Statistics for the Social Sciences
Psychology 340
Fall 2006
Using t-tests
Outline (for 2 weeks)
Statistics for the
Social Sciences
• Review t-tests
– One sample, related samples, independent samples
– Additional assumptions
• Levene’s test
Testing Hypotheses
Statistics for the
Social Sciences
• Hypothesis testing: a five step program
–
–
–
–
Step 1: State your hypotheses
Step 2: Set your decision criteria
Step 3: Collect your data
Step 4: Compute your test statistics
• Compute your estimated standard error
• Compute your t-statistic
• Compute your degrees of freedom
– Step 5: Make a decision about your null hypothesis
t Test for Dependent Means
Statistics for the
Social Sciences
• Unknown population mean and variance
• Two situations
– One sample, two scores for each person
• Repeated measures design
– Two samples, but individuals in the samples are related
• Same procedure as t test for single sample, except
– Use difference scores
– Assume that the population mean is 0
Statistics for the
Social Sciences
Statistical analysis
follows design
• The related-samples ttest can be used when:
– 1 sample
– Two scores per subject
D  D
t
sD
Statistics for the
Social Sciences
Statistical analysis
follows design
• The related-samples ttest can be used when:
– 1 sample
– Two scores per subject
- OR – 2 samples
– Scores are related
D  D
t
sD
Performing your statistical test
Statistics for the
Social Sciences
test statistic 
• Difference scores
– For each person,
subtract one score

from the other
– Carry out
hypothesis testing
with the difference
scores
• Population of
difference scores with a
mean of 0
– Population 2 has a
mean of 0
observed difference
difference expected by chance
What are all of these “D’s” referring to?
Mean of the
differences
D  D
t
sD
Estimated
standard error of
the differences
sD
sD 
nD
Number of
difference scores
df  nD 1


Test statistic
Diff.
Expected by
chance
Performing your statistical test
Statistics for the
Social Sciences
test statistic 
observed difference
difference expected by chance
What are all of these “D’s” referring to?
(Pre-test) - (Post-test)
Difference
D  D
t
sD

Person Pre-test
Post-test scores
1
2
3
4
45
55
40
60
43
49
35
51
2
6
5
9
22
H0: There is no difference
between pre-test and posttest

D = 0
HA: There is a difference
between pre-test and posttest
D ≠ 0
Performing your statistical test
Statistics for the
Social Sciences
test statistic 
observed difference
difference expected by chance
What are all of these “D’s” referring to?
(Pre-test) - (Post-test)
Difference
D  D
t
sD

Person Pre-test
Post-test scores
1
2
3
4
nD  4
45
55
40
60
43
49
35
51
2
6
5
9
22
 D  D = 5.5
nD

Performing your statistical test
Statistics for the
Social Sciences
test statistic 
observed difference
difference expected by chance
What are all of these “D’s” referring to?
Difference
D  D 5.5   D
t

sD
sD

Person Pre-test
Post-testscores
1
2
3
4
nD  4
45
55
40
60
43
49
35
51
2
6
5
9
22
D = 5.5

Performing your statistical test
Statistics for the
Social Sciences
test statistic 
observed difference
difference expected by chance
What are all of these “D’s” referring to?
Difference
D  D 5.5   D
t

2- 5.5 = -3.5 12.25
sD
sD
6- 5.5 = 0.5 0.25
sD
5- 5.5 = -0.5 0.25 sD 
nD

Person Pre-test
Post-test scores
1
2
3
4
nD  4
45
55
40
60
43
49
35
51
D - D (D - D)2
9- 5.5 = 3.5 12.25

22
25 = SSD
sD 
D = 5.5


SSD
25

 2.9
n D 1 4  1
Performing your statistical test
Statistics for the
Social Sciences
test statistic 
observed difference
difference expected by chance
What are all of these “D’s” referring to?
Difference

Person Pre-test
Post-test scores
1
2
3
4
nD  4
45
55
40
60
43
49
35
51
2
6
5
9
22
D = 5.5
D  D 5.5   D
t

-3.5 12.25
sD
sD
0.5 0.25
sD
-0.5 0.25 sD 
nD
D - D (D - D)2
3.5 12.25

25 = SSD
sD 


SSD
25

 2.9
n D 1 4  1
Performing your statistical test
Statistics for the
Social Sciences
test statistic 
observed difference
difference expected by chance
What are all of these “D’s” referring to?
Difference

Person Pre-test
Post-test scores
1
2
3
4
nD  4
45
55
40
60
43
49
35
51
2
6
5
9
22
D = 5.5
D  D 5.5   D
t

