Statistical Inference on Two Populations

1. Statistical Inference on Two Populations • Many real world problems involve the comparison of 2 populations • Examples: • Medical trial in which participants are randomly assigned a drug or a placebo • Quality control testing that compares lifetime of product A to product B • Educational testing to compare the performance of men to women on a standardized test

2. Independent Samples • Independent samples are those samples taken from the same population, or different populations, which have no effect on each another. • Can have independent samples when: • Similar subjects receive different treatments • Dissimilar subjects receive the same treatment

3. Dependent Samples • Dependent samples have some relationship so that each value in one sample is paired with a corresponding value in the other sample. • Called matched pairs or paired samples • Can occur when values are clearly paired, or when the same attribute is measured twice on each subject (i.e. pre/post or before/after

4. Inference About Two Means • Appropriate hypothesis for comparing 2 means: • Equivalent hypothesis are:

5. Comparing Two MeansDependent Samples • Dependent data is collected in pairs (x1,x2) • The measure of interest is usually the difference, d= x1-x2. • Hypotheses

6. Comparing Two MeansDependent Samples • Test Statistic: • Rejection Region (3 cases of H1) • Two-tailed: For H1: μd ≠ 0, Reject H0 for |t| ≥ tα/2 • Left-tailed: For H1: μd < 0, Reject H0 for t ≤ -tα • Right-tailed: For H1: μd > 0, Reject H0 for t ≥ tα

7. Comparing Two MeansDependent Samples • Confidence Interval:

8. Comparing Two MeansIndependent Samples (n1,n2 ≥ 30) • Test Statistic: • Rejection Region (3 cases of H1): • Two-tailed: For H1: μ1 ≠ μ2, Reject H0 for |Z| ≥ zα/2 • Left-tailed: For H1: μ1 < μ2, Reject H0 for Z ≤ -zα • Right-tailed: For H1: μ1 > μ2, Reject H0 for Z ≥ zα

9. Comparing Two MeansIndependent Samples (n1,n2 ≥ 30) • Confidence Interval:

10. Comparing Two MeansIndependent Samples (n1 or n2 < 30) • Test Statistic: • Rejection Region (3 cases of H1): • Two-tailed: For H1: μ1 ≠ μ2, Reject H0 for |T| ≥ tα/2 • Left-tailed: For H1: μ1 < μ2, Reject H0 for T ≤ -tα • Right-tailed: For H1: μ1 > μ2, Reject H0 for T ≥ tα

11. Comparing Two MeansIndependent Samples (n1 or n2 < 30) • Degrees of freedom:

12. Comparing Two MeansIndependent Samples (n1 or n2 < 30) • Confidence Interval:

13. Inference About Two Proportions • Appropriate hypothesis for comparing 2 proportions: • Equivalent hypothesis are:

14. Comparing Two Population Proportions • Test Statistic: • Rejection Region (3 cases of H1) • Two-tailed: For H1: p1 ≠ p2, Reject H0 for |Z| ≥ zα/2 • Left-tailed: For H1: p1 < p2, Reject H0 for Z ≤ -zα • Right-tailed: For H1: p1 > p2, Reject H0 for Z ≥ zα

15. Comparing Two Population Proportions • Confidence Interval:

16. Inference About Two Variances • Appropriate hypothesis for comparing 2 variances or standard deviations: • Equivalent hypothesis are:

17. Comparing Two Population Variances • Test Statistic:

18. Comparing Two Population Variances • Rejection Region (3 cases of H1) • Two-tailed: For H1: s1 ≠ s2, Reject H0 for F ≤ F1-α/2, or F ≥ Fα/2 • Left-tailed: For H1: s1 < s2, Reject H0 for F ≤ F1-α • Right-tailed: For H1: s1 > s2, Reject H0 for F ≥ Fα

19. Comparing Two Population Variances • Confidence Interval: