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Two Independent Means Unit 8 HS 167 8: Comparing Two Means 1 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 167 8: Comparing Two Means 2 One sample SRS from one population Comparisons made to an external reference population HS 167 8: Comparing Two Means 3 Paired Sample Two samples with each observation in sample 1 matched to a unique observation in sample 2 Just like a one-sample problem except inferences directed toward within-pair differences DELTA HS 167 8: Comparing Two Means 4 Independent sample inference Independent samples from two populations No matching or pairing HS 167 8: Comparing Two Means 5 What type of sampling method? 1. Measure vitamin content in loaves of bread and see 2. 3. if the average meets national standards. Compare vitamin content of bread immediately after baking versus 3 days later (same loaves are used on day one and 3 days later) 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 167 8: Comparing Two Means 6 Illustrative example: independent samples Goal: compare response variable in two groups 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 HS 167 8: Comparing Two Means 7 Data setup for independent samples Two columns Response variable in one column Explanatory variable in other column HS 167 8: Comparing Two Means 8 Side-by-side boxplots Compare locations, spreads, and shapes 400 Interpretation: 21 20 (1) Different locations (group 1 > group 2) 300 (2) Different spreads (group 1 < group 2) 200 100 N= 20 20 1 2 (3) Shape: fairly symmetrical (but both with outside values) GROUP HS 167 8: Comparing Two Means 9 Summary statistics by group If no major departures from Normality, report means and standard deviations (and sample sizes) Group n mean std dev 1 2 20 20 245.05 210.30 36.64 48.34 Take time to look at your results. HS 167 8: Comparing Two Means 10 Notation for independent samples Parameters (population) Group 1 N1 µ1 σ1 Group 2 N2 µ2 σ2 Group 1 n1 s1 Group 2 n2 x1 x2 Statistics (sample) s2 x1 x2 is the sample mean difference x1 x2 estimates 1 2 HS 167 8: Comparing Two Means 11 Sampling distribution of mean difference The sampling distribution of the mean difference is key to inference x1 x2 ~ N ( 1 2 , SE x1 x2 ) {FIGURE DRAWN ON BOARD} The SDM difference tends to be Normal with expectation μ1 − μ2 and standard deviation SE; (SE discussed next slide) HS 167 8: Comparing Two Means 12 Pooled Standard Error Illustrative data (summary statistics) Group ni si xbari 1 20 36.64 245.05 2 20 48.34 210.30 df1 n1 1 20 1 19 s 2 pooled df 2 n2 1 20 1 19 df df1 df 2 19 19 38 (df1 )( s12 ) (df 2 )( s22 ) df (19)(36.64 2 ) (19)( 48.34 2 ) 38 1839.623 1 1 1 1 SEx1 x2 s 2pooled 1839.623 13.56 20 20 n1 n2 HS 167 8: Comparing Two Means 13 Confidence interval for µ1 – µ2 (1−αlpha)100% confidence interval for µ1 – µ2 ( x1 x2 ) t df ,1 SEx1 x2 2 Illustrative example (Cholesterol in type A and B men) ( x1 x2 ) (t n 1,.975 )( SE x1 x2 ) (245.05 210.30) (2.02)(13.56) 34.75 27.39 (7.36, 62.14) HS 167 8: Comparing Two Means 14 Comparison of CI formulas (point estimate) (t*)( SE ) Type of sample single paired independent HS 167 point estimate df for t* n 1 x nd 1 xd x1 x2 (n1 1) (n2 1) 8: Comparing Two Means SE n delta n 1 1 s 2pooled n1 n2 15 Independent t test A. H0: µ1 = µ2 vs. H1: µ1 > µ2 or H1: µ1 < µ2 or H1: µ1 µ2 Pooled t statistic tstat ( x1 x2 ) SE x1 x2 with df df1 df 2 B. Independent t statistic C. P-value – use t table or software utility to convert tstat to P- value D. Significance level (n1 1) (n2 2) Illustrative example SE x1 x2 13.56 x1 x2 245.05 210.30 34.75 df 19 19 38 tstat x1 x2 34.75 2.56 SE x1 x2 13.56 One - sided P between 0.01 and 0.005 Two - sided P between 0.01 and 0.02 HS 167 8: Comparing Two Means 16 SPSS output These are the pooled (equal variance) statistics calculated in HS 167 HS 167 8: Comparing Two Means 17 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 HS 167 Sampling independence Normality Equal variance 8: Comparing Two Means 18 Sample size requirements for confidence intervals 1.96 n d 2 This will restrict the margin of error to no bigger than plus or minus d HS 167 8: Comparing Two Means 19 Sample size requirement for CI Suppose, you have a variable with = 15 4 152 For d 5, use n 2 36 5 4 152 For d 2.5, use n 144 2 2.5 4 152 For d 1, use n 2 900 1 HS 167 8: Comparing Two Means Sample size requirements increases when you need precision 20 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. HS 167 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. 8: Comparing Two Means 21 Determinants of sample size requirements “Difference worth detecting” () Standard deviation of data () Type I error rate () We consider only .05 two-sided Power of test (we consider on 80% power) HS 167 8: Comparing Two Means 22 Sample size requirements for test Approx. sample size needed for 80% power at alpha = .05 (two-sided) to detect a difference of Δ: 16 n 1 2 2 Illustrative example: Suppose Δ = 25 and = 45 … 2 16 45 n 1 52.8 53 2 25 HS 167 8: Comparing Two Means 23