1. Statistics for the Social Sciences
Psychology 340
Spring 2010
Using t-tests (related samples)

2. t Test for Dependent Means
Unknown population mean and variance
More realistic, these designs are commonly found in research,
Based on having two related groups of observations
Two situations
One sample, two scores for each person
Repeated measures design
Two samples, but individuals in the samples are related
Related samples, Dependent samples, matched samples

3. Statistical analysis follows design
1 sample
Two scores per subject
The related-samples t-test can be used when:

4. Within-Groups Factor Sometimes called “repeated measures” design
2-levels, All of the participants are in both levels of the IV
levels
participants
Pre-test
Post-test
Test
Memory patients before getting the treatment
Memory patients after getting the treatment

5. Statistical analysis follows design
The related-samples t-test can be used when:
2 samples
Scores are related
1 sample
Two scores per subject
- OR -

6. Matching groups Group A Group B Matched groups
Trying to create equivalent groups
Also trying to reduce some of the overall variability
Eliminating variability from the variables that you matched people on
Red
Short
21yrs
matched
Red
Short
21yrs
matched
Blue
tall
23yrs
Blue
tall
23yrs
matched
Green
average
22yrs
Green
average
22yrs
Color
Height
Age
matched
Brown
tall
22yrs
Brown
tall
22yrs

7. Testing Hypotheses 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
Very similar to one sample t-test from earlier

8. Performing your statistical test
What are we doing when we test the hypotheses?
Computing a test statistic: Generic test
Compares the differences between groups of related observations to the difference expected by the null hypothesis
Could be difference between a sample and a population, or between different samples
Based on standard error or an estimate of the standard error
Because the groups of observations come from same individuals or matched individuals, the variability is typically reduced

9. Performing your statistical test
Difference scores
For each person, subtract one score from the other
Carry out hypothesis testing with the difference scores
H0 Population of difference scores has a mean = 0
What are all of these “D’s” referring to?
Mean of the differences
Test statistic
Diff. Expected by chance
Estimated standard error of the differences
Number of difference scores

10. Comparing t-tests Independent-samples t One-sample t Related samples t
Observed (sample) means
Difference between
Observed (sample) means

11. Comparing t-tests Independent-samples t One-sample t Related samples t
Hypothesized population means
from the Null hypothesis
Hypothesized difference between
Population means
from the Null hypothesis

12. Comparing t-tests Independent-samples t One-sample t Related samples t
Hypothesized difference between
Population means
from the Null hypothesis
H0:
Memory performance by the treatment group is equal to memory performance by the no treatment group.
So:

13. Comparing t-tests Independent-samples t One-sample t Related samples t
Hypothesized difference between
Population means
from the Null hypothesis
The numerator’s of both the independent samples and related samples t-tests are numerically identical. I’ve used notational differences to illustrate that one is based on two sets of observations (from the two samples), while the other is based on difference scores.
Hypothesized population means
from the Null hypothesis
= 0
= 0
Is equal to
H0:
Memory performance by the treatment group is equal to memory performance by the no treatment group.
The major difference between the two tests comes from how the estimated standard errors are computed.
So:

14. Performing your statistical test
(Pre-test) - (Post-test)
What are all of these “D’s” referring to?
Difference
scores
Person
Pre-test
Post-test 1 2 3 4 45 55 40 60 43 49 35 51 2
H0:
There is no difference between pre-test and post-test 6 5 9
μD = 0 22
HA:
There is a difference between pre-test and post-test
μD ≠ 0

15. Performing your statistical test
(Pre-test) - (Post-test)
What are all of these “D’s” referring to?
Difference
scores
Person
Pre-test
Post-test 1 2 3 4 45 55 40 60 43 49 35 51 2 6 5 9 22
= 5.5

16. Performing your statistical test
What are all of these “D’s” referring to?
Difference
scores
Person
Pre-test
Post-test 1 45 43 2 2 55 49 6 3 40 35 5 4 60 51 9 22
D = 5.5

17. Performing your statistical test
What are all of these “D’s” referring to?
Difference
scores
Person
Pre-test
Post-test 45 55 40 60 43 49 35 51
D - D
(D - D)2 1 2
- 5.5 =
-3.5
12.25 2 6
0.5
- 5.5 =
0.25 3 5
-0.5
- 5.5 =
0.25 4 9
3.5
- 5.5 =
12.25 22
25 = SSD
D = 5.5

