Chapter 12: Comparing Independent Means

1. Chapter 12: Comparing Independent Means

2. In Chapter 12: 12.1 Paired and Independent Samples 12.2 Exploratory and Descriptive Statistics 12.3 Inference About the Mean Difference 12.4 Equal Variance t Procedure (Optional) 12.5 Conditions for Inference 12.6 Sample Size and Power

3. Types of Samples • Single sample. One group; no concurrent control group • Paired samples. Two samples; data points uniquely matched • Two independent samples. Two samples, separate (unrelated) groups.

4. What Type of Sample? • Measure vitamin content in loaves of bread and see if the average meets national standards • Compare vitamin content of loaves immediately after baking versus content in same loaves 3 days later • Compare vitamin content of bread immediately after baking versus loaves that have been on shelf for 3 days Answers: 1 = single sample 2 = paired samples 3 = independent samples

5. Experimental vs. Observational Groups Independent samples can • Experimental –an intervention or treatment is assigned as part of the study protocol • Non-experimental (observational) – groups defined by a innate characteristics or self-selected exposure “Two Groups” by Pieter Bruegel the Elder (c. 1525 – 1569)

6. Illustrative Data* * Data set WCGS.sav (p. 49) Type A personality men (n = 20)233, 291, 312, 250, 246, 197, 268, 224, 239, 239, 254, 276, 234, 181, 248, 252, 202, 218, 212, 325 Type B personality men (n = 20)344, 185, 263, 246, 224, 212, 188, 250, 148, 169, 226, 175, 242, 252, 153, 183, 137, 202, 194, 213 Do means from these populations differ? If so, by how much?

7. Notation

8. Illustrative DataCholesterol levels (mg / dL) Type A men in the sample have higher average cholesterol by 35 mg/dL

9. Standard Error To address this question, calculate the standard error of the mean difference:

10. Degrees of Freedom • Two ways to estimate degrees of freedom: • dfWelch[complex formula on p. 244 of text] • dfconserv. = the smaller of (n1 – 1) or (n2 – 1) For the illustrative data: dfWelch = 35.4 (via SPSS) dfWelch = 35.4 (via SPSS) dfconserv. = smaller of (n1–1) or (n2 – 1) = 20 – 1 = 19 dfconserv. = smaller of (n1–1) or (n2 – 1) = 20 – 1 = 19

11. (1 – α)100% CI for µ1–µ2 Note: (point estimate) ± (t)(SE) margin of error

12. Comparison of CI Formulas

13. Example

14. Interpretation The CI interval aims for µ1− µ2 with (1– α)100% confidence

15. HypothesisTest • Test claim of “no difference in populations” • Note: widely different sample means can arise just by chance • Null hypothesis: H0: μ1 – μ2 = 0 (equivalently H0: μ1 = μ2) • Alternative hypothesis Ha: μ1 – μ2 ≠ 0 (two-sided) OR Ha: μ1 – μ2 > 0 (“right-sided”) ORHa: μ1 – μ2 < 0 (“left-sided”)

16. Test Statistic dfWelch= 35.4 (via SPSS) dfconserv. = 19

17. P-value via Table C tstat = 2.56 with 19 df • One-tailed P between .01 and .005 • Two-tailed P between .02 and .01 (i.e., less than .02) • .01 < P < .02 provides good evidence against H0  observed difference is statistically significant

18. SPSS Response variable (chol) in one column Explanatory variable (group) in a different column

19. Equal Variance Not AssumedPreferred method (§12.3) Equal variance t procedure (§12.4) SPSS Output

20. Summary of independent t test • H0: μ1 –μ2 = 0 C. P-value from Table C or computer(Interpret in usual fashion)

21. Hypothesis Test with the CI • H0: μ1 – μ2 = 0 can be tested at α-level of significance with the (1 – α)100% CI • Example: 95% CI for μ1 – μ2 = (6.4 to 63.1)  excludes μ1 – μ2 = 0  Significant difference at α = .05

22. Hypothesis Test with the CI • H0: μ1 – μ2 = 0 • 99% CI for μ1 – μ2 is (-2.2 to 71.7), which includes μ1 – μ2 = 0  Not Significant at α = .01