1-1 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 24, Slide 1 Chapter 25 Paired Samples and Blocks

1-2 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 24, Slide 2 Chapter 25 Introduction In Olympic speed skating, skaters are paired up with one opponent during a race. One runner starts out in the inner lane whilst the other starts in the outer and then they switch midway. Some people think this creates an unfair advantage for the person who starts out in the outer lane. We could perform hypothesis testing on the two groups and see if we should believe if there is actually a difference in the true means. Let’s talk about how we would go about doing this.

1-3 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 24, Slide 3 Chapter 24 Introduction Thus, if you realize that your two sets of data are not independent of each other, but are rather results that have been paired together to form two groups, then we can not use the methods from Chapter 24, thus we must come up with another way of inference. The difficulty here is not in performing the inference, it is simply in knowing when data has been paired.

1-4 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 24, Slide 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 or nawh? We randomly select 20 males and 20 females and compare the average time they spend watching TV. We randomly select 20 couples and compare the time the husbands and wives spend watching TV. Suppose a teacher counts the number of homework assignments each student turned in for a particular unit, and then pairs this number with each student’s percentage on the unit test. Concentration and TV’s. Please remember, Mr. Hardin.

1-5 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 24, Slide 5 Paired Data 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. Quick note: the order of the pairs is vitally important.

1-6 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 24, Slide 6 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. You just gotta think about it. 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.

1-7 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 24, Slide 7 Paired Data (cont.) In other words, because this is important, we will just compute the differences of the two groups. This will create one set of data. 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. Your calculator can quickly and easily make a difference list by subtracting two lists!

1-8 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 24, Slide 8 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. differences 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.

1-9 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 24, Slide 9 The Paired t-Test So, ultimately we’re trying to see if there is a difference in the two groups or nawh (usually). Thus, when the conditions are met, we are ready to test whether the paired differences differ significantly from zero. We test the hypothesis H 0 : μ d = μ 0, where the d’s are the pairwise differences and μ 0 is almost always 0.

1-10 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 24, Slide 10 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 Student’s t-model on n – 1 degrees of freedom, so we can use that model to obtain a P-value.

1-11 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 24, Slide 11 What to Write Write both hypotheses. H 0 : μ d = μ 0 Verify 3 conditions, giving a simple histogram of the differences of the pairs, noting things. Then, say that “we can use the t model with d.f. = n-1, a mean of the reported mu & a SD of to perform a paired t-test.” Draw the t curve with reported mu, statistic, and shaded region given by the alternative hypothesis. Perform the testing using the calculator and give the t- score of the statistic and the P-value of this test value. State your conclusion, referring back to any noteworthy stuff, if necessary. n = # of pairs

1-12 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 24, Slide 12 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.

1-13 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 24, Slide 13 WHAT IS NEEDED Verify the 3 conditions, same as hyp. testing. Say you can use the t-distribution model (and DRAW IT) with d.f. of n-1, a mean of the sample mean and standard error of to create a paired t-inverval. Write out the confidence interval in this form and then fill in the critical value and the SE info: where the critical value depends on the confidence level and the d.f. Use the calculator to calculate the actual interval. Interpret in context.

1-14 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 24, Slide 14 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.

1-15 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 24, Slide 15 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:

1-16 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 24, Slide 16 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.

1-17 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 24, Slide 17 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.

1-18 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 24, Slide 18 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.

1-19 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 24, Slide 19 AP Tips Make sure to carefully define your symbols. You can subtract the pairs in either order, but you need to communicate which order you picked. Watch your direction! For 1-sided tests, the order you subtract changes the in the H A. Read carefully! One of the most common AP errors is treating a 2-sample test like matched pairs, or vice versa.