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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
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Testing Differences in Means (t-tests)
Dr. Richard Jackson jackson_r@mercer.edu
© Mercer University 2005© Mercer University 2005All Rights ReservedAll 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 21 231 32 243 44 121 21 229 39 4
Repeated Measures (Paired) t
(See Table I)Patient Before After Differenc
e1 120 117 32 100 96 43 110 105 54 90 84 65 130 123 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 A177200251239190180210185
Diet B142155141205147171213164
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 One-Tail 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
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