies have recently become very concerned about the style of their cans. n There are cans with fluted and embossed sides and cans with six-color graphics and holograms. n Coca-Cola is even experimenting with a contoured can, shaped like the old fashioned Coke bottle minus the neck. n Evidently, these companies believe the styles of the can make a difference to consumers, which presumably translates into higher sales

4. 10.110.1 | 10.1a | 10.2 | 10.3 | 10.5 | 10.6 | 10.7 | 10.8 | 10.910.1a10.210.310.510.610.710.810.9 Background Information -- continued n Assume that a soft-drink company is considering a style change to its current can, which has been the company’s trademark for many years. n To determine whether this new style is popular with consumers, the company runs a number of focus group sessions around the country. n At each of these sessions, randomly selected consumers are allowed to examine the new and traditional styles, exchange ideas, and offer their opinions.

5. 10.110.1 | 10.1a | 10.2 | 10.3 | 10.5 | 10.6 | 10.7 | 10.8 | 10.910.1a10.210.310.510.610.710.810.9 Background Information -- continued n Eventually, they fill out a form where, among other items, they are asked to respond to the following items, each on a 1 to 7 scale with 7 being the best: –Rate the attractiveness of the traditional-style can. –Rate the attractiveness of the new-style can. –Rate the likelihood that you would buy the product with the traditional-style can. –Rate the likelihood that you would buy the product with the new-style can. n The results over all focus groups are shown on the next slide.

6. 10.110.1 | 10.1a | 10.2 | 10.3 | 10.5 | 10.6 | 10.7 | 10.8 | 10.910.1a10.210.310.510.610.710.810.9 CANS.XLS n This file contains the focus group data. n What can the company conclude from these data? Are hypothesis tests appropriate?

7. 10.110.1 | 10.1a | 10.2 | 10.3 | 10.5 | 10.6 | 10.7 | 10.8 | 10.910.1a10.210.310.510.610.710.810.9 Solution n First, it is a good idea to examine summary statistics for the data. –The averages indicate some support for the new-style can. –Also we might expect the ratings for a given consumer to be correlated. This turn out to be the case as shown by the relatively large correlations in the table of correlations. The large positive correlations indicate that if we want to examine differences between survey items, a paired-sample procedure will make the most efficient use of the data.

8. 10.110.1 | 10.1a | 10.2 | 10.3 | 10.5 | 10.6 | 10.7 | 10.8 | 10.910.1a10.210.310.510.610.710.810.9 Solution -- continued n Of course, a paired-procedure also makes sense because each consumer answers each item on the form. n There are several differences of interest. The two most obvious are – the difference between the attractiveness ratings of the two styles - that is, column B minus column C and –the differences between the likelihoods of buying the two styles - that is, column D minus Column E.

9. 10.110.1 | 10.1a | 10.2 | 10.3 | 10.5 | 10.6 | 10.7 | 10.8 | 10.910.1a10.210.310.510.610.710.810.9 Solution -- continued n A third difference of interest is –the difference between the attractiveness ratings of the new style and the likelihoods of buying the new can - this is column C minus Column E. n Finally, a fourth difference that might be of interest is –the difference between the third difference (column C minus column E) and the similar difference for the old style (column B minus column D). n All of these differences appear next to the original data in the following figure.

10. 10.110.1 | 10.1a | 10.2 | 10.3 | 10.5 | 10.6 | 10.7 | 10.8 | 10.910.1a10.210.310.510.610.710.810.9 Original and Difference Variables for Soft Drink Data

11. 10.110.1 | 10.1a | 10.2 | 10.3 | 10.5 | 10.6 | 10.7 | 10.8 | 10.910.1a10.210.310.510.610.710.810.9 Solution -- continued n The differences are defined as: –Diff1: Column B - Column C –Diff2: Column D - Column E –Diff 3: Column C - Column E –Diff4: Column B - Column D –Diff5: Column H - I n We generate all of them automatically with StatPro/Statistical Inference/Paired Sample Analysis menu item.

12. 10.110.1 | 10.1a | 10.2 | 10.3 | 10.5 | 10.6 | 10.7 | 10.8 | 10.910.1a10.210.310.510.610.710.810.9 Solution -- continued n For each of the differences, we test the mean difference over all potential consumers with a paired- sample analysis. n We treat each difference variable as a single sample and run the same t test. n In each case the hypothesized difference D 0 is 0. n The only question is whether to run one-tailed or two- tailed tests. We propose that Diff1,Diff2, Diff5 tests be two-tailed and that the test on Diff3 be a one-tailed test with the alternative of the “greater than” variety.

