### 1. Matched pairs analysis

Mastery Quiz Prep

#### Matched Pairs Hypothesis Testing

• Distinguish between quantitative 2-sample and quantitative matched pairs data and choose the appropriate interval / test procedures.
• If the study uses the same individuals more than once or specially-related individuals, then it is matched pairs.  Use the list of differences to do an interval or test of a single quantitative variable
• If not (if it is a random or block design), then use a 1-quantitative, 1-categorical variable test/interval.
• Find and interpret a confidence interval for a matched-pairs study.
• Decide which order to subtract.  Subtract all pairs.  Use the resulting list of differences in a confidence interval of a single mean.
• Interpret: I am _(percent)_% confident that _(each pair of individuals)_(have/succeed)_ an average of _(low end of interval)_ to _(high end of interval)_ more _(response variable units)_ with _(better treatment)_  than _(worse treatment)_..
• For example (single individual in both treatments): I am 90% confident that each local girls basketball player makes an average of 5.1 to 7.4 more free throws with their right-hand than their left hand (out of 25).
• For example (separate paired individuals): I am 99% confident that husbands make an average of 0.8 to 4.4 more free throws than their wives (out of 25).
• OR plus-minus form: I am 99% confident that husbands make an average of 2.6 more free throws than their wives (out of 25) with a margin of error of ±1.8 free throws.
• Perform and interpret a significance test for a matched-pairs study.
• Decide which order to subtract.  Subtract all pairs.  Use the resulting list of differences in a test.
• The null hypothesis is always that the average of the list of differences = 0.  Write as μgroupA-groupB = 0.  This makes it easy to know what μ stands for.  Note that with matched pairs, there is only one μ.

Practice problems:

1. Explain why the following situation is NOT a matched-pairs problem:
A group of disc golfers wants to compare two different putting techniques.  In one, the player throws the disc with the forehand, just like a normal Frisbee.  In the other, the player uses a backhand flick to throw.  To test the methods, they get 20 volunteers to throw 8 discs each from 30 feet away.  Half of the volunteers are randomly assigned to one of the techniques, and the rest to the other, before the throwing starts.  The group wants to prove that throwing forehand will work better on average than throwing backhand.  Results (number of shots made out of 8):
Backhand: 2, 7, 3, 8, 3, 2, 1, 4, 3, 4
Forehand: 3, 3, 7, 6, 8, 4, 5, 6, 4, 8

2. Explain why the following situation is NOT a matched-pairs problem:
A group of robot enthusiasts who were not impressed with the frisbee shooters made two t-shirt launchers.  The first cannon, the larger one, has a 3" diameter barrel.  The smaller one has a 2" barrel.  A pile of 20 t-shirts is randomly divided between between the two cannons and their launch distances (in feet) are recorded.  The group is convinced that the large cannon is better.
Small cannon trials: 80, 56, 78, 61, 31, 64, 72, 66, 69, 78
Large cannon trials: 84, 97, 88, 77, 91, 83, 43, 89, 79, 67

3. Explain why the following situation is NOT a matched-pairs problem:
A student group decided to compare how well players did using two different strategies of building a card tower.  The subjects were first instructed on a specific method they needed to use for their tower and told them it was required to use this strategy.  Since the group didn’t want players to mix strategies, they tested two completely separate groups of people.  People volunteered to play and were randomly assigned to a strategy using a coin flip on the day of the experiment.  The results:
Strategy 1 (seconds required to build the tower): 33, 42, 59, 68, 73, 91, 33, 45
Strategy 2 (seconds required to build the tower): 73, 33, 49, 62, 65, 48, 47, 66

4-5: Answer the following for questions for each scenario:
a)    What is the explanatory variable in this scenario. What are its options (treatments)?
b)    What is measured for each individual (the response variable in this scenario)? Is it quantitative or categorical?
c)    If it is quantitative, is there matched pairs or two distinct samples?
d)    Based on the last two responses, what type of interval/test will you perform in StatKey?
e)    What is the null hypothesis in this scenario? Use the correct symbols (μ or p) and use subscripts so it is clear what each symbol means.
f)    What is the alternative hypothesis in this scenario?  Re-read the problem to see if there is an intended direction.
g)    What is the p-value of your test?
h)    Is this an observational study or experiment?  Based on this and your p-value, what can you conclude?
i)    What is the estimated difference between the two groups (95% interval)?
j)    Convert this interval to plus-minus form.
k)    Interpret the confidence interval of the difference in a sentence.  Use plus-minus form because it is often far more readable in a sentence.

4. After taking some criticism from their study design, the disc golfers tried a new approach: each player throws 10 discs with one technique and 10 discs with the other technique.  There were 8 volunteer players.  The order of which technique comes first is randomized for each player.  Again, they still want to prove that throwing forehand will work better.  Results:
 Player # 1 2 3 4 5 6 7 8 Backhand 7 3 4 1 4 2 5 6 Forehand 9 2 7 6 4 5 3 9

5  Another team compared the ability to accurately kick a soccer ball into a goal with their left and right foot.  Each person kicked the ball 10 times per foot to see how many they could get in the goal.  The organizing team assumed that more people would succeed using their right, so they want to verify this in their test.  Each person’s data is vertically stacked above each other.
Right: 8 8 4 4 9
Left: 3 2 5 3 8

Free Response Prep

Explain how StatKey simulates tests of independence for matched-pairs data (same question, all versions of quiz):
Before entering, you need to take the difference of each pair and enter the list of differences.  It resamples this list with replacement to produce a bootstrap sample.  It then plots the average of each of these samples and finds the range of the middle 95% (or whatever percentage you choose).

Practice solutions

1. There are two distinct groups of players.  They have no special pairing between the members of both sample groups, and they are not the same person trying both methods.

2. There are two distinct groups of shirts.  They have no special pairing between the shirts in both sample groups, and they are not the same shirt being fired in each cannon.

3. There are two distinct groups of participants.  They have no special pairing between the members of both sample groups, and they are not the same person trying both strategies.

4. Frisbee #2 (video solution below):
a) Explanatory Variable: Method of throwing
Treatment Groups: Forehand and backhand
b) number of shots made; quantitative
c) Matched Pairs (each person is doing both forehand and backhand)
d) Test for a single mean because once you find the difference in the two trials, it will become one number for each person instead of two.
e) H0: μf-b = 0
f)    HA: μf-b  >0
g) p= .02
h) experiment; we can reject the null because p< 0.05. We can conclude the throwing forehand instead of backhand causes a higher amount of shots made.
i) .125 to 3.125
j) 1.625 ± 1.5 more forehand
k) I am 95% confident that each Frisbee player makes an average of 1.6 ± 1.5 more shots with their forehand than their backhand.

5. Soccer kicks:
• a) Explanatory Variable: What foot is used to kick the soccer ball
Treatment Groups: right foot or left foot
• b) Number of goals made; Quantitative
• c) Matched Pairs
• d) Test for a single mean
• e)   Null: people kick equally well with both feet
H0: μright - left = 0 (this is ALWAYS the null for a matched pairs)
• f)  Alt: people kick better with their right foot
HA: μright - left > 0
• g) p= 0.027
• h) Depends how it is run.  If there is no randomization in which order people kick with, it will be an observational study.  If it is randomized, it would be an experiment, but since people know which leg they are using, there might still be lurking variables that affect the experiment. We can conclude that kicking with the right foot causes people to make more goals.
• i) .200 to 4.800
• j) 2.5 ± 2.3
• k) I am 95% confident that soccer players make 2.5 ± 2.3 more goals (out of 10) with their right foot than their left foot

Notes