1. 1 Experimental Statistics - week 3 Statistical Inference 2-sample Hypothesis Tests Review Continued Chapter 8: Inferences about More Than 2 Population Central Values

2. 2 Note: Thursday we will have class in the computer lab (Room 15 Clements - basement. Enter from North side, west stairs.) I suggest that you download the file “car.dat” from my internet site onto a 3 1/4” diskette and bring that to lab.

3. 3 Two Independent Samples Assumptions: Measurements from Each Population are –Mutually Independent  Independent within Each Sample  Independent Between Samples –Normally Distributed (or the Central Limit Theorem can be Invoked) Analysis Differs Based on Whether the Two Populations Have the Same Standard Deviation

4. 4 Two Types of Independent Samples Population Standard Deviations Equal –Can Obtain a Better Estimate of the Common Standard Deviation by Combining or “Pooling” Individual Estimates Population Standard Deviations Different –Must Estimate Each Standard Deviation –Very Good Approximate Tests are Available If Unsure, Do Not Assume Equal Standard Deviations

5. 5 Equal Population Standard Deviations Test Statistic df = n 1 + n 2 - 2 where

6. 6 Behrens-Fisher Problem

7. 7 Satterthwaite’s Approximate t Statistic Approximate t df (i.e. approximate t)

8. 8 Often-Recommended Strategy for Tests on Means Test Whether  1 =  2 (F-test ) –If the test is not rejected, use the 2-sample t statistics, assuming equal standard deviations –If the test is rejected, use Satterthwaite’s approximate t statistic NOTE: This is Not a Wise Strategy –the F-test is highly susceptible to non-normality Recommended Strategy: –If uncertain about whether the standard deviations are equal, use Satterthwaite’s approximate t statistic

9. 9 Example 3: Comparing the Mean Breaking Strengths of 2 Plastics Plastic A: Plastic B: Assumptions: Mutually independent measurements Normal distributions for measurements from each type of plastic Equal population standard deviations

10. 10 New diet -- Is it effective? Design: 50 people: randomly assign 25 to go on diet and 25 to eat normally for next month. Assess results by comparing weights at end of 1 month. Diet: No Diet: Diet: No Diet: Run 2-sample t-test using guidelines we have discussed. Is this a good design?

11. 11 Better Design: Randomly select subjects and measure them before and after 1-month on the diet. Subject Before After 1 150 147 2 210 195 : : : n 187 190 Difference 3 15 : -3 Procedure: Calculate differences, and analyze differences using a 1-sample test “Paired t-Test”

12. 12 Example 4: International Gymnastics Judging Question: Do judges from a contestant’s country rate their own contestant higher than do foreign judges? Data:

13. 13 Assignment -- Due Tuesday, Feb. 1 Problems in Ott and Longnecker: # 5.57, page 241 -- parts (a), (b), and (c). # 6.71, page 330 # 6.83, page 334 (a) For the hypothesis tests, run the tests using the 4-step procedure I gave in class. Also, in each case, find the p-value.

14. 14 Hypothetical Sample Data Scenario A Pop 1 Pop 2 5 8 79 66 38 49 Scenario B Pop 1 Pop 2 3 7 10 4 312 1 4 813 For one scenario, | t | = 1.17 For the other scenario, | t | = 3.35

15. 15 In general, for 2-sample t-tests: To show significance, we want the difference between groups compared to the variability within groups

16. 16 Begin Thursday, Jan 27 lecture

17. 17 Completely Randomized Design 1-Factor Analysis of Variance (ANOVA) Setting (Assumptions): - t populations - populations are normal - mutually independent random samples are taken from the populations

18. 18     1-Factor ANOVA  ...

19. 19 Question: Notes: - not directional i.e. no “1-sided / 2-sided” issues - alternative doesn’t say that all means are distinct

20. 20 Completely Randomized Design 1-Factor Analysis of Variance Example data setup where t = 5 and n = 4

21. 21 Notation:

22. 22 A Sum-of-Squares Identity Note: This is for the case in which all sample sizes are equal ( n ) In words: T otal SS = SS between samples + within sample SS Note: Formula for unequal sample sizes given on page 388

23. 23 In words: TSS(t otal SS) = total sample variability SSB (SS between samples) = variability due to factor effects SSW (within sample SS) = variability due to uncontrolled error

24. 24 Pop 1 5 5 5 5 Pop 2 9 9 9 9 Pop 3 7 7 7 7

25. 25 Pop 1 4 8 3 9 Pop 2 6 10 2 6 Pop 3 5 8 7 4

26. 26 To show significance, we want the difference between groups compared to the variability within groups Recall: For 2-sample t-test to test we use

27. 27 Note: Our test statistic for testing will be of the form This has an F distribution Question: What type of F values lead you to believe the null is NOT TRUE?

28. 28 Analysis of Variance Table Note:

29. 29 Note:

30. 30 CAR DATA Example For this analysis, 5 gasoline types (A - E) were to be tested. Twenty cars were selected for testing and were assigned randomly to the groups (i.e. the gasoline types). Thus, in the analysis, each gasoline type was tested on 4 cars. A performance-based octane reading was obtained for each car, and the question is whether the gasolines differ with respect to this octane reading. A 91.7 91.2 90.9 90.6 B 91.7 91.9 90.9 C 92.4 91.2 91.6 91.0 D 91.8 92.2 92.0 91.4 E 93.1 92.9 92.4

31. 31 ANOVA Table Output - car data Source SS df MS F p-value Between 6.108 4 1.527 6.80 0.0025 samples Within 3.370 15 0.225 samples Totals 9.478 19

32. 32 F-table -- p.1106

33. 33 Extracted from From Ex. 8.2, page 390-391 3 Methods for Reducing Hostility 12 students displaying similar hostility were randomly assigned to 3 treatment methods. Scores (HLT) at end of study recorded. Method 1 96 79 91 85 Method 2 77 76 74 73 Method 3 66 73 69 66 Test:

34. 34 ANOVA Table Output - hostility data Source SS df MS F p-value Between samples Within samples Totals