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Chapter 25

Chapter 25. Paired Samples and Blocks. Independent vs. Dependent Samples.

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Chapter 25

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  1. Chapter 25 Paired Samples and Blocks

  2. Independent vs. Dependent Samples • We would like to determine if students taking an ACT prep course will score better than students not taking the course. A random sample of 25 students was chosen who took the course and a random sample of another 25 students was chosen who did not take the course. At the end of the prep course, both groups were given the ACT.

  3. Independent vs. Dependent Samples • Group #1 – score on ACT from students taking prep course • Group #2 – score on ACT from students NOT taking prep course • Observations taken from two independent samples

  4. Independent vs. Dependent Samples • We would like to determine if students can improve their ACT score by taking a prep course. A random sample of 25 students was chosen. They first took the ACT test. Then they spent 6 weeks taking the prep course. At the end of the 6 weeks, they took the ACT test again.

  5. Independent vs. Dependent Samples • Group #1 – score on ACT before prep course • Group #2 – score on ACT after prep course • Two observations taken for each subject

  6. Important Difference • If the values come from: • two independent samples • use inference for (μ1 – μ2) = difference in means for two groups • dependent samples (e.g. values collected twice from same subject) • use inference for μd = mean difference between two values • The second situation is called Matched Pairs

  7. Inference for μd • d = pairwise difference • Confidence Interval • Hypothesis Test (degrees of freedom = n-1)

  8. Pairwise Differences • ACT Prep Course: Does the course improve scores?

  9. Matched Pairs Hypothesis Test Example • Measure effectiveness of exercise program in lowering blood cholesterol levels • SRS of men from a population • Measure cholesterol before program starts • Go through exercise program for 12 weeks • Measure cholesterol after program ends • Is the exercise program effective?

  10. Matched Pairs Hypothesis Test Example - Data

  11. Matched Pairs Hypothesis TestExample • Two observations from each subject • Post value depends on pre value • Observations are dependent • Do not have independent samples • Cannot use inference for μ1 - μ2 • Violates independent samples assumption

  12. Matched Pairs Hypothesis Test Example - Data • Look at pre-program levels minus post-program levels

  13. Matched Pairs • Because it is the differences we are interested in, we will treat them as the data and ignore the original two groups. • A matched pairs t-test is just a one-sample t-test (from Chapter 23) for the mean of the pairwise differences.

  14. Matched Pairs Hypothesis Test Example • Let μd = mean difference in cholesterol levels in population • Sample mean difference is • Sample standard deviation of differences is sd = 17.99 • n = number of differences = 10

  15. Matched Pairs Hypothesis Test Example • If exercise program has no effect, mean difference will be 0 • If exercise program is effective, mean difference will be positive • Step 1: • HO: μd = 0 • HA: μd > 0

  16. Matched Pairs Hypothesis Test Example • Step 2: • Assumptions: • Random sample • Nearly Normal Population • Test statistic:

  17. Matched Pairs Hypothesis Test Example • Step 3: • P-value = P(t9 > 1.42) = between 0.05 and 0.10

  18. Matched Pairs Hypothesis Test Example • Step 4: • Decision: Since p-value > 0.05 = α, we will fail to reject the null hypothesis. • Conclusion: There is no evidence that the exercise program reduced the mean cholesterol level of men in this population.

  19. Matched Pairs Confidence Interval Example • For a random sample of 12 European cities, the average high temperatures in January and July are given on the following slide. • Find a 90% confidence interval for the mean temperature difference between summer and winter in Europe. Assume the Nearly Normal assumption is satisfied.

  20. Matched Pairs Confidence Interval Example

  21. Matched Pairs Confidence Interval Example • Check assumptions • Random • ok • Nearly Normal • ok

  22. Matched Pairs Confidence Interval Example

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