18. Performing your statistical test
What are all of these “D’s” referring to?
Difference
scores
Person
Pre-test
Post-test 45 55 40 60 43 49 35 51
D - D
(D - D)2 1 2
-3.5
12.25 2 6
0.5
0.25 3 5
-0.5
0.25 4 9
3.5
12.25 22
25 = SSD
D = 5.5

19. Performing your statistical test
What are all of these “D’s” referring to?
Difference
scores
Person
Pre-test
Post-test 45 55 40 60 43 49 35 51
D - D
(D - D)2 1 2
-3.5
12.25 2 6
0.5
0.25 3 5
-0.5
0.25 4 9
3.5
12.25 22
25 = SSD
D = 5.5
2.9 = sD

20. Performing your statistical test
What are all of these “D’s” referring to?
Difference
scores
Person
Pre-test
Post-test 45 55 40 60 43 49 35 51
D - D
(D - D)2 ? 1 2
-3.5
12.25 2 6
0.5
0.25 3 5
-0.5
0.25 4 9
3.5
12.25
Think back to the null hypotheses 22
25 = SSD
D = 5.5
2.9 = sD
1.45 = sD

21. Performing your statistical test
What are all of these “D’s” referring to?
Difference
scores
Person
Pre-test
Post-test 45 55 40 60 43 49 35 51
D - D
(D - D)2 1 2
-3.5
12.25 2 6
0.5
0.25
H0: Memory performance at the post-test are equal to memory performance at the pre-test. 3 5
-0.5
0.25 4 9
3.5
12.25 22
25 = SSD
D = 5.5
2.9 = sD
1.45 = sD

22. Performing your statistical test
What are all of these “D’s” referring to?
Difference
scores
Person
Pre-test
Post-test 45 55 40 60 43 49 35 51
D - D
(D - D)2 1 2
-3.5
12.25 2 6
0.5
0.25
This is our tobs 3 5
-0.5
0.25 4 9
3.5
12.25 22
25 = SSD
D = 5.5
2.9 = sD
1.45 = sD

23. Performing your statistical test
What are all of these “D’s” referring to?
Difference
scores
Person
Pre-test
Post-test 45 55 40 60 43 49 35 51
D - D
(D - D)2 1 2
-3.5
12.25 2 6
0.5
0.25
tobs 3 5
-0.5
0.25
tcrit
α= 0.05
Two-tailed 4 9
3.5
12.25 22
25 = SSD
D = 5.5
2.9 = sD
1.45 = sD

24. Performing your statistical test
What are all of these “D’s” referring to?
Difference
scores
Person
Pre-test
Post-test 45 55 40 60 43 49 35 51
D - D
(D - D)2 1 2
-3.5
12.25 2 6
0.5
0.25
tobs 3 5
-0.5
0.25
tcrit
α= 0.05
Two-tailed 4 9
3.5
12.25
+3.18 = tcrit 22
25 = SSD
tobs=3.8
D = 5.5
2.9 = sD
- Reject H0
1.45 = sD

25. Performing your statistical test
What are all of these “D’s” referring to?
Difference
scores
Person
Pre-test
Post-test 45 55 40 60 43 49 35 51
D - D
(D - D)2 1 2
-3.5
12.25 2 6
0.5
0.25
tobs 3 5
-0.5
0.25
tcrit
α= 0.05
Two-tailed 4 9
3.5
12.25 22
25 = SSD
Tobs > tcrit so we reject the H0
D = 5.5
2.9 = sD
1.45 = sD

26. Statistical Tests Summary
Design
Statistical test
(Estimated) Standard error
One sample, σ known
One sample, σ unknown
Two related samples, σ unknown
Two independent samples, σ unknown

27. Effect Sizes & Power for t Test for Dependent Means
7-11
Remember we don’t know these

28. Approximate Sample Size Needed for 80% Power (.05 significance level)
Using Power and effect sizes to determine how many participants you need
7-12

29. Using SPSS: Related samples t
Person
Pre-test
Post-test
Entering the data
Different groups of observations go into separate columns
e.g., pre-test in one column, post-test in a separate column 1 45 43 2 55 49 3 40 35 4 60 51
Performing the analysis
Analyze -> Compare means -> paired samples t-test
Identify which columns are the related samples of obs
Reading the output
Means of the different groups, the mean difference, the computed-t, degrees of freedom, p-value (Sig.)