13. 10.110.1 | 10.1a | 10.2 | 10.3 | 10.5 | 10.6 | 10.7 | 10.8 | 10.910.1a10.210.310.510.610.710.810.9 Solution -- continued n The reasoning is that the company has no idea which way the differences (Diff1, Diff2, and Diff5) will go, whereas it expects that Diff3 will be positive on average. n That is, it expects that consumers’ ratings of the attractiveness of the new design will, on average, be larger than their likelihoods of purchasing the product. n Any of these tests could be either one- or two- tailed.

14. 10.110.1 | 10.1a | 10.2 | 10.3 | 10.5 | 10.6 | 10.7 | 10.8 | 10.910.1a10.210.310.510.610.710.810.9 Results n The results from the four tests appear in the following charts.

15. 10.110.1 | 10.1a | 10.2 | 10.3 | 10.5 | 10.6 | 10.7 | 10.8 | 10.910.1a10.210.310.510.610.710.810.9 Results -- continued n These outputs also include 99% confidence intervals for the corresponding mean differences.

16. 10.110.1 | 10.1a | 10.2 | 10.3 | 10.5 | 10.6 | 10.7 | 10.8 | 10.910.1a10.210.310.510.610.710.810.9 Results -- continued n The results can be summarized as follows. –From the output for the Diff1 variable there is overwhelming evidence that consumers, on average, rate the attractiveness of the new design higher than the attractiveness of the current design and the corresponding p- value for a two-tailed test of the mean difference is 0.000. A 99% confidence interval for the mean difference extends from -0.801 to -0.277. The 99% confidence interval does not include the hypothesized value 0.

17. 10.110.1 | 10.1a | 10.2 | 10.3 | 10.5 | 10.6 | 10.7 | 10.8 | 10.910.1a10.210.310.510.610.710.810.9 Results -- continued –The results are basically the same fore the difference between consumers’ likelihoods of buying the product with the two styles. Again, consumers are definitely more likely, on average, to buy the product with the new-style can. A 99% confidence interval for the mean difference extends from -0.739 to -0.216. –The company’s hypothesis that consumers’ ratings of attractiveness ratings and the likelihoods of buying the product with this style can is confirmed. The test statistics for this one-tailed test is 3.705 and the corresponding p-value is 0.000. A 99% confidence interval for the mean difference extends from 0.182 to 1.041.

18. 10.110.1 | 10.1a | 10.2 | 10.3 | 10.5 | 10.6 | 10.7 | 10.8 | 10.910.1a10.210.310.510.610.710.810.9 Results -- continued –There is no evidence that the difference between attractiveness ratings and the likelihood of buying is any different for the new-style can than for the current-style can. The test statistic for a two-tailed test of this difference is 0.401 and the corresponding p-value, 0.689, isn’t close to any of the traditional significance levels. A 99% confidence interval for the mean difference extends from -0.336 to 0.458.

19. 10.110.1 | 10.1a | 10.2 | 10.3 | 10.5 | 10.6 | 10.7 | 10.8 | 10.910.1a10.210.310.510.610.710.810.9 Results -- continued n These results are further confirmed by these histograms of the difference variables.

20. 10.110.1 | 10.1a | 10.2 | 10.3 | 10.5 | 10.6 | 10.7 | 10.8 | 10.910.1a10.210.310.510.610.710.810.9 Results -- continued

21. 10.110.1 | 10.1a | 10.2 | 10.3 | 10.5 | 10.6 | 10.7 | 10.8 | 10.910.1a10.210.310.510.610.710.810.9 Results -- continued n This example illustrates once again how hypothesis tests and confidence intervals provide complementary information, although the confidence intervals are arguably the more useful of the two here. n We conclude this example by recalling the distinction between practical significance and statistical significance. –Due to the extremely low p-values, the results leave no doubt as to statistical significance.

22. 10.110.1 | 10.1a | 10.2 | 10.3 | 10.5 | 10.6 | 10.7 | 10.8 | 10.910.1a10.210.310.510.610.710.810.9 Results -- continued –The soft-drink company, on the other hand, is more interested in knowing whether the observed differences are of any practical importance. It is a question of what differences are important for the business. n We suspect that the company would indeed be quite impressed with the observed differences in the sample - and might very well switch to the new-style can.