1. 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. 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 H 0 :  d =  0, where the d’s are the pairwise differences and  0 is almost always 0.

8. 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.

9. 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?

10. Solution Parameter μ d : the mean differences between men and women employed in selected professions Hypothesis H 0 : μ d =0The mean gender difference in the workplace is 0. H a : μ d ≠0The mean gender difference in the workplace is not 0.

11. 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).

12. Solution

13. 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.

14. 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.

15. Example: Confidence Intervals for Matched Pairs Over the ten-year period 1990-2000, 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.

16. Solution Parameter μ d : 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.

17. Solution

18. Conclusion in context We are 90% confident that the mean difference in the unemployment rates between AU and UK is between.369 and 1.013.