1. Class 14 Testing Hypotheses about Means Paired samples 10.3 p 419-425

2. Weight (in pounds) of 72 anorexic patients before and after treatment Weight beforeafter beforeafter beforeafter 80.780.2 72.388.2 80.282.6 89.481.0 89.078.8 87.8100.4 91.886.4 80.582.2 83.385.2 74.086.3 84.985.6 79.783.6 78.176.1 81.581.4 84.584.6 88.378.1 82.681.9 80.886.2 87.375.1 79.976.4 87.486.7 75.186.7 88.7103.6 83.695.2 80.673.5 94.998.4 83.394.3 78.484.6 76.393.4 86.091.5 77.677.4 81.073.4 82.591.9 88.779.5 80.582.1 86.7100.3 81.389.6 85.096.7 79.676.7 78.181.4 89.295.3 76.976.8 70.581.8 81.382.4 94.2101.6 77.3 76.572.5 73.494.9 85.284.2 70.090.9 80.575.2 86.075.4 80.471.3 81.677.3 81.479.5 83.385.4 82.195.5 79.773.0 83.081.6 77.690.7 85.588.3 87.789.1 83.592.5 84.484.7 84.283.9 89.993.8 79.081.4 86.482.7 86.091.7 77.581.2 76.575.7 87.398.0

3. Data/Data Analysis/ Descriptive Statistics/Summary Statistics and Confidence Level for Mean Before After Mean82.36Mean85.04 Standard Error0.61Standard Error0.93 Median81.85Median84.05 Mode86Mode81.4 Standard Deviation5.184Standard Deviation7.927 Sample Variance26.875Sample Variance62.838 Kurtosis-0.007Kurtosis-0.614 Skewness-0.022Skewness0.408 Range24.9Range32.3 Minimum70Minimum71.3 Maximum94.9Maximum103.6 Sum5929.9Sum6122.8 Count72Count72 Confidence Level(95.0%)1.218Confidence Level(95.0%)1.863 s/n^.5 7.9/72^.5 82.36 +/- 1.218 is the 95% confidence interval for the mean.

4. H0: μ b = μ a Ha: μ a > μ b Test Statistic P-value = t.dist.rt(2.40,142) = 0.0088

5. H0: μ b = μ a Ha: μ a > μ b t-Test: Two-Sample Assuming Equal Variances AfterBefore Mean85.03982.360 Variance62.83826.875 Observations72 Pooled Variance44.857 Hypothesized Mean Difference0.000 df142 t Stat2.400 P(T<=t) one-tail0.00884 t Critical one-tail1.656 P(T<=t) two-tail0.018 t Critical two-tail1.977 Same as previous slide! Data must be in two columns. If this is all you want, =t.test() is for you!

6. The 2-sample t-test we just did is VALID. But we can do better….. By taking advantage of our paired data.

7. Paired Data n1 must equal n2 For each of the before values, there must be a corresponding after value for the same element. – Here the data elements are the patients. And the paired nature of the data is OBVIOUS. Using a paired test when the data are paired USUALLY leads to a valid and LOWER p-value. – Because s1 and s2 (the standard deviations of each group) do NOT enter into the “equation” – Instead, we use the sample standard deviation of the n differences…which is usually “pretty” small. Instead of dealing with the variation in weights across the patients (s1 and s2), we deal only with the variation in pounds gained. – 90 to 92 and 45 to 47 are both gains of 2.

8. H0: μ b = μ a Ha: μ a > μ b Better than before! t-Test: Paired Two Sample for Means AfterBefore Mean85.03982.36 Variance62.83826.875 Observations72 Pearson Correlation0.3498 Hypothesized Mean Difference0 df71 t Stat2.9116 P(T<=t) one-tail0.0024 t Critical one-tail1.6666 P(T<=t) two-tail0.0048 t Critical two-tail1.9939

9. H0: μ b = μ a Ha: μ a > μ b The = t.dist(array1,array2,1,1) takes you directly to the p-value 1 for 1-tail 1 for paired If all you want is the p-value…..

10. H0: μ b = μ a Ha: μ a > μ b IDGroupBeforeAfterAft-Before 1180.780.2-0.5 2189.481-8.4 3191.886.4-5.4 417486.312.3 5178.176.1-2 6188.378.1-10.2 67382.195.513.4 68377.690.713.1 69383.592.59 70389.993.83.9 7138691.75.7 72387.39810.7 Average2.679167 count72 stdev7.807796 standard error0.920158 t-stat2.911639 dof71 p-value0.002401 A paired two-sample t-test for means Is equivalent to A one-sample t-test of H0: μ A-B = 0. 2.68/.92

11. Case: The Sophomore Jinx

13. The Data…. Exhibit 1 American League Rookie Award Data, Non Pitchers Rookie YearSophomore Year YearPlayerGABBASAGABBASA 1949Roy Sievers140471306471113370238395 1950Walter Dropo13655932258399360239369 1951Gilbert McDougald131402306488152555263369 1953Harvey Kuenn155679308386155656306390 1998Ben Grieve155583288458148486265481 1999Carlos Beltran15666329345498372247366 2001Ichiro S uzuki 157692350457157647321425 2002Eric Hinske151566279481124449243437 2003Angel Berroa158567287451134512262385 Exhibit 2 National League Non-Pitchers Rookie YearSophomore Year YearPlayerGABBASAGABBASA 1950Samuel Jethroe141582273442148572280460 1951Willie Mays12146427447234127236409 1953James Gilliam151605278415146607282418 1954Wallace Moon151635304435152593295459 1955William Virdon144534281433157580319445 1996Todd Hollandsworth149478291437106296247368 1997Scott Rolen156561283469160601290532 2000Rafael Furcal13145529538279324275370 2001Albert Pujols161590329610157590314561

14. H0: Ha: Test Statistic P-value and Conclusion

15. additional notes….