Chapter 25 Paired Samples：25章配對樣本

1、.Chapter 25: Paired SamplesPaired DataData 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 be

2、fore and after a treatment.o When pairs arise from an experiment, the pairing is a type of blocking.o When they arise from an observational study, it is a form of matching.· How do you know if the data are paired?o consider how they were collected and what they mean (check the Ws).o There is no

3、 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.· Now that we have only one set of

4、 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.Assumptions and Conditions· Paired Data Assumption: The data must be paired.&#

5、183; Independence Assumption: o Independence Assumption: The differences must be independent of each other. o Randomization Condition: Randomness can arise in many ways. What we want to know usually focuses our attention on where the randomness should be. For example, if it was an observational stud

6、y, we need to have a random sample of individuals. If it was an experiment, we need to have random assignment of subjects to treatments.o 10% Condition: When a sample is obviously small, we may not explicitly check this condition.· Normal Population Assumption: We need to assume that the popula

7、tion of differences follows a Normal model.o Nearly Normal Condition: Check this with a histogram.Paired t-TestWe test the hypothesis H0: md = D0, where the ds are the pairwise differences and D0 is almost always 0. We use the test statistic: where n is the number of pairs. is the ordinary standard

8、error for the mean applied to the differences.Confidence Interval for Matched Pairs (Paired t-interval)The confidence interval is:Example:Before you took this course, you probably heard many stories about statistics courses. Oftentimes parents of students have had bad experiences with statistics cou

9、rses and pass on their anxieties to their children. To test whether actually taking statistics decreases students anxieties about the subject, a statistics instructor gave a test to rate student anxiety at the beginning and end of his course. Anxiety levels were measured on a scale of 0-10. Here are

10、 the data for 16 randomly chosen students from a class of 180 students: Pre-course anxiety level 7 6 9 5 6 7 5 7 6 4 3 2 1342Post-course anxiety level 4 3 7 3 4 5 4 6 5 3 2 2 1343Difference (Post Pre) -3-3-2-2-2-2-1-1-1-1 -1 0 0001 1. Do the data indicate that anxiety levels about statistics decreas

11、es after students take a statistics course? Test an appropriate hypothesis and state your conclusion. Hypotheses:Assumptions:Test Statistic:Decision:Conclusion:2. Create and interpret a 90% confidence interval. The table below gives the average high temperatures in January and July for several Europ

12、ean cities. Calculate a 90% CI for the mean temperature difference between summer and winter in Europe. Be sure to check the assumptions and to interpret your confidence interval.Mean High Temperatures in degrees Fahrenheit City Jan. JulyVienna3475Copenhagen3672Paris4276Berlin3574Athens5490Rome5488A

13、msterdam4069Madrid4787London4473Edinburgh2176Moscow2176Belgrade3784A company institutes an exercise break for its workers to see if this will improve job satisfaction, as measured by a questionnaire that assesses workers satisfaction. Scores for 10 randomly selected workers before and after the implementation of the exercise program a