HS 1678: Comparing Two Means1 Two Independent Means Unit 8

HS 1678: Comparing Two Means2 Sampling Considerations One sample or two? If two samples, paired or independent? Is the response variable quantitative or categorical? Am I interested in the mean difference? This chapter → two independent samples → quantitative response → interest in mean difference

HS 1678: Comparing Two Means3 One sample Comparisons made to an external reference population SRS from one population

HS 1678: Comparing Two Means4 Paired Sample Just like a one-sample problem except inferences directed toward within-pair differences DELTA Two samples with each observation in sample 1 matched to a unique observation in sample 2

HS 1678: Comparing Two Means5 Independent sample inference No matching or pairing Independent samples from two populations

HS 1678: Comparing Two Means6 What type of sampling method? 1. Measure vitamin content in loaves of bread and see if the average meets national standards. 2. Compare vitamin content of bread immediately after baking versus 3 days later (same loaves are used on day one and 3 days later) 3. Compare vitamin content of bread immediately after baking versus loaves that have been on shelf for 3 days 1 = single sample 2 = paired samples 3 = independent samples

HS 1678: Comparing Two Means7 Illustrative example: independent samples Fasting cholesterol (mg/dl) 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 Goal: compare response variable in two groups

HS 1678: Comparing Two Means8 Data setup for independent samples Two columns Response variable in one column Explanatory variable in other column

HS 1678: Comparing Two Means9 Side-by-side boxplots Interpretation: (1)Different locations (group 1 > group 2) (2)Different spreads (group 1 < group 2) (3)Shape: fairly symmetrical (but both with outside values) Compare locations, spreads, and shapes

HS 1678: Comparing Two Means10 Summary statistics by group Groupnmeanstd dev Take time to look at your results. If no major departures from Normality, report means and standard deviations (and sample sizes)

HS 1678: Comparing Two Means11 Notation for independent samples Parameters (population) Group 1N1N1 µ1µ1 σ1σ1 Group 2N2N2 µ2µ2 σ2σ2 Statistics (sample) Group 1n1n1 s1s1 Group 2n2n2 s2s2

HS 1678: Comparing Two Means12 Sampling distribution of mean difference The sampling distribution of the mean difference is key to inference {FIGURE DRAWN ON BOARD} The SDM difference tends to be Normal with expectation μ 1 − μ 2 and standard deviation SE; (SE discussed next slide)

HS 1678: Comparing Two Means13 Pooled Standard Error Illustrative data (summary statistics) Groupnini sisi xbar i

HS 1678: Comparing Two Means14 Confidence interval for µ 1 – µ 2 Illustrative example (Cholesterol in type A and B men) (1−αlpha)100% confidence interval for µ 1 – µ 2

HS 1678: Comparing Two Means15 Comparison of CI formulas Type of sample point estimate df for t*SE single paired independent

HS 1678: Comparing Two Means16 Independent t test A. H 0 : µ 1 = µ 2 vs. H 1 : µ 1 > µ 2 or H 1 : µ 1 < µ 2 or H 1 : µ 1  µ 2 B. Independent t statistic C. P-value – use t table or software utility to convert t stat to P- value D. Significance level Pooled t statistic Illustrative example

HS 1678: Comparing Two Means17 SPSS output These are the pooled (equal variance) statistics calculated in HS 167

HS 1678: Comparing Two Means18 Conditions necessary for t procedures Validity assumptions good information (no information bias) good sample (“no selection bias”) good comparison (“no confounding” – no lurking variables) Distributional assumptions Sampling independence Normality Equal variance

HS 1678: Comparing Two Means19 Sample size requirements for confidence intervals This will restrict the margin of error to no bigger than plus or minus d

HS 1678: Comparing Two Means20 Sample size requirement for CI Suppose, you have a variable with  = 15 Sample size requirements increases when you need precision

HS 1678: Comparing Two Means21 Sample size for significance test Goal: to conduct a significance test with adequate power to detect “a difference worth detecting” The difference worth detecting is a difference difference worth finding. In a study of an anti-hypertensives for instance, a drop of 10 mm Hg might be worth detecting, while a drop of 1 mm Hg might not be worth detecting. In a study on weight loss, a drop of 5 pounds might be meaningful in a population of runway models, but may be meaningless in a morbidly obese population.

HS 1678: Comparing Two Means22 Determinants of sample size requirements “Difference worth detecting” (  ) Standard deviation of data (  ) Type I error rate (  We consider only  two-sided Power of test (we consider on 80% power)

HS 1678: Comparing Two Means23 Sample size requirements for test Approx. sample size needed for 80% power at alpha =.05 (two-sided) to detect a difference of Δ: Illustrative example: Suppose Δ = 25 and  = 45 …