2. Copyright © 2009 Pearson Education, Inc. Chapter 25 Paired Samples and Blocks

3. Copyright © 2009 Pearson Education, Inc. Slide 1- 3 Objectives: The student will be able to: Find and interpret in context a paired confidence interval, checking the necessary assumptions. Perform a paired t-test to include: writing appropriate hypotheses, checking the necessary assumptions, drawing an appropriate diagram, computing the P-value, making a decision, and interpreting the results in the context of the problem.

4. Copyright © 2009 Pearson Education, Inc. Slide 1- 4 Paired Data Data are paired when the observations are collected in pairs or the observations in one group are naturally related to observations in the other group. Paired data arise in a number of ways. Perhaps the most common is to compare subjects with themselves before and after a treatment. When pairs arise from an experiment, the pairing is a type of blocking. When they arise from an observational study, it is a form of matching.

5. Copyright © 2009 Pearson Education, Inc. Slide 1- 5 Paired Data (cont.) If you know the data are paired, you can (and must!) take advantage of it. To decide if the data are paired, consider how they were collected and what they mean (check the W’s). There is no test to determine whether the data are paired. Once we know the data are paired, we can examine the pairwise differences. Because it is the differences we care about, we treat them as if they were the data and ignore the original two sets of data.

6. Copyright © 2009 Pearson Education, Inc. Slide 1- 6 Paired Data (cont.) Now that we have only one set of data to consider, we can return to the simple one-sample t-test. Mechanically, a paired t-test is just a one-sample t-test for the mean of the pairwise differences. The sample size is the number of pairs.

7. Copyright © 2009 Pearson Education, Inc. Slide 1- 7 Assumptions and Conditions Paired Data Assumption: Paired data Assumption: The data must be paired. Independence Assumption: Independence Assumption: The differences must be independent of each other. Randomization Condition: Randomness can arise in many ways. What we want to know usually focuses our attention on where the randomness should be. 10% Condition: When a sample is obviously small, we may not explicitly check this condition. Normal Population Assumption: We need to assume that the population of differences follows a Normal model. Nearly Normal Condition: Check this with a histogram or Normal probability plot of the differences.

8. Copyright © 2009 Pearson Education, Inc. Slide 1- 8 A Paired t-Test When the conditions are met, we are ready to test whether the paired differences differ significantly from zero. We test the hypothesis H 0 :  d =  0, where the d’s are the pairwise differences and  0 is almost always 0. Hypotheses In general H 0 :  d = 0 and H A :  d ≠, >, or < 0 The paired t-test

9. Copyright © 2009 Pearson Education, Inc. Slide 1- 9 We use the statistic where is the mean of the pairwise differences and n is the number of pairs. is the ordinary standard error for the mean applied to the differences. When the conditions are met and the null hypothesis is true, this statistic follows a Student’s t-model with n – 1 degrees of freedom, so we can use that model to obtain a P-value. A Paired t-Test (cont.) The paired t-test

10. Copyright © 2009 Pearson Education, Inc. Slide 1- 10 When the conditions are met, we are ready to find the confidence interval for the mean of the paired differences. The confidence interval is where the standard error of the mean difference is The critical value t* depends on the particular confidence level, C, that you specify and on the degrees of freedom, n – 1, which is based on the number of pairs, n. Confidence Intervals for Matched Pairs Paired t-interval

11. Copyright © 2009 Pearson Education, Inc. Using the TI for hypothesis testing a matched sample- dependent sample A psychologist who is interested in testing the relationship between stress and short- term memory administered a test to 10 subjects prior to their exposure to a stressful situation and then retested them after the stress situation. From the following data can we conclude that the stress situation decreases one’s performance on a tests that measures short-term memory? Use an alpha of 0.05 Subject 1 2 3 4 5 6 7 8 9 10 Pre stress 13 15 9 13 15 17 13 16 11 13 Post stress 10 14 7 15 11 14 13 14 9 14 This is a matched sample problem.. same group of subjects doing two different things…. they are dependent. We are interested in their mean difference. We want to look at everyone's difference in memory before the stress and after the stress. We then will look at the average of all of their differences. Since the question is to determine if stress decreases short-term memory, we need to discover if pre-stress memory>post- stress memory. So the difference we will take is pre - post. H 0 : μ 0 = 0 H A : μ 0 > 0 (i.e. pre-stress > post-stress) In L1 put the pre stress measures and in L2 put the post stress measures. Go to the very top of L3, (actually go on top of "L3") and type in 2nd L1 - 2nd L2 enter. You should see data in L3 which represents each subject’s difference. Now do a T-test on L3 using Data. Slide 1- 11

