1. Testing Differences in Means (t-tests) Dr. Richard Jackson jackson_r@mercer.edu © Mercer University 2005 All Rights Reserved

2. Student t test A parametric statistic Tests difference in 2 means William Gossett

3. Steps in Research State Null Hypothesis. State alternative Hypothesis. Determine Significance Level Collect Data Calculate Test Statistic (example = t) Accept or Reject Null Hypothesis Make Conclusions

4. Requirements of the t test 2 means Continuous Data Normally distributed

5. Hypothesis Associated with t H 0 : m 1 = m 2 H 1 : m 2  m 2

6. Types of Samples Associated with t Repeated Measures of Paired (See Table I) Independent (See Table III)

7. If Requirements Not Met, Use Non-Parametric Counterparts Repeated Measures – Wilcoxon Signed Rank or Sign Test Independent – Mann Whitney U.

8. Formula for t t = X 1 - X 2 S DX Similar to Z A “ Difference ” / A Standard Deviation

9. Standard of Difference in Means Similar to Standard Error of Mean Replicate Study to Determine Difference in 2 Groups Many Times

10. Standard Error of Difference In Means XXX 1 -X 2 23212 31322 43441 21 2 29394

11. Repeated Measures (Paired) t (See Table I) PatientBeforeAfterDifference 11201173 2100964 31101055 490846 51301237

12. Null Hypothesis H o : m b =m a X b =110 X a =105

13. Calculation of t Using Statistix (See Table II) Mean Difference is 5 STD Error of Difference is 0.7071 t = -7.07 p = 0.0021

14. Conclusion A priori significance label set at 0.05 p = 0.0021 Reject H o (p < 0.05) Conclusion: “ Significant ” difference in before and after

15. Independent Sample t (See Table III) Diet A 177 200 251 239 190 180 210 185 Diet B 142 155 141 205 147 171 213 164

16. Hypothesis H o : m a = m b H 1 : m a  m b X a = 204; X b = 167.3

17. Calculation of t Using Statistix (See Table IV) Test for Equality of Variances (p=0.49) Use T for Equal Variances T = 2.65, p = 0.0191 Reject H o (p < 0.05) Conclusion: Difference is “ Significant ”

18. Use of t Table (See Table V) Compare Calculated t with Tabled t Calculated t > Table t : Reject H o Calculated t  Table t : Accept H o

19. Degrees of Freedom (Sample Size) (See Table V) Independent (N 1 + N 2 – 2) Repeated (N – 1)

20. One–Tail Versus Two-Tail Test (See Table V) H   m, <m 2 Prior Knowledge of Difference

21. One-Tail Versus Two-Tail (See Table V) When in Doubt, use Two-Tail Two-Tail More Conservative

22. Significance Level Access Top Most Times Use 0.05

23. Example Using Repeated Measures t Degrees of Freedom = N-1 = 5-1 = 4 Two-Tail Test Significance Level = 0.05 Tabled Value = 2.776 Calculated Value = -7.07 Conclusion Reject H o

24. Example Using Independent t Degrees of Freedom = N 1 +N 2 -2 = 14 Two-Tail Test Significance Level = 0.05 Tabled Value = 2.145 Calculated t = 2.65 Conclusion: Reject H o

25. Observations About t Table As Sample Size Increases, Tables Value Decreases As Significance Level Decreases, Tabled Value Increases Two-Tail Tabled Value Larger than One- Tail Tabled Value for Some Significance Level

26. Sample Size Determination Power Desired (Average = 0.80) Variability of Groups How Small Difference Detect

27. Example Sample Size for t N = 16 S 2 /D 2 S = Standard Deviation of subjects D = Smallest difference to detect

28. Example Sample Size for t Cholesterol Levels in 2 groups Range Estimate = 170-230 = 60 60/6 = 10 = S D Estimated at 10 N = 16(10) 2 /(10) 2 = 16

29. Summary for t Difference in 2 means Data Continuous and Normally Distributed Calculated t with p value allows Researcher to Accept/Reject H o p-Value Provides Probability of Type I Error if Reject

30. Computer Exercise: t Tests See exercise at end of module. Using the Statistix software, analyze the data in each of the problems. See instructions in next slide.

31. How to Perform t Tests Using Statistix Enter Variables and Data Select Statistics Select One, Two, Multi-Sample Tests Select Paired t Test or Two-Sample t Test For Paired t: Select Variables then OK For Two-Sample t: Select “ Table ” Under Model Specification, Select Variables then OK