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Testing Differences in
Means (t-tests)
Dr. Richard Jackson
[email protected]
© Mercer University 2005
All Rights Reserved
Student t test
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A parametric statistic
Tests difference in 2 means
William Gossett
Steps in Research
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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
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2 means
Continuous Data
Normally distributed
Hypothesis Associated with t
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H0: m1= m2
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H1: m2
m2
Types of Samples Associated
with t
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Repeated Measures of Paired (See
Table I)
Independent (See Table III)
If Requirements Not Met, Use
Non-Parametric Counterparts
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Repeated Measures – Wilcoxon Signed
Rank or Sign Test
Independent – Mann Whitney U.
Formula for t
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t = X1- X2
SDX
Similar to Z
A “Difference” / A Standard Deviation
Standard of Difference in
Means
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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
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Ho: mb=ma
Xb=110
Xa=105
Calculation of t Using Statistix
(See Table II)
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Mean Difference is 5
STD Error of Difference is 0.7071
t = -7.07
p = 0.0021
Conclusion
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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
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Ho : ma = mb
H1 : ma mb
Xa = 204; Xb = 167.3
Calculation of t Using Statistix
(See Table IV)
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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)
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Compare Calculated t with Tabled t
Calculated t > Table t : Reject Ho
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Calculated t
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Table t : Accept Ho
Degrees of Freedom
(Sample Size)
(See Table V)
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Independent (N1 + N2 – 2)
Repeated (N – 1)
One–Tail Versus Two-Tail Test
(See Table V)
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H
m, <m2
Prior Knowledge of Difference
One-Tail Versus Two-Tail
(See Table V)
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When in Doubt, use Two-Tail
Two-Tail More Conservative
Significance Level
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Access Top
Most Times Use 0.05
Example Using Repeated
Measures t
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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
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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
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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
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Power Desired (Average = 0.80)
Variability of Groups
How Small Difference Detect
Example Sample Size for t
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N = 16S2/D2
S = Standard Deviation of subjects
D = Smallest difference to detect
Example Sample Size for t
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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
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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
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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
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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