1. 1 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Chapter 21 Paired Samples and Blocks

2. 2 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Objectives 70. Find and interpret in context a paired confidence interval, checking the necessary assumptions. 71. 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.

3. 3 Copyright © 2014, 2012, 2009 Pearson Education, Inc. 21.1 Paired Data

4. 4 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Pairing Speed-skating races run in pairs. One starts in the inner lane, the other in the outer lane. Above are some of the randomly assigned pairs. The data are paired rather than independent. Blocking involves pairing arising from an experiment. Matching involves pairing arising from an observational study. With pairing, we look at the differences.

5. 5 Copyright © 2014, 2012, 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. 6 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Comparing Mileage Do flexible schedules reduce total mileage? The table shows some workers’ mileage before and after the company switched to a 4-day work week. Why are these data paired? Each driver’s mileage recorded before and after. Not independent since we expect a high “before” mileage to predict a high “after” mileage. Paired because the differences such as Jeff: 2798 – 2914 = –116 have meaning.

7. 7 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Differences for Speed-Skater Pairs For paired data, create a new data set of the differences. We can now look only at the differences. Ignoring the original data, we now have a single data set. Proceed with a one-sample t-test. This process is called a paired t-test.

8. 8 Copyright © 2014, 2012, 2009 Pearson Education, Inc. 21.2 Assumptions and Conditions

9. 9 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Assumptions and Conditions Paired Data Condition The data must be paired. Only use pairing if there is a natural matching. The two-sample t-test and the paired t-test are not interchangeable. Independence Assumption For paired data, the groups are never independent. Need differences independent, not individuals Randomization ensures independence.

10. 10 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Assumptions and Conditions (Continued) Normal Population Assumption Need to assume the differences follow a Normal model. Nearly Normal Condition: Sketch a histogram and normal probability plot of the differences. Normality less important for larger sample sizes. Even if the individual measurements are skewed, bimodal or have outliers, the differences may still be Normal.

11. 11 Copyright © 2014, 2012, 2009 Pearson Education, Inc. 4-Day vs. 5-Day Mileage Is it okay to use these data to test whether the new schedule changed the amount of driving? Paired Data Condition: Before and after study Independence Assumption: The driving behavior of one worker is independent of another. Randomization Condition: The work trips randomly occurred. Nearly Normal Condition: The sample size is small, but the histogram is unimodal and symmetric. The paired t-test can be used.

12. 12 Copyright © 2014, 2012, 2009 Pearson Education, Inc. The Paired t-Test When the conditions are met, we can test whether the mean differences significantly differ from 0. H 0 :  d =  0 (  0 is usually 0) and s d are the mean and standard deviation of the pairwise differences and n is the number of pairs. Use the Student’s t-model with n-1 degrees of freedom and find the P-value.

13. 13 Copyright © 2014, 2012, 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- 13

14. 14 Copyright © 2014, 2012, 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- 14

15. 15 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Speed-Skating Comparisons Was there a difference between speeds between inner and outer speed-skating lanes? Think → Plan: I have data for 17 pairs of racers. Hypotheses: H 0 :  d = 0 H A :  d ≠ 0

16. 16 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Speed-Skating Comparisons Think → Model: Paired Data Condition: The racers compete in pairs. Independence Assumption: Each race was independent of the others. Racers were assigned to lanes randomly. Nearly Normal Condition: The histogram of the differences is unimodal and symmetric. No outliers. Use Student’s t-model with 16 df. Perform a paired t-test.

17. 17 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Speed-Skating Comparisons Show → Mechanics: n = 6 pairs Create a List (L 3 ) of differences: L 3 =L 1 -L 2 Perform a T-Test on L 3

18. 18 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Speed-Skating Comparisons Tell → Conclusion: P-value is large. I can’t conclude that the observed difference isn’t due simply to random chance. It appears the fans may have interpreted a random fluctuation in the data as favoring one lane. There’s insufficient evidence to declare any lack of fairness.

