1. CH 25 Paired Samples and Blocks

2. Paired Data 1. Observations that are collected in pairs (data on age differences between husbands and wives, for instance). 2. Observations in one group are naturally related to the observations in another (comparing mileage driven over 4 days and 5 days for a group of 11 field workers)

3. Paired Data The most common form of pairing is “before and after” a treatment of some sort. Pairs that arise from an observational study are called matched pairs. Pairs that arise from an experiment are called blocked pairs.

4. CONDITIONS: 1) The data are paired. The differences are what must be independent (the difference in driving mileage for one person does not influence the difference in driving mileage for another person, for example). 2) The sample distribution of differences is approximately normal - the populations of differences is known to be normal, or - the number of sample differences is large (n  30), or - graph data to show approximately normal 3) 10% rule – The sample of differences is not more than 10% of the population of differences. This would not apply to randomized experiments. Differences of Paired Means (Matched Pairs)

5. Paired Data LARGE POINT: If data are paired, then the two groups are not independent…this means that two-sample techniques are not and will not be appropriate. Hypothesis testing for paired differences will be done just as a one-sample t-test would have been done…the number of pairs would be our sample size.

6. Of the following situations, decide which should be analyzed using one-sample matched pair procedure and which should be analyzed using two-sample procedures? A pharmaceutical company wants to test its new weight-loss drug. Before giving the drug to a random sample, company researchers take a weight measurement on each person. After a month of using the drug, each person’s weight is measured again. Matched pair

7. Of the following situations, decide which should be analyzed using one-sample matched pair procedure and which should be analyzed using two-sample procedures? A researcher wants to know if a population of brown rats on one city has a greater mean length than a population in another city. She randomly selects rats from each city and measures the lengths of their tails. Two independent samples

8. Of the following situations, decide which should be analyzed using one-sample matched pair procedure and which should be analyzed using two-sample procedures? A researcher wants to know if a new vitamin supplement will make the tails of brown rats grow longer. She takes 50 rats and divides them into 25 pairs matched by gender and age. Within each pair, she randomly selects one rat to receive the new vitamin. After six months, she measures the length of the rat’s tail. Matched pair

9. Of the following situations, decide which should be analyzed using one-sample matched pair procedure and which should be analyzed using two-sample procedures? A college wants to see if there’s a difference in time it took last year’s class to find a job after graduation and the time it took the class from five years ago to find work after graduation. Researchers take a random sample from both classes and measure the number of days between graduation and first day of employment Two independent samples

10. Matched Pairs (Special type of one- sample means)

11. Differences of Paired Means (Matched Pairs)

12. Hypothesis Statements: H 0 :  d = hypothesized value H a :  d < hypothesized value H a :  d > hypothesized value H a :  d ≠ hypothesized value Differences of Paired Means (Matched Pairs) Parameter:  d = true mean difference in …

13. Hypothesis Test: Differences of Paired Means (Matched Pairs)

14. Parameters and Hypotheses Having done poorly on their Math semester exams in December, six students repeat the course in a credit recovery class in the spring and take another exam in April. If we consider these students to be representative of all students who might take this credit recovery class in other years, do the following results provide evidence that the credit recovery class is worthwhile? December: 54 49 68 66 62 62 April: 50 65 74 64 68 72 μ d = the true mean difference in scores between December and April for students who repeated the course in the credit recovery class H o : μ d = 0 H a : μ d > 0 Page 590: 18 (context modified slightly)

15. Assumptions (Conditions) Since the conditions are met, a t-test for the matched pairs is appropriate. 1) The samples must be paired and random. The samples are from the same student so they are paired and we will assume the sample differences are a random sample of the population of differences.. 3) The sample should be less than 10% of the population. The population should be at least 60 students, which we will assume. 2) The sample distribution should be approximately normal. 4)  is unknown The boxplot shows no outliers and is roughly symmetric, so we will assume that the sample distribution of differences is approximately normal.

16. Calculations  = 0.05 5.333 7.4475 p-value =.069 >.05

17. Conclusion: Decision: Since p-value > , I fail to reject the null hypothesis at the.05 level. There is not sufficient evidence to suggest that the true mean difference in scores has risen from December to April. This suggests that program may not be worthwhile.