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

2. Objectives:
Paired Data
Paired t-Test
Paired t-Confidence Interval

3. 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.

4. Paired Data (cont.)
If you know the data are paired, you can (and must!) take advantage of it.
To decide if the data are paired, consider how they were collected and what they mean (check the W’s).
There is no test to determine whether the data are paired.
We cannot use a two-sample t-test for paired data because paired data come from samples that are not Independently chosen.
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.

5. 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 means of the pairwise differences.
The sample size is the number of pairs.

6. Assumptions and Conditions
Paired Data Assumption:
Paired data Assumption: The data must be paired, depentent.
Independence Assumption:
Randomization Condition: Randomness can arise in many ways. What we want to know usually focuses our attention on where the randomness should be.
10% Condition: When a sample is obviously small, we may not explicitly check this 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.

7. 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.

8. The Paired t-Test (cont.)
P (define parameter)
: Mean difference between __ and ___.
H (write hypotheses)
(this value is usually 0)
A (check assumptions)
Matched Pairs Assumption: Every data value in one sample is paired with one in the other sample.
2. Randomization Condition:(using Paired data)
3. Nearly Normal Condition:
Small: (n<15) approx. normal, no outliers,unimodal, symmetric
Medium: (15<n<40) Unimodal, symmetric
 Large: (n>40) not necessary to check shape because of CLT
4. 10% ConditionSample size is less than 10% of population.

9. T (calculate test statistic)
N (name procedure)
If all of the necessary assumptions and conditions have been met, we may proceed with the paired t-test.
T (calculate test statistic)
O (obtain p-value) sketch
M (make decision) __
since the p-value is _____
S (state conclusion) If
were true, we would expect to see a sample result at least
as extreme as the one we observed in about _____ out of every _____
samples of this size by chance. This __________ strong enough evidence to conclude .

10. Example: Matched Pairs t Hypothesis Test
The following chart shows the number of men and women employed in randomly selected professions. Do these data suggest that there is a significant difference in gender in the workplace?

11. Solution
Parameter
μd: the mean differences between men and women employed in selected professions
Hypothesis
H0: μd=0 The mean gender difference in the workplace is 0.
Ha: μd≠0 The mean gender difference in the workplace is not 0.

12. Solution
Assumptions
Paired Data Assumption: data paired because they count employees in the same occupations.
Randomization Condition: occupations were randomly selected
10% Condition: there are more than 80 occupations.
Nearly Normal Condition: n=8, a small sample size, must check for skew and outliers (unimodal and approx. symmetric, some sight skewness).

13. Solution
Name Matched pairs hypothesis test, so use one sample t-test (on 𝑥 𝑑 ) Test Statistic 𝑛=8, 𝑑𝑓=7, 𝑥 𝑑 = , 𝑠 𝑑 = 𝑔𝑒𝑡 𝑥 𝑑 𝑎𝑛𝑑 𝑠 𝑑 𝑓𝑟𝑜𝑚 1−𝑉𝑎𝑟 𝑆𝑡𝑎𝑡𝑠 𝑜𝑛 𝑑𝑖𝑓𝑓. 𝑑𝑎𝑡𝑎

14. Solution
Obtain p-value p-value = 2P(t ≥ 1.373) = .212 Make decision p-value is very large, so fail to reject the null hypothesis State conclusion in context The p-value is too large at any commonly accepted levels to be able to reject the null hypothesis, therefore we fail to reject the null hypothesis. There is insufficient evidence to support a difference in gender in the workplace.

15. 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.

16. Example: Confidence Intervals for Matched Pairs
Over the ten-year period , the unemployment rates for Australia and the United Kingdom were reported as follows.
Find a 90% confidence interval for the mean difference in unemployment rates for Australia and the United Kingdom.

17. Solution
Parameter
μd: mean difference in unemployment rates for Australia and the United Kingdom
Assumptions
Matched Pairs Assumption: Yes, matched by year.
Randomization Condition: Yes, no reason to believe that this sequence of years is not representative of the mean of the differences in unemployment rates of these 2 countries.
10% Condition: n = 11, 110 years of employment records for both countries.
Nearly Normal Condition: n < 15, check for skew and outliers. There is some sight skewness and no outliers, use t- distribution.

18. Solution
Name Matched pairs confidence interval, so use one sample t-interval (on 𝑥 𝑑 ) Interval 𝑛=11, 𝑑𝑓=10, 𝑥 𝑑 =.691, 𝑠 𝑑 =.589 𝑔𝑒𝑡 𝑥 𝑑 𝑎𝑛𝑑 𝑠 𝑑 𝑓𝑟𝑜𝑚 1−𝑉𝑎𝑟 𝑆𝑡𝑎𝑡𝑠 𝑜𝑛 𝑑𝑖𝑓𝑓. 𝑑𝑎𝑡𝑎 (.369, 1.013)

19. Solution
Conclusion in context We are 90% confident that the mean difference in the unemployment rates between AU and UK is between .369 and