19. 19 Copyright © 2014, 2012, 2009 Pearson Education, Inc. 21.3 Confidence Intervals for Matched Pairs

20. 20 Copyright © 2014, 2012, 2009 Pearson Education, Inc. How Much Older is the Husband? 170 couples selected at random provided their ages. Goal: Find a confidence interval for the mean difference.

21. 21 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Paired t-interval When the conditions are met, the confidence interval for the mean paired difference is The critical value t* from the Student’s t-model depends on the confidence level C and df = n – 1. (n is the number of pairs.)

22. 22 Copyright © 2014, 2012, 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- 22

23. 23 Copyright © 2014, 2012, 2009 Pearson Education, Inc. How Much Older is the Husband? How big a difference is there, on average, between husbands and wives? Think → Plan: I want the mean difference in age. Have a random sample of 200 couples with 170 providing ages their ages. n = 170 Model: Paired Data Condition: Married couples make a natural pairing. Independence Assumption: The survey was randomized.

24. 24 Copyright © 2014, 2012, 2009 Pearson Education, Inc. How Much Older is the Husband? Think → Model: Nearly Normal Condition: The histogram is unimodal and symmetric. The conditions are met, so I can use the Student’s t-model with n – 1 = 169 degrees of freedom and find a paired t-interval.

25. 25 Copyright © 2014, 2012, 2009 Pearson Education, Inc. How Much Older is the Husband? Show → Mechanics: n = 170, = 2.2 years, s d = 4.1 years t 169 = 1.97 (from StatCrunch) Or Use Stat->Tests->Tinterval on L 3 The 95% CI for the difference between the means is 2.2 ± 0.61 = 1.6 to 2.8 years.

26. 26 Copyright © 2014, 2012, 2009 Pearson Education, Inc. How Much Older is the Husband? Tell → Conclusion: I am 95% confident that husbands are, on average, 1.6 to 2.8 years older than their wives.

27. 27 Copyright © 2014, 2012, 2009 Pearson Education, Inc. 21.4 Practice

28. 28 Copyright © 2014, 2012, 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- 28 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

29. 29 Copyright © 2014, 2012, 2009 Pearson Education, Inc. How large is the effect size of hypnosis on pain scale ratings? … Paired T-Interval (T-Interval on Pairwise Differences) Slide 1- 29 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

30. 30 Copyright © 2014, 2012, 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- 30

31. 31 Copyright © 2014, 2012, 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: (lower numbers indicate more relief) 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- 31 Subject123456789 Drug283117221232241825 Placebo323319261730261925

32. 32 Copyright © 2014, 2012, 2009 Pearson Education, Inc. 21.4 Blocking

33. 33 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Disadvantages and Advantages of Pairing Disadvantage Fewer degrees of freedom. Each pair considered as a single data value instead of two values. Advantage Can significantly reduce variability by focusing just on what is being compared. Pairing is an example of effective blocking. The advantage outweighs the disadvantage, so consider pairing if possible.

34. 34 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Comparing Pairs and No Pairs Pairing With pairing husbands and wives, there was a clear difference in ages. 95% CI: (1.6, 2.8). Without Pairing Without pairing, this would not have been as conclusive. From the box plots the difference is hard to detect.

35. 35 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Dependence and Independence Previous methods required independence both between and within groups. With pairing, the samples are always dependent. With pairing, we need the pairs to be independent of each other. Paired methods are sometimes called methods for “dependent” samples.

36. 36 Copyright © 2014, 2012, 2009 Pearson Education, Inc. What Can Go Wrong? Don’t use a two-sample t-test when you have paired data. Two-sample t-test and paired t-tests are not the same. Don’t use a paired t method when the samples aren’t paired. Always check for a natural pairing such as husband/wife, before/after, younger sister/older sister.

37. 37 Copyright © 2014, 2012, 2009 Pearson Education, Inc. What Can Go Wrong? Don’t forget outliers. Outliers of the differences not individuals. Make a plot of the differences to check for outliers. Don’t look for the difference between the means of paired groups with side-by-side box plots. The box plots of each group still contain the variation that pairing can remove. Comparing box plots is likely to be misleading.