Warm Up

1. Warm Up • Of all Geometry students at NRHS a sample was taken to try to test the claim that their mean score on the last quiz was an 87. The following are the sample scores: 86, 85, 90, 91, 74, 87, 82, 70, 58, 78, 81 • Test if the true mean score is different from the claim. • If claim is rejected, estimate the true mean score with 95% confidence.

2. Comparing Two Means Section 11.2

3. Two Sample Problems • Compare the responses to two treatments or compare the characteristics of two populations. • Used when we have a separate sample from each treatment or population. • Examples: • Randomized comparative experiment in medical research (placebo vs drug). • Test of social insight (men vs women). • Compare incentives to use a credit card.

4. Comparing Two Population Means • First compare graphically. • Assumptions: • Both samples are SRS’s from 2 distinct populations. • Samples are Independent. • Both populations are normally distributed with µ and σ unknown.

5. Two Sample Notation Population variable mean standard deviation 1 x1 µ1σ1 2 x2 µ2σ2 sample sample sample Population size mean standard deviation 1 n1 x1 s1 2 n2 x2 s2

6. Two Sample Inference • To compare the two population means. • Give a confidence interval to estimate the difference between the means, µ1 - µ2 • Test hypothesis that there is no difference, H0: µ1 = µ2

7. Sampling Distribution of x1 – x2 • Mean of sampling distribution = µ1 - µ2 , so unbiased estimator. • Variance of the difference is sum of variances of each sample mean: • If both populations are normal then sampling distribution is normal. • If normal, we could standardize and do 2-sample z test. • But we don’t know σ1 or σ2 so we use t-test.

8. Two Sample t Procedures • Use smaller of the two sample’s degrees of freedom • Confidence Interval:

9. Two Sample t Procedures • Hypothesis Testing: Test the claim H0: µ1 = µ2 • Since we use smaller df our results will be conservative: true p-value ≤ calculated value, and CI has higher confidence level than C

10. Robustness • Two-sample t procedures are more robust than 1-sample when distribution is not symmetric. • When 2 samples are same size with similar shapes t results are accurate even with small samples: n’s = 5 (different shapes need larger samples). • Same guidelines as 1-sample t procedures but replace “sample size” with “sum of sample sizes”

11. 2-Sample Mean t Procedure Guidelines • SRS is more important than normal population, except with small samples. • Sample size sums < 15: use t procedures if data are close to normal; do not use if clearly nonnormal or outliers present. • Sample size sums ≥ 15: ok to use t procedures unless outliers or strong skewness. • Large sample sums ≥ 40: ok to use t procedures even if clearly skewed.

12. Example: Does Calcium reduce blood pressure? • 21 men, 10 given calcium and 11 given placebo. (double blind). • Response variable is the decrease in systolic blood pressure after 12 weeks. (increase is negative) • Group 1 (calcium): 7 -4 18 17 -3 -5 1 10 11 -2 • Group 2 (placebo): -1 12 -1 -3 3 -5 5 2 -11 -1 -3

13. Example: Does Calcium reduce blood pressure? • Calculate Summary Statistics: Group Treatment n x s 1 Calcium 10 5.000 8.743 2 Placebo 11 -.273 5.901 • Hypothesis Test: • Hypotheses and ID populations and parameter of interest • Assumptions: SRS? Normal?~Graph p.621 • t statistic, p-value • Summary • 90% Confidence Interval • Calculator

14. Calcium Example

15. Example: Social Insight Test • Scores range from 0-41; male and female liberal arts college students are tested. • Results: Group Sex n x s 1 Male 133 25.34 5.05 2 Female 162 24.94 5.44 • Do males and females differ in social insight? • Hypothesis Test • Hypotheses: • Assumptions: SRS? Normal? • Test: Calculator: 2 sample t test, 2-sided • Summary

16. Social Insight Example