Class 14 Testing Hypotheses about Means Paired samples 10.3 p

Weight (in pounds) of 72 anorexic patients before and after treatment Weight beforeafter beforeafter beforeafter

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 Variance Kurtosis-0.007Kurtosis Skewness-0.022Skewness0.408 Range24.9Range32.3 Minimum70Minimum71.3 Maximum94.9Maximum103.6 Sum5929.9Sum Count72Count72 Confidence Level(95.0%)1.218Confidence Level(95.0%)1.863 s/n^.5 7.9/72^ / is the 95% confidence interval for the mean.

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

H0: μ b = μ a Ha: μ a > μ b t-Test: Two-Sample Assuming Equal Variances AfterBefore Mean Variance Observations72 Pooled Variance Hypothesized Mean Difference0.000 df142 t Stat2.400 P(T<=t) one-tail 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!

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

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.

H0: μ b = μ a Ha: μ a > μ b Better than before! t-Test: Paired Two Sample for Means AfterBefore Mean Variance Observations72 Pearson Correlation Hypothesized Mean Difference0 df71 t Stat P(T<=t) one-tail t Critical one-tail P(T<=t) two-tail t Critical two-tail1.9939

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…..

H0: μ b = μ a Ha: μ a > μ b IDGroupBeforeAfterAft-Before Average count72 stdev standard error t-stat dof71 p-value A paired two-sample t-test for means Is equivalent to A one-sample t-test of H0: μ A-B = /.92

Case: The Sophomore Jinx

The Data…. Exhibit 1 American League Rookie Award Data, Non Pitchers Rookie YearSophomore Year YearPlayerGABBASAGABBASA 1949Roy Sievers Walter Dropo Gilbert McDougald Harvey Kuenn Ben Grieve Carlos Beltran Ichiro S uzuki Eric Hinske Angel Berroa Exhibit 2 National League Non-Pitchers Rookie YearSophomore Year YearPlayerGABBASAGABBASA 1950Samuel Jethroe Willie Mays James Gilliam Wallace Moon William Virdon Todd Hollandsworth Scott Rolen Rafael Furcal Albert Pujols

H0: Ha: Test Statistic P-value and Conclusion

additional notes….