S519: Evaluation of
Information Systems
Social Statistics
Inferential Statistics
Chapter 9: t test
This week
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What is t test
Types of t test
TTEST function
T-test ToolPak
Why not z-test
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In most cases, the z-test requires more
information than we have available
We do inferential statistics to learn about the
unknown population but, ironically, we need to
know characteristics of the population to make
inferences about it
Enter the t-test: “estimate what you don’t know”
William S. Gossett
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Employed by Guinness Brewery,
Dublin, Ireland, from 1899 to
1935.
Developed t-test around 1905, for
dealing with small samples in
brewing quality control.
Published in 1908 under
pseudonym “Student” (“Student’s
t-test”)
Types of t test
Degrees of Freedom and t test
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Degrees of freedom describes the number of
scores in a sample that are free to vary.
degrees of freedom = df = n-1
The larger, the better
One sample t test
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Very similar like z test
Use sample statistics instead of population
parameters (mean and standard deviation)
Evaluate the result through t test table (Table
B2) instead of z test table (Table B1)
An example
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We show 26 babies the two pictures at the same time
(one w/ his/her mother, the other a scenery picture) for
60 seconds, and measure how long they look at the
facial configuration.
Our null assumption is that they will not look at it for
longer than half the time, μ = 30
Our alternate hypothesis is that they will look at the face
stimulus longer b/c face recognition is hardwired in their
brain, not learned (directional)
Our sample of n = 26 babies looks at the face stimulus
for M = 35 seconds, s = 16 seconds
Test our hypotheses (α = .05, one-tailed)
Step 1: Hypotheses
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Sentence:
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Null: Babies look at the face stimulus for less than
or equal to half the time
Alternate: Babies look at the face stimulus for
more than half the time
Code Symbols:
Step 2: Determine Critical
Region
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Population variance is not known, so use
sample variance to estimate
n = 26 babies; df = n-1 = 25
Look up values for t at the limits of the critical
region from our critical values of t table
Set α = .05; one-tailed
tcrit = +1.708
Step 3: Calculate t statistic
from sample
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Central Limit Theorem
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μ = 30
sM=s/ n =16/
26 =
3.14
Step 4: Decision and
Conclusion
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The tobt=1.59 does not exceed tcrit=1.708
∴ We must retain the null hypothesis
Conclusion: Babies do not look at the face
stimulus more often than chance, t(25) =
+1.59, n.s., one-tailed. Our results do not
support the hypothesis that face processing
is innate.
Independent t test
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A research design that uses a separate
sample for each treatment condition is called
an independent-measures (or betweensubjects) research design.
t Statistic for IndependentMeasures Design
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The goal of an independent-measures
research study:
To evaluate the difference of the means
between two populations.
Mean of first population: μ1
Mean of second population: μ2
Difference between the means: μ1- μ2
t Statistic for IndependentMeasures Design
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Null hypothesis: “no change = no effect = no
difference”
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H0: μ1- μ2 = 0
Alternative hypothesis: “there is a difference”
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H1: μ1- μ2 ≠ 0
T test formula
x1  x2
t
 (n1  1) s12  (n2  1) s22   n1  n2 
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n1  n2  2
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  n1n2 
x1
x2
n1
n2
s12
s22
: is the mean for Group 1
: is the mean for Group 2
: is the number of participants in Group 1
: is the number of participants in Group 2
: is the variance for Group 1
: is the variance for Group 2
Value for degrees of freedom: df = df1 + df2
An Example
Group 1
7
5
3
4
3
6
2
10
3
10
8
5
8
1
5
1
8
4
5
3
5
7
1
9
2
5
2
12
15
4
Group 2
5
4
4
5
5
7
8
8
9
8
3
2
5
4
4
6
7
7
5
6
4
3
2
7
6
2
8
9
7
6
T test steps
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Step 1: A statement of the null and research
hypotheses.
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Null hypothesis: there is no difference between
two groups
H 0 : 1  2
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Research hypothesis: there is a difference
between the two groups
H1 : 1  2
T test steps
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Step 2: setting the level of risk (or the level of
significance or Type I error) associated with
the null hypothesis
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0.05
T test steps
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Step 3: Selection of the appropriate test
statistic
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Determine which test statistic is good for your
research
Independent t test
T test steps
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Step 4: computation of the test statistic value
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t= 0.14
T test steps
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Step 5: determination of the value needed for
the rejection of the null hypothesis
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Table B2 in Appendix B (S-p360)
Degrees of freedom (df): approximates the sample
size
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Group 1 sample size -1 + group 2 sample size -1
Our test df= 58
Two-tailed or one-tailed
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Directed research hypothesis  one-tailed
Non-directed research hypothesis  two-tailed
T test steps
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Step 6: A comparison of the obtained value
and the critical value
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0.14 and 2.001
If the obtained value > the critical value, reject the
null hypothesis
If the obtained value < the critical value, retain the
null hypothesis
T test steps
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Step 7 and 8: make a decision
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What is your decision and why?
Interpretation
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How to interpret t(58) = 0.14, p>0.05, n.s.
Excel: TTEST function
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TTEST (array1, array2, tails, type)
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array1 = the cell address for the first set of data
array2 = the cell address for the second set of
data
tails: 1 = one-tailed, 2 = two-tailed
type: 1 = a paired t test; 2 = a two-sample test
(independent with equal variances); 3 = a twosample test with unequal variances
Excel: TTEST function
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It does not compute the t value
It returns the likelihood that the resulting t
value is due to chance (the possibility of the
difference of two groups is due to chance)
Excel ToolPak
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Select t-test: two sample assuming equal
variances
t-Test: Two-Sample Assuming Equal Variances
Mean
Variance
Observations
Pooled Variance
Hypothesized Mean Difference
df
t Stat
P(T<=t) one-tail
t Critical one-tail
P(T<=t) two-tail
t Critical two-tail
Variable 1
5.433333333
11.70229885
30
7.979885057
0
58
-0.137103112
0.44571206
1.671552763
0.891424121
2.001717468
Variable 2
5.533333333
4.257471264
30
Effect size
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If two groups are different, how to measure
the difference among them
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Effect size
X1  X 2
ES 
SD
ES: effect size
X : the mean for Group 1
X : the mean for Group 2
SD: the standard deviation from either group
1
2
Effect size
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A small effect size ranges from 0.0 ~ 0.2
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A medium effect size ranges from 0.2 ~ 0.5
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The two groups are different
A large effect size is any value above 0.50
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Both groups tend to be very similar and overlap a lot
The two groups are quite different
ES=0the two groups have no difference and overlap entirely
ES=1the two groups overlap about 45%