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

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Presentation on theme: "Class 14 Testing Hypotheses about Means Paired samples 10.3 p 419-425."— Presentation transcript:

1 Class 14 Testing Hypotheses about Means Paired samples 10.3 p

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

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 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.

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

5 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!

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 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

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 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

11 Case: The Sophomore Jinx


13 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

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

15 additional notes….

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