INFERENCE WITH MATCHED PAIRS a special type of t-inference AP Statistics Chapter 25 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. a) 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”… b) 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… 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. a) 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… b) 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… 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… Why matched pairs? s1 (why not stick with 2-sample t?) Student 1 2 3 4 5 2 n1 6 7 8 615 s2 2 ? n2 Before 550 520 500 600 640 730 590 s1 = 73.04 After 600 570 550 650 690 665 780 640 s2 = 73.04 +50 +50 +50 +50 +50 +50 +50 +50 sd = 0 • Is there variance in the “before” scores? • Is there variance in the “after” scores? • Is there variance in the So instead of s1 2 n1 s2 improvements? 2 n2 we use sd n Last flap of our “means” foldable (outside) matched pairs! paired t-interval and paired t-test Update your foldables (inside, top half) Define md (“true mean difference…”) ***define which way you are subtracting!!! Conditions: • • • • Paired data??? Random sample (pairs) (10%) Nearly Normal Condition o n > 30 (number of PAIRS!!!) o boxplots/histogram of DIFFERENCES!!! ***do NOT graph BOTH boxplots!!! So we may use a t-distribution, df = n – 1 (“n” is the number of pairs) Update your foldables (inside, bottom half) on calculator, just do “t-test” or “t-interval” with the differences paired t-interval: x d t * df s d n d paired t-test: H O: m d 0 > H A: m d < 0 ≠ t xd 0 sd nd 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) 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 not consecutive) Day 1 2 Morning 9 Afternoon 10 3 4 5 6 7 8 7 10 10 2 5 7 6 9 9 7 9 6 8 4 Since you have two values for each day, they are dependent on the day – making this data matched pairs You may subtract either way – just be careful when writing Ha Day 1 2 Morning 9 7 Afternoon 10 Differences -1 Conditions: 3 4 5 6 7 8 10 10 2 5 7 6 9 9 8 4 7 9 6 -2 1 2 -2 -2 -2 0 You need to state assumptions using the differences! • 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 whalewatching days Day Differences 1 2 -1 -2 3 4 5 6 7 1 2 -2 -2 -2 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 8 0 Day Differences 1 2 -1 -2 3 4 5 6 7 1 2 -2 -2 -2 8 0 At the 5% significance level, is there evidence that more whales are sighted in the afternoon? H0: mD = 0 Ha: mD < 0 Be careful writing your Ha! Think about how you subtracted: M–A If you If subtract afternoon – afternoon is more, should the differences + or -? morning; then Ha: mD be >0 (Don’t look at numbers!!!!) mD = true mean difference in whale sightings, define which way you are subtracting!!! morning – afternoon Day 1 Differences 2 -1 -2 3 4 5 6 7 1 2 -2 -2 -2 8 0 finishing the hypothesis test: t xm s n In your calculator, perform a t-test . 75 0 Notice that if 1 . 3416 using the you subtracted 1 . 5811 differences 3) A-M, then(L your 8 p . 1108 df 7 a . 05 test statistic t = + 1.3416, but p-value would be the same Since p-value (.1108) > a (.05), we fail to reject H0. We lack sufficient evidence to suggest that more whales are sighted in the afternoon than in the morning. …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) x d t df df = 8 – 1 = 7 * sd nd ( 0 . 75 ) 1 . 895 (1 . 5811 ) 8 (-1.809, 0.3091) since this isthat really We are 90% confident theatrue mean difference in whale sightings is from interval, 1.809 fewer in the the morning to 0.3091 more in 1-sample get the morning. t* value from the t-table We can’t really say that it matters when you go whaling! whale watching!