1. Copyright © 2010 Pearson Education, Inc. Slide 25 - 1

2. Copyright © 2010 Pearson Education, Inc. Slide 25 - 2 Solution: C

3. Copyright © 2010 Pearson Education, Inc. Chapter 25 Paired Samples and Blocks

4. Copyright © 2010 Pearson Education, Inc. Slide 25 - 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. 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.

5. Copyright © 2010 Pearson Education, Inc. Slide 25 - 5 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 means of the pairwise differences. The sample size is the number of pairs.

6. Copyright © 2010 Pearson Education, Inc. Slide 25 - 6 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.

7. Copyright © 2010 Pearson Education, Inc. Slide 25 - 7 The 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 ds are the pairwise differences and 0 is almost always 0.

8. Copyright © 2010 Pearson Education, Inc. Slide 25 - 8 The Paired t-Test (cont.) We use the statistic where 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 Students t-model on n – 1 degrees of freedom, so we can use that model to obtain a P-value.

9. Copyright © 2010 Pearson Education, Inc. Slide 25 - 9 Confidence Intervals for Matched Pairs 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.

10. Copyright © 2010 Pearson Education, Inc. Slide 25 - 10 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 husbandswe care about the paired differences.

11. Copyright © 2010 Pearson Education, Inc. Slide 25 - 11 Blocking (cont.) In this case, we have paired dataeach husband is paired with his respective wife. The display we are interested in is the difference in ages:

12. Copyright © 2010 Pearson Education, Inc. Slide 25 - 12 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.

13. Copyright © 2010 Pearson Education, Inc. Slide 25 - 13 Example: Do flexible schedules reduce the demand for resources? The Lake County, Illinois Health Department experimented with a flexible four-day workweek. For a year, the department recorded the mileage driven by 11 field workers on an ordinary 5-day workweek. Then it changed to a flexible four-day workweek and recorded the mileage for another year. Why are these data paired? Name5-day mileage 4-day mileage Jeff27982914 Betty77246112 Roger75056177 Tom8381102 Aimee45923281 Greg81074997 Larry G.12281695 Tad87186606 Larry M.10971063 Leslie80896392 Lee38073362

14. Copyright © 2010 Pearson Education, Inc. Slide 25 - 14 Example: Is it okay to use these data to test whether the new schedule changed the amount of driving? Example: Is there evidence that a four-day workweek would change how many miles workers drive? Example: By how much, on average, might a change in workweek schedule reduce the amount driven by workers?

15. Copyright © 2010 Pearson Education, Inc. Slide 25 - 15 Homework: Pg. 587 1-15 (13 & 15 compute formulas by hand and check with calculator) (Day 2) Pg. 587 17-23 (watch out! They mix different types of problems from chpt. 23, 24 and 25! On problem 19 ignore what is in (). On 21 eliminate the outlier, 23b, eliminate car #4 as an outlier.)