1. 1 Chapter 14 Statistical Inference: Other Two- Sample Test Statistics ITwo-Sample F Test for Population Variances Using Independent Samples A.Statistical Hypotheses

2. 2 B.F Test Statistic for Two Variances (Independent Samples) 1.The degrees of freedom for the numerator and denominator are, respectively,

3. 3 C.F Sampling Distribution Figure 1. Sampling distribution of F. For the critical region is in the upper tail of the sampling distribution. The critical value in the upper tail is denoted by

4. 4 D.Assumptions of the F Test 1.Samples are independent 2.Populations are normally distributed 3.Participants are random samples or have been randomly assigned to the conditions E.Robustness 1.The F test is not robust with respect to violation of normality assumption.

5. 5 Table 1. Upper Percentage Points of the F Distribution.251.321.301.291.28 30.101.721.671.641.61.052.011.931.891.84.012.702.552.472.39.251.301.281.261.25 40.101.661.611.571.54.051.921.841.791.74.012.522.372.292.20.251.271.251.241.22 60.101.601.541.511.48.051.841.751.701.65.012.352.202.122.03

6. 6 F.Computational Example 1.Statistical hypotheses

7. 7 G.Lower Tail Critical Values 1.Lower tail critical values are denoted by 2.Only upper tail critical values are given in Appendix Table D.5. 3.Appendix Table D.5 can be used to obtain lower tail critical values as follows.

8. 8 IIConfidence Interval for Two Variances (Independent Samples) A. Two-Sided Confidence Interval 1.A two-sided 100(1 –  )% confidence interval is where and are the values of F that cut off the upper  /2 region of the sampling distribution of F for 1 = n 1 – 1 and 2 = n 2 – 1.

9. 9 B. One-Sided Confidence Interval 1.Lower 100(1 –  )% confidence interval 2.Upper 100(1 –  )% confidence interval

10. 10 C.Computational Example

11. 11 IIITwo-Sample t Test for Population Variances Using Dependent Samples A.Statistical Hypotheses B.t Test Statistic

12. 12 C.Assumptions of the t Test 1.Samples are dependent 2.Populations are normally distributed 3.Participants are random samples or have been randomly assigned to the conditions

13. 13 IVConfidence Interval for Two Variances (Dependent Samples) A. Two-Sided Confidence Interval

14. 14 B.One-Sided Confidence Interval 1.Lower 100(1 –  )% confidence interval 2.Upper 100(1 –  )% confidence interval

15. 15 3.Margin of error, m, associated with the difference is Vz Test for Two Proportions Using Independent Samples A.Statistical Hypotheses H 0 : p 1 = p 2 H 0 : p 1 ≤ p 2 H 0 : p 1 ≥ p 2 H 1 : p 1 ≠ p 2 H 1 : p 1 > p 2 H 1 : p 1 < p 2

16. 16 B.z Test Statistic 1.The denominator of z is an estimator of the standard error of the difference between two population proportions.

17. 17 C. Assumptions of the z Test 1.Samples are independent 2.Binomial distributions 3.Participants are random samples or have been randomly assigned to the conditions 4. 5.Both populations are at least 10 larger than their respective samples

18. 18 D.Computational Example: Landmark Study of the Effects of Aspirin On Heart Attacks 1.22,071 men physicians were randomly assigned to an aspirin group (n 1 = 11,037) and a placebo group (n 2 = 11,034) in a double-blind study. 2.Proportion of participants who suffered one or more heart attacks in the aspirin group = 139/11,037 =.01259 and the placebo group = 239/11,034 =.02166.

19. 19 3.Statistical hypotheses H 0 : p 1 = p 2 H 1 : p 1 ≠ p 2 4.z statistic

20. 20 5.The difference between the aspirin and placebo groups was.01259 –.02166 =.0091. The researchers concluded that the use of aspirin would result in almost one million fewer heart attacks over a five-year period. 6.The difference,.01259 –.02166 =.0091, was described as “a statistically extreme beneficial effect.”

21. 21 VIz Confidence Interval for Two Proportions (Independent Samples) A. Two-Sided 100(1 –  Confidence Interval

22. 22 B. One-Sided Confidence Interval 1.Lower 100(1 –  )% confidence interval 2.Upper 100(1 –  )% confidence interval

23. 23 C. Assumptions of the z Confidence Interval 1.Samples are independent 2.Binomial distributions 3.Participants are random samples or have been randomly assigned to the conditions 4. 5.Both populations are at least 10 larger than their respective samples

24. 24 D.Confidence Interval for the Aspirin Experiment

25. 25 1.Margin of error associated with the difference

26. 26 VIIz Test for Two Proportions Using Dependent Samples A.Statistical Hypotheses H 0 : p 1 = p 2 H 0 : p 1 ≤ p 2 H 0 : p 1 ≥ p 2 H 1 : p 1 ≠ p 2 H 1 : p 1 > p 2 H 1 : p 1 < p 2 1.To test one of the null hypotheses, n elements of two dependent samples are placed in a 2 by 2 table.

27. 27 Sample 2 Category 0Category 1 Category 1aba + b Sample 1 Category 0cdc + d a + cb + dn

28. 28 B.z Test Statistic

29. 29 C. Assumptions of the z Test 1.Samples are dependent 2.Binomial distributions 3.Participants are random samples or have been randomly assigned to the conditions 4.a + d ≥ 10 for two-tailed test and ≥ 30 for one-tailed test

30. 30 D.Computational Example 1.A random sample of n = 200 students are polled about whether they approve or disapprove of capital punishment before and after seeing a film on the effects of crime on victims. 2.The statistical hypotheses are H 0 : p 1 ≥ p 2 H 1 : p 1 < p 2

31. 31 Sample 2 DisapproveApprove Approvea = 5b = 25a + b = 30 Sample 1 Disapprovec = 140d = 30c + d = 170 a + c = 145b + d = 55n = 200

32. 32 VIIIz Confidence Interval for Two Proportions (Dependent Samples) A.Two-Sided 100(1 –  Confidence Interval

33. 33 B. One-Sided Confidence Interval 1.Lower 100(1 –  )% confidence interval 2.Upper 100(1 –  )% confidence interval

34. 34 C.Confidence Interval for the Attitude Experiment

35. 35 1.Margin of error associated with the difference

36. 36 D. Assumptions of the z Confidence Interval 1.Samples are dependent 2.Binomial distributions 3.Participants are random samples or have been randomly assigned to the conditions 4.a + d ≥ 10 for two-tailed test and ≥ 30 for one-tailed test