12. Copyright © 2009 Pearson Education, Inc. Using the TI for hypothesis testing a matched sample- dependent sample (cont) H 0 : μ d = 0 H A : μ d > 0 (i.e. pre-stress > post-stress) In L1 put the pre stress measures and in L2 put the post stress measures. Go to the very top of L3, (actually go on top of "L3") and type in 2nd L1 - 2nd L2 enter. You should see data in L3 which represents each subject’s difference. Now do a T-test on L3 using Data. Now go to Stat -> Tests -> T-Test (same as we did for single samples) μ 0 = 0 List: L3 Freq: 1 μ = 0 : use > Calculate You find a test statistic of 2.33333 which is beyond the critical cutoff of 1.83 indicating a rejection of the null hypothesis. A p value of.02 also indicates a rejection of the null hypothesis. Therefore, our conclusion would be that stressful situations can cause a decrease in short-term memory. Slide 1- 12

13. Copyright © 2009 Pearson Education, Inc. Using your TI to find a confidence interval of the difference in means of paired data To find the confidence interval on μ d using paired data Place the before figures in L1 and the after figures in L2. Go to the very top of L3 and punch in L1 – L2 then enter. Their differences between the entries in L1 and the entries in L2 should show up in L3. Then do a TInterval using the data in L3. This now becomes test of ONE mean. Slide 1- 13

14. Copyright © 2009 Pearson Education, Inc. 30. Use the following data (representing hospital admissions from motor vehicle crashes) and an alpha of 0.05 to test the claim that Friday the 13ths are unlucky (Triola 2008): Friday the 6 th (immediately preceding the 13 th )Friday the 13 th 913 612 1114 1110 3434 512 H0: µd=0 HA: µd>0 Test Statistic (t) = 2.71 P-Value = 0.02 Conclusion: Since the P-Value is less than alpha, we can reject the null. The statistical evidence appears to show that Friday the 13 th is indeed unlucky (more data is probably needed to conclusively show that Friday the 13ths are unlucky). Slide 1- 14

15. Copyright © 2009 Pearson Education, Inc. A study was conducted to investigate the effectiveness of hypnotism in reducing pain. Results for randomly selected subjects are given below. The measurements represent a pain scale (where higher #’s indicate more pain). Use an alpha of 0.05 to test the claim that hypnosis lowers pain. H0: µd=0 HA: µd>0 Test Statistic (t) = 3.04 P-Value = 0.009 Conclusion: Since the P-Value is less than alpha, we can reject the null. The statistical evidence appears to show that the hypnosis treatment is effective in reducing pain. Slide 1- 15 Before HypnosisAfter Hypnosis 6.66.8 6.52.4 9.07.4 10.38.5 11.38.1 8.16.1 6.33.4 11.62.0

16. Copyright © 2009 Pearson Education, Inc. 32. To test the effectiveness of a drug to relieve asthma, a group of subjects was randomly given a drug and placebo on two different occasions. After 1 hour an asthmatic relief index was obtained for each subject, with these results: Use 0.05 for alpha. Is the drug effective? H0: µd=0 HA: µd>0 Test Statistic (t) =-2.75 P-Value = 0.01 Conclusion: Since the P-Value is less than alpha, we can reject the null. The statistical evidence appears to confirm that the drug is effective. Slide 1- 16 Subject123456789 Drug283117221232241825 Placebo323319261730261925

17. Copyright © 2009 Pearson Education, Inc. Slide 1- 17 Blocking Consider estimating the mean difference in age between husbands and wives. The following display is worthless. It does no good to compare all the wives as a group with all the husbands—we care about the paired differences.

18. Copyright © 2009 Pearson Education, Inc. Slide 1- 18 Blocking (cont.) In this case, we have paired data—each husband is paired with his respective wife. The display we are interested in is the difference in ages:

19. Copyright © 2009 Pearson Education, Inc. Slide 1- 19 Blocking (cont.) Pairing removes the extra variation that we saw in the side-by-side boxplots and allows us to concentrate on the variation associated with the difference in age for each pair. A paired design is an example of blocking.

20. Copyright © 2009 Pearson Education, Inc. Slide 1- 20 What Can Go Wrong? Don’t use a two-sample t-test for paired data. Don’t use a paired-t method when the samples aren’t paired. Don’t forget outliers—the outliers we care about now are in the differences. Don’t look for the difference between means of paired groups with side-by-side boxplots.

21. Copyright © 2009 Pearson Education, Inc. Slide 1- 21 What have we learned? Pairing can be a very effective strategy. Because pairing can help control variability between individual subjects, paired methods are usually more powerful than methods that compare independent groups. Analyzing data from matched pairs requires different inference procedures. Paired t-methods look at pairwise differences. We test hypotheses and generate confidence intervals based on these differences. We learned to Think about the design of the study that collected the data before we proceed with inference.