### t test

```Social Statistics: t test
This week
What is t test
 Types of t test
 TTEST function
 T-test ToolPak
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Why not z-test
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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
Enter the t-test: “estimate what you don’t know”
William S. Gossett
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”)
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Types of t test
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Degrees of Freedom and t test
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
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One sample t test
Very similar like z test
 Use sample statistics instead of population
parameters (mean and standard deviation)
 Evaluate the result through t test table instead
of z test table
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An example
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We show 26 babies the two pictures at the same time (one
with 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 and 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

Sentence:
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
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Code Symbols:
Step 2: Determine Critical Region
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
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Step 3: Calculate t statistic from
sample

Central Limit Theorem
μ = 30
 sM=s/
=16/

n
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= 3.14
26
Step 4: Decision and Conclusion
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.
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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 between-subjects)
research design.
t Statistic for IndependentMeasures Design

The goal of an independent-measures research
study:
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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
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t Statistic for IndependentMeasures Design
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Null hypothesis: “no change = no effect = no
difference”
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Alternative hypothesis: “there is a difference”
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H0: μ1- μ2 = 0
H1: μ1- μ2 ≠ 0
T test formula
t
x1  x2
 (n1  1) s12  (n2  1) s22   n1  n2 




n1  n2  2

  n1n2 
1 : is the mean for Group 1
2 : is the mean for Group 2
1 : is the number of participants in Group 1
2 : is the number of participants in Group 2
1 2 : is the variance for Group 1
2 2 : is the variance for Group 20
Value for degrees of freedom: df = df1 + df2
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An Example
Group 1
7
5
3
4
3
6
2
10
3
10
8
5
8
1
5
1
8
4
5
3
17
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
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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
Research hypothesis: there is a difference between
the two groups
H1 : 1  2
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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
Determine which test statistic is good for your
research
 Independent t test
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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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T Distribution Critical Values Table
T test steps
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Step 5: (cont.)
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Degrees of freedom (df): approximates the sample size
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Two-tailed or one-tailed
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Group 1 sample size -1 + group 2 sample size -1
Our test df = 58
Directed research hypothesis  one-tailed
Non-directed research hypothesis  two-tailed
T test steps

Step 6: A comparison of the obtained value
and the critical value
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
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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: T.TEST function
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T.TEST (array1, array2, tails, type)
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
two-sample test with unequal variances
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Excel: T.TEST function
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)
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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
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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
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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
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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%
```