1. C HAPTER 8 Matched Pairs

2. I NTRODUCTION In the last two chapter, we looked at comparing two means taken from independent samples. Measures from one sample had no relationship (or were independent from) to any measures from the other sample. Suppose we wanted to see if people lose weight on a new diet. We have pre-weights and post- weights. Are any measures from one of these samples related to a measure from the other?

3. P AIRED VS. U NPAIRED D ATA (1) Unpaired data: How much warmer are females than males, on average? Independent samples (No reason to match up a specific female temperature with a specific male temperature.) Paired data: How much faster do people’s hearts beat before and after doing 30 jumping jacks? Dependent samples (We will look at the differences in peoples heart rates before and after exercising.)

4. P AIRED VS. U NPAIRED D ATA (2) For unpaired data: Use the two sample t-test. We cannot use a paired samples test. It would make no sense to pair data arbitrarily. When we are able to pair up data, we take quite a bit of variability out of the picture. Think of pre-weight and post-weight. The variability in people’s initial weights is removed.

5. M ANY N AMES FOR THE SAME THING All of these mean the same: Paired Data Dependent Samples Matched Pairs

6. C OMPARING P AIRED S AMPLES In Section 8.1 we will use simulation-based techniques. We used coin flipping and an applet to conduct the simulations. In Section 8.2 we will use theory-based techniques. We will use Fathom to examine the data and determine the p-value.

7. E XAMPLE 8.1: S WIMMING IN S YRUP ?

8. Researches thought that swimming in something thicker than water would be slower because of added resistance. However, the swimmers should be able to generate more force with each stroke thus adding to their speed. Null: The average swimming speed in water and guar do not differ. Alternative: The average swimming speed in water and guar do differ.

9. E XAMPLE 8.1: S WIMMING IN S YRUP ? Swimmer ID12345 Speed in guar (cm/s) 99.498.8123.9122.5129.5 Speed in water (cm/s) 91.5103.5118.5125.0127.0 Results With five swimmers being tested, The average speed in guar was 114.82 cm/sec The average speed in water was 113.10 cm/sec

10. E XAMPLE 8.1: S WIMMING IN S YRUP ? Swimmer ID12345 Speed in guar (cm/s) 99.498.8123.9122.5129.5 Speed in water (cm/s) 91.5103.5118.5125.0127.0 Diff in speed = guar - water 7.9-4.75.4-2.52.5

11. E XAMPLE 8.1: S WIMMING IN S YRUP ?

12. Is it possible to see such a difference in swimming speeds due to chance alone, even if swimming speeds in guar and water are the same, on average? Is it unlikely? The null hypothesis basically says the medium in which they are swimming doesn’t matter. How can we model this?

13. E XAMPLE 8.1: S WIMMING IN S YRUP ? If swimming in guar or water does not matter than we can randomly choose swimmers (by flipping a coin) and switch their times. I did this once and got a mean difference of 1.44 Swimmer ID12345 Speed in guar (cm/s) 91.5103.5123.9125.0129.5 Speed in water (cm/s) 99.4 98.8 118.5 122.5 127.0 Diff in speed = guar - water -7.94.75.42.5 H = switchHHTH T

14. E XAMPLE 8.1: S WIMMING IN S YRUP ? I will repeat the process many, many times and create a null distribution with my simulated mean differences. (Notice the center)

15. E XAMPLE 8.1: S WIMMING IN S YRUP ? Our original mean difference was 1.72, so I need to count the number of mean differences above 1.72 and below -1.72 (remember this is a two- sided test). In doing so, I get a p-value of 0.528

16. E XAMPLE 8.1: S WIMMING IN S YRUP ? So what’s our conclusion? We cannot conclude the average swimming speed in guar and water differ. Our sample showed that swimming in guar was faster, but the results aren’t significant. It is tough to show significance with a sample of just five. It would be nice to have a larger data set.

17. E XPLORATION 8.2: D OES S TANDING R AISE YOUR H EART R ATE ?

18. S ECTION 8.2: T HEORY - BASED A PPROACH TO A NALYZING D ATA FROM P AIRED S AMPLES In the last section, our null distributions were centered at zero. Why? The heart rate null distribution was fairly normal shaped. The swimming in syrup one was not. (The sample size was very small.)

19. E XAMPLE 8.2: H OW M ANY M&M S W OULD Y OU L IKE ? Does bowl or plate size effect how much you eat? To test this, researchers randomly assigned either a small bowl or a large bowl to student subjects. They were told to take as many M&Ms as they wanted to eat for the day. The number of M&Ms they took were counted. During the next session, bowl sizes were switched. (Those that had large bowls now received small bowls and vice versa.) The number of M&M’s they took were again counted. Why are these samples dependent?

20. E XAMPLE 8.2: H OW M ANY M&M S W OULD Y OU L IKE ?

21. Bowl size M&Ms count Small 33 24 35 24 40 33 88 36 65 38 28 50 26 34 51 25 26 Large 41 92 61 19 21 35 42 50 11 104 97 36 43 62 33 62 32 Results

22. E XAMPLE 8.2: H OW M ANY M&M S W OULD Y OU L IKE ? Bowl sizeSample sizeSample average Small17 Large17 Difference = Small - Large 17

23. E XAMPLE 8.2: H OW M ANY M&M S W OULD Y OU L IKE ? Null distribution assuming bowl makes no difference.

24. E XAMPLE 8.2: H OW M ANY M&M S W OULD Y OU L IKE ? We got a p-value of 0.252, so we cannot conclude that bowl size has an effect on how many M&Ms are taken.

25. E XAMPLE 8.2: H OW M ANY M&M S W OULD Y OU L IKE ? Our null distribution appeared fairly normal so it would seem that a theory-based test would work. For small sample sizes (less than 20) we should not have highly skewed data or large outliers. Our results aren’t the greatest, but should be okay. Let’s run a theory-based test (as well as find a confidence interval) for this in Fathom.

26. E XPLORATION 8.2: B UYING B ESTSELLERS FOR B ETTER P RICES