1. November 15

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. Sample Types (for Comparing Means) Single sample. One group; no concurrent control group, comparisons made to external population (Ch 11) Paired samples. Two samples w/ each data point in one sample uniquely matched to a point in the other; analyze within-pair differences (Ch 11) Two independent samples. Two separate groups; no matching or pairing; compare separate groups

4. Quantitative outcome One sample §11.1 – §11.4 Two samples Paired samples §11.5 Independent samples Chapter 12

5. What Type of Sample? 1.Measure vitamin content in loaves of bread and see if the average meets national standards 2.Compare vitamin content of bread loaves immediately after baking versus values in same loaves 3 days later 3.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

6. Illustrative Example: Cholesterol and Type A & B Personality Group 1 (Type A personality): 233, 291, 312, 250, 246, 197, 268, 224, 239, 239, 254, 276, 234, 181, 248, 252, 202, 218, 212, 325 Group 2 (Type B personality): 344, 185, 263, 246, 224, 212, 188, 250, 148, 169, 226, 175, 242, 252, 153, 183, 137, 202, 194, 213 Do fasting cholesterol levels differ in Type A and Type B personality men? Data (mg/dl) are a subset from the Western Collaborative Group Study* * Data set is documented on p. 49 in the text.

7. SPSS Data Table One column for the response variable (chol) One column for the explanatory variable (group)

8. §12.2: Exploratory & Descriptive Methods Start with EDA Compare group shapes, locations and spreads Examples of applicable techniques Side-by-side stemplots (right) Side-by-side boxplots (next slide) Group 1 | | Group 2 -------------------- |1t|3 |1f|45 |1s|67 98|1.|8889 110|2*|011 33332|2t|22 55544|2f|4455 76|2s|6 9|2.| 21|3*| |3t| |3f|4 (×100)

9. Side-by-Side Boxplots Interpretation : Location: group 1 > group 2 Spreads: group 1 < group 2 Shapes : Both fairly symmetrical, outside values in each; no major departures from Normality

10. Summary Statistics Groupnmeanstd dev 120245.0536.64 220210.3048.34

11. §12.3 Inference About Mean Difference (Notation) Parameters (population) Group 1N1N1 µ1µ1 σ1σ1 Group 2N2N2 µ2µ2 σ2σ2 Statistics (sample) Group 1n1n1 s1s1 Group 2n2n2 s2s2

12. Standard Error of Mean Difference Standard error of the mean difference

13. There are two ways to estimate the degrees of freedom for this SE: df Welch = formula on p. 244 [calculate w/ computer] df conservative = the smaller of (n 1 – 1) or (n 2 – 1) For the cholesterol comparison data: df conservative = smaller of (n 1 –1) or (n 2 – 1) = 20 – 1 = 19

14. Confidence Interval for µ 1 –µ 2 (1−α)100% confidence interval for µ 1 – µ 2 = For the cholesterol comparison data:

15. Comparison of CI Formulas Type of sample point estimate df for t*SE singlen – 1 pairedn – 1 independent smaller of n 1 −1 or n 2 −1

16. Hypothesis Test A.Hypotheses. H 0 : μ 1 = μ 2 against H a : μ 1 ≠ μ 2 (two-sided) [H a : μ 1 > μ 2 (right-sided) H a : μ 1 < μ 2 (left-sided) ] B.Test statistic. C. P-value. Convert the t stat to P-value with t table or software. Interpret. D. Significance level (optional). Compare P to prior specified α level.

17. Hypothesis Test – Example A. Hypotheses. H 0 : μ 1 = μ 2 vs. H a : μ 1 ≠ μ 2 B. Test stat. In prior analyses we calculated sample mean difference = 34.75 mg/dL, SE = 13.563 and df conserv = 19. C. P-value. P = 0.019 → good evidence against H 0 (“significant difference”). D. Significance level (optional). The evidence against H 0 is significant at α = 0.02 but not at α = 0.01.

18. Equal variance t procedure (§12.4) Preferred method (§12.3) SPSS Output

19. 12.4 Equal Variance t Procedure (Optional) Also called pooled variance t procedure Not as robust as prior method, but… Historically important Calculated by software programs Leads to advanced ANOVA techniques

20. We start by calculating this pooled estimate of variance Pooled variance procedure

21. The pooled variance is used to calculate this standard error estimate: Confidence Interval Test statistic All with df = df 1 + df 2 = (n 1 −1) + (n 2 −1)

22. Pooled Variance t Confidence Interval Groupnini sisi xbar i 12036.64245.05 22048.34210.30 Data

23. Pooled Variance t Test Data: Groupnini sisi xbar i 12036.64245.05 22048.34210.30

24. §12.5 Conditions for Inference Conditions required for t procedures: “Validity conditions” a. Good information (no information bias) b. Good sample (“no selection bias”) c. “No confounding” “Sampling conditions” a. Independence b. Normal sampling distribution (§9.5, §11.6)