1. Paired Samples and Blocks
Chapter 25
Paired Samples and Blocks

2. 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 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.

3. Paired Data (cont.)
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.

4. Paired Data (cont.)
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 mean of the pairwise differences.
The sample size is the number of pairs.

5. Assumptions and Conditions
Paired Data Assumption: The data must be paired.
Independence Assumption: The differences must be independent of each other. Check the:
Randomization Condition
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.

6. The Paired t-Test
When the conditions are met, we are ready to test whether the paired differences differ significantly from zero.
We test the hypothesis H0: d = 0, where the d’s are the pairwise differences and 0 is almost always 0.

7. 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.

8. 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.

9. EXAMPLE One indicator of physical fitness is resting pulse
rate. Ten men volunteered to test an exercise
device advertised on television by using it three
times a week for 20 minutes. Their resting
pulse rates were measured before the test
began and then again after six weeks. Is there
evidence that this kind of exercise can reduce
resting pulse rates?

10. Subject
Before
After
Difference
Allen 73
Brandon 83 79 -4
Carlos 85 81
David 87 86 -1
Edwin 91
Franco 99 -8
Greg 84 -3
Hans -2
Ivan 1
Jorge 76

11. 1. State the hypotheses
2. Model

12. 1. State the hypotheses
where is after - before
2. Model
Independent Differences
Paired Data – before and after data
Nearly Normal – histogram is symmetric with no outliers.

13. 3. Mechanics

14. 3. Mechanics

15. 4. Conclusion

16. 4. Conclusion
Since our p-value of is less than
alpha of 0.05, we reject the null hypothesis.
We have evidence to suggest the resting
pulse rate for this type of exercise is
lowered.

17. Quiz Chapter 24 Next Class Investigative Task Due 2/9
ASSIGNMENT
A#3/5 p. 571#25, 28
p. 586 #5-8, 13, 21, 24
Quiz Chapter 24 Next Class
Investigative Task Due 2/9