-3.5 12.25
sD
sD
0.5 0.25
sD
2.9

 1.45
s

-0.5 0.25
D
4
nD
D - D (D - D)2
3.5 12.25

25 = SSD
2.9 = sD

Performing your statistical test
Statistics for the
Social Sciences
test statistic 
observed difference
difference expected by chance
What are all of these “D’s” referring to?
Difference

Person Pre-test
Post-test scores
1
2
3
4
nD  4
45
55
40
60
43
49
35
51
2
6
5
9
22
D = 5.5
D - D (D - D)2
-3.5 12.25
0.5 0.25
-0.5 0.25
3.5 12.25

25 = SSD
2.9 = sD
1.45 = sD
D  D 5.5   D
t

sD
1.45
?
Think back to the null
hypotheses
Performing your statistical test
Statistics for the
Social Sciences
test statistic 
observed difference
difference expected by chance
What are all of these “D’s” referring to?
Difference

Person Pre-test
Post-test scores
1
2
3
4
nD  4
45
55
40
60
43
49
35
51
2
6
5
9
22
D = 5.5
D - D (D - D)2
-3.5 12.25
0.5 0.25
-0.5 0.25
3.5 12.25

25 = SSD
2.9 = sD
1.45 = sD
D  D 5.5   D
t

sD
1.45
H0: Memory performance at
the post-test are equal to
memory performance at the
pre-test.
D  0

Performing your statistical test
Statistics for the
Social Sciences
test statistic 
observed difference
difference expected by chance
What are all of these “D’s” referring to?
Difference

Person Pre-test
Post-test scores
1
2
3
4
nD  4
45
55
40
60
43
49
35
51
2
6
5
9
22
D = 5.5
D - D (D - D)2
-3.5 12.25
0.5 0.25
-0.5 0.25
3.5 12.25

25 = SSD
2.9 = sD
1.45 = sD
D  D
t
sD

5.5  0
1.45
This is our tobs 3.8
Performing your statistical test
Statistics for the
Social Sciences
test statistic 
observed difference
difference expected by chance
What are all of these “D’s” referring to?
Difference

Person Pre-test
Post-test scores
1
2
3
4
nD  4
45
55
40
60
43
49
35
51
2
6
5
9
22
D = 5.5
df  nD 1
D - D (D - D)2
D  D
t
sD

5.5  0
1.45
-3.5 12.25
0.5 0.25
tobs 3.8
-0.5 0.25
= 0.05 Two-tailed tcrit 3.18
3.5 12.25
Proportion in one tail

25 = SSD
0.10
0.05
0.025
0.01
Proportion in two tails
df
0.20
0.10
0.05
0.02
2.9 = sD
1
3.078
6.314
12.706
31.821
2
1.886
2.920
4.303
6.965
1.45 = sD
3
1.638
2.353
3.182
4.541
4
1.533
2.132
2.776
3.747
0.005
0.01
63.657
9.925
5.841
4.604
Performing your statistical test
Statistics for the
Social Sciences
test statistic 
observed difference
difference expected by chance
What are all of these “D’s” referring to?
Difference

Person Pre-test
Post-test scores
1
2
3
4
nD  4
45
55
40
60
43
49
35
51
2
6
5
9
22
D = 5.5
df  nD 1
D - D (D - D)2
D  D
t
sD

5.5  0
1.45
-3.5 12.25
0.5 0.25
tobs 3.8
-0.5 0.25
= 0.05 Two-tailed tcrit 3.18
3.5 12.25

25 = SSD
tobs=3.8
2.9 = sD
- Reject H0
1.45 = sD
+3.18 = tcrit
Performing your statistical test
Statistics for the
Social Sciences
test statistic 
observed difference
difference expected by chance
What are all of these “D’s” referring to?
Difference

Person Pre-test
Post-test scores
1
2
3
4
nD  4
45
55
40
60
43
49
35
51
2
6
5
9
22
D = 5.5
df  nD 1
D - D (D - D)2
D  D
t
sD

5.5  0
1.45
-3.5 12.25
0.5 0.25
tobs 3.8
-0.5 0.25
= 0.05 Two-tailed tcrit 3.18
3.5 12.25

25 = SSD
Tobs > tcrit so we
2.9 = sD
reject the H0
1.45 = sD
Statistics for the
Social Sciences
• Using SPSS
Person Pre-test Post-test
1
2
3
4
45
55
40
60
43
49
35
51
Next time
Statistics for the
Social Sciences
• Independent samples t-tests