INFERENCE WITH MATCHED PAIRS

1. INFERENCE WITH MATCHED PAIRS a special type of t-inference AP Statistics Chapter 25

2. Which situation requires a 2-sample t procedure, and which requires matched pairs? • A researcher wishes to determine whether listening to music affects students' performance on memory test. • He randomly selects 50 students and has each student perform a memory test once while listening to music and once without listening to music. He then compares the two scores for each student, “with music” vs. “without music”… • He takes 50 students, and has half of them perform a memory test without listening to music, and the other half perform the memory test while listening to music. He then obtains the means and standard deviations of the “with music” and the “without music” groups… matched pairs 2-sample t

3. Which situation requires a 2-sample t procedure, and which requires matched pairs? • A manufacturer has designed athletic footwear which it hopes will improve the performance of athletes running the 100-meter sprint. • 30 athletes are selected for this study. One group of 15 runs the sprint wearing the new footwear, and the other group of 15 runs with their normal footwear. He then compares the mean sprint times between the two groups… • 30 athletes are selected, each of them runs one sprint wearing the new footwear, and also one sprint with their normal footwear. Randomization (flipping a coin for each athlete) determines which footwear they run with first. The two times for each athlete are compared… 2-sample t matched pairs

4. gasp! DESIGNING STUDIES in tonight’s HW!!! A couple of tips(reminders?): • For an experiment, you don’t need a random sample – volunteers are okay! But use randomization to split subjects into groups (use a RNG… or flip a coin for each person… it is OKAY for groups to be DIFFERENT sizes) • When designing a matched-pairs procedure, EVERYONE gets both “treatments” – so randomize the order!!!(if practical. For “before-after” scenarios, you can’t really do this…) 30 athletes are selected… randomization (flipping a coin for each athlete) determines which footwear they run with first. The two times for each athlete are compared… Some run with the new shoes FIRST… Some run with the new shoes SECOND.

5. Why matched pairs? (why not stick with 2-sample t?) s1 = 73.04 sd = 0 +50 +50 +50 +50 +50 +50 +50 +50 s2 = 73.04 • Is there variance in the “before” scores? • Is there variance in the “after” scores? • Is there variance in the improvements? we use So instead of

6. Last flap of our “means” foldable (outside) matched pairs! pairedt-interval and pairedt-test

7. Update your foldables (inside, top half) • Define md(“true mean difference…”) • Conditions: • Paired data??? • Random sample (pairs) • (10%) • Nearly Normal Condition • n > 30 (number of PAIRS!!!) • boxplots/histogram of DIFFERENCES!!! • So we may use a t-distribution, • df = n – 1 (“n” is the number of pairs) ***define which way you are subtracting!!! ***do NOT graph BOTH boxplots!!!

8. Update your foldables (inside, bottom half) on calculator, just do “t-test” or “t-interval” with the differences paired t-interval: paired t-test: > < ≠

9. SOME TIPS ON HOW TO TELL MATCHED PAIRS… (WARNING: THIS IS NOT A COMPLETE LIST) • The two sets of data MUST have the same number of elements… • HOWEVER, just because both sets of data have the same count does NOT NECESSARILY make it matched pairs (so be careful!) • Is each PAIR of numbers linked somehow? (sometimes this is very difficult to determine)

10. BIG PICTURE: BLOCKING/STRATIFYING AND INFERENCE 2-sample t(no blocking) larger p-value • Reduces the variability (spread) of our data (sampling model) • With LESS variability, we are MORE likely to reject Ho. • Makes it easier to detect an “effect” (a “change” or “difference” or “improvement”, etc.) Matched pairs (blocking) smaller p-value

11. A whale-watching company noticed that many customers wanted to know whether it was better to book an excursion in the morning or the afternoon. To test this question, the company collected the following data (number of whales sighted) on 8 randomly selected days over the past month. (Note: days were notconsecutive) You may subtract either way – just be careful when writing Ha Since you have two values for each day, they are dependenton the day – making this data matched pairs

12. You need to state assumptions using the differences! • Conditions: • The data are paired by day since whale-watching conditions may change from day to day • We have a random sample of days for whale-watching • n = 8 days is certainly less than 10% of all whale-watching days

13. Nearly Normal Condition: • The box plot of differences is skewed, but has no outliers, so normality is plausible (especially with this small a sample size)(remember, you can also do a dot plot!!!) • We may use a t-distribution w/ df = 7

14. At the 5% significance level, is there evidence that more whales are sighted in the afternoon? Be careful writing your Ha! Think about how you subtracted: M – A If afternoon is more, should the differences be + or -? (Don’t look at numbers!!!!) H0: mD = 0 Ha: mD < 0 mD = true mean difference in whale sightings, morning – afternoon If you subtract afternoon – morning; then Ha: mD> 0 define which way you are subtracting!!!

15. finishing the hypothesis test: Since p-value (.1108) > a (.05), we fail to reject H0. We lacksufficient evidence to suggest that more whales are sighted in the afternoon than in the morning. In your calculator, perform a t-test using the differences (L3) Notice that if you subtracted A-M, then your test statistic t = + 1.3416, but p-value would be the same

16. …and now, a paired t-interval Develop a 90% confidence interval for the true average difference in number of whales sighted (morning – afternoon) statistic  (critical value) (SE) df = 8 – 1 = 7 (-1.809, 0.3091) since this is really a 1-sample interval, get the t* value from the t-table We are 90% confident that the true mean difference in whale sightings is from 1.809 fewer in the morning to 0.3091 morein the morning. We can’t really say that it matters when you go whaling! whale watching!

17. (here is the problem we did in class, but without the work, if you wish to give it a shot and check with someone later)

18. IS CAFFEINE DEPENDENCE REAL? The table below contains data on the subjects’ scores on a depression test. Higher scores show more symptoms of depression. a) Do the data from this study provide statistical evidence at the 5% level of significance that caffeine deprivation leads to an increase in depression?

19. IS CAFFEINE DEPENDENCE REAL? The table below contains data on the subjects’ scores on a depression test. Higher scores show more symptoms of depression. b) Use a 90% confidence interval to estimate the true mean increase in depression scores that results from being deprived of caffeine.