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Testing Differences in Means (t-tests) Dr. Richard Jackson [email protected] © Mercer University 2005 All Rights Reserved Student t test A parametric statistic Tests difference in 2 means William Gossett 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 Requirements of the t test 2 means Continuous Data Normally distributed Hypothesis Associated with t H0: m1= m2 H1: m2 m2 Types of Samples Associated with t Repeated Measures of Paired (See Table I) Independent (See Table III) If Requirements Not Met, Use Non-Parametric Counterparts Repeated Measures – Wilcoxon Signed Rank or Sign Test Independent – Mann Whitney U. Formula for t t = X1- X2 SDX Similar to Z A “Difference” / A Standard Deviation Standard of Difference in Means Similar to Standard Error of Mean Replicate Study to Determine Difference in 2 Groups Many Times Standard Error of Difference In Means X X X1-X2 23 21 2 31 32 2 43 44 1 21 21 2 29 39 4 Repeated Measures (Paired) t (See Table I) Patient Before After Difference 1 120 117 3 2 100 96 4 3 110 105 5 4 90 84 6 5 130 123 7 Null Hypothesis Ho: mb=ma Xb=110 Xa=105 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 Conclusion A priori significance label set at 0.05 p = 0.0021 Reject Ho (p < 0.05) Conclusion: “Significant” difference in before and after Independent Sample t (See Table III) Diet A Diet B 177 200 251 239 190 180 210 185 142 155 141 205 147 171 213 164 Hypothesis Ho : ma = mb H1 : ma mb Xa = 204; Xb = 167.3 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 Ho (p < 0.05) Conclusion: Difference is “Significant” Use of t Table (See Table V) Compare Calculated t with Tabled t Calculated t > Table t : Reject Ho Calculated t Table t : Accept Ho Degrees of Freedom (Sample Size) (See Table V) Independent (N1 + N2 – 2) Repeated (N – 1) One–Tail Versus Two-Tail Test (See Table V) H m, <m2 Prior Knowledge of Difference One-Tail Versus Two-Tail (See Table V) When in Doubt, use Two-Tail Two-Tail More Conservative Significance Level Access Top Most Times Use 0.05 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 Ho Example Using Independent t Degrees of Freedom = N1+N2-2 = 14 Two-Tail Test Significance Level = 0.05 Tabled Value = 2.145 Calculated t = 2.65 Conclusion: Reject Ho Observations About t Table As Sample Size Increases, Tables Value Decreases As Significance Level Decreases, Tabled Value Increases Two-Tail Tabled Value Larger than OneTail Tabled Value for Some Significance Level Sample Size Determination Power Desired (Average = 0.80) Variability of Groups How Small Difference Detect Example Sample Size for t N = 16S2/D2 S = Standard Deviation of subjects D = Smallest difference to detect 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 Summary for t Difference in 2 means Data Continuous and Normally Distributed Calculated t with p value allows Researcher to Accept/Reject Ho p-Value Provides Probability of Type I Error if Reject 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. 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