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Statistics for the Social Sciences
Psychology 340Fall 2006
Using t-tests
Statistics for the Social Sciences
Outline (for 2 weeks)
• Review t-tests– One sample, related samples, independent samples
– Additional assumptions• Levene’s test
Statistics for the Social Sciences
Testing Hypotheses
– 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
• Hypothesis testing: a five step program
Statistics for the Social Sciences
t Test for Dependent Means
• 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 t-test can be used when:
– 1 sample
t =D−μD
sD
– Two scores per subject
Statistics for the Social Sciences
Statistical analysis follows design
• The related-samples t-test can be used when:
– 1 sample– Two scores per subject
t =D−μD
sD
– 2 samples
– Scores are related
- OR -
Statistics for the Social Sciences
Performing your statistical test
€
test statistic =observed difference
difference expected by chance
€
sD
=sD
nD
Diff. Expected by chance
Estimated standard error of the differences
€
df = nD −1
Test statistic
€
t =D − μ
D
sD
What are all of these “D’s” referring to?
Mean of the differences
Number of difference scores
• 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
Statistics for the Social Sciences
Performing your statistical test
€
test statistic =observed difference
difference expected by chance
What are all of these “D’s” referring to?
Differencescores
2
65
9
22 €
t =D − μ
D
sD
PersonPre-testPost-test
1
23
4
45
5540
60
43
4935
51
(Pre-test) - (Post-test)
H0
:There is no difference between pre-test and post-test
HA: There is a difference between pre-test and post-test
μD = 0
μD ≠ 0
Statistics for the Social Sciences
Performing your statistical test
€
test statistic =observed difference
difference expected by chance
What are all of these “D’s” referring to?
Differencescores
2
65
9
22= 5.5
€
t =D − μ
D
sD
PersonPre-testPost-test
1
23
4
45
5540
60
43
4935
51
€
nD = 4
(Pre-test) - (Post-test)
D∑nD
=D
Statistics for the Social Sciences
Performing your statistical test
€
test statistic =observed difference
difference expected by chance
What are all of these “D’s” referring to?
PersonPre-testPost-testDifferencescores
1
23
4
45
5540
60
43
4935
51
2
65
9
22D = 5.5
€
t =D − μ
D
sD D
D
s
μ−=
5.5
€
nD = 4
Statistics for the Social Sciences
Performing your statistical test
€
test statistic =observed difference
difference expected by chance
What are all of these “D’s” referring to?
PersonDifferencescores
1
23
4
Pre-testPost-test
45
5540
60
43
4935
51
2
65
9
22D = 5.5
€
sD
=sD
nD
€
t =D − μ
D
sD
-3.5- 5.5 =
0.5- 5.5 =-0.5- 5.5 =
3.5- 5.5 =
12.25
0.250.25
12.25
25 = SSD
D - D(D - D)2
€
sD =SSD
nD −19.2
14
25=
−=
D
D
s
μ−=
5.5
€
nD = 4
Statistics for the Social Sciences
Performing your statistical test
€
test statistic =observed difference
difference expected by chance
What are all of these “D’s” referring to?
Person
1
23
4
Pre-testPost-test
45
5540
60
43
4935
51
2
65
9
22D = 5.5
€
sD
=sD
nD
€
t =D − μ
D
sD
25 = SSD
(D - D)2
€
sD =SSD
nD −19.2
14
25=
−=
D
D
s
μ−=
5.5 -3.5
0.5-0.5
3.5
D - DDifferencescores
12.25
0.250.25
12.25
€
nD = 4
Statistics for the Social Sciences
Performing your statistical test
€
test statistic =observed difference
difference expected by chance
What are all of these “D’s” referring to?
Person
1
23
4
Pre-testPost-test
45
5540
60
43
4935
51
2
65
9
22D = 5.5
€
t =D − μ
D
sD
25 = SSD
(D - D)2
€
sD
=sD
nD
45.14
9.2==
2.9 = sD
D
D
s
μ−=
5.5 -3.5
0.5-0.5
3.5
D - DDifferencescores
12.25
0.250.25
12.25
€
nD = 4
Statistics for the Social Sciences
Performing your statistical test
€
test statistic =observed difference
difference expected by chance
What are all of these “D’s” referring to?
Person
1
23
4
Pre-testPost-test
45
5540
60
43
4935
51
2
65
9
22D = 5.5
€
t =D − μ
D
sD
25 = SSD
(D - D)2
2.9 = sD
45.1
5.5 Dμ−=
1.45 = sD
?Think back to the null hypotheses
-3.5
0.5-0.5
3.5
D - DDifferencescores
12.25
0.250.25
12.25
€
nD = 4
Statistics for the Social Sciences
Performing your statistical test
€
test statistic =observed difference
difference expected by chance
What are all of these “D’s” referring to?
Person
1
23
4
Pre-testPost-test
45
5540
60
43
4935
51
2
65
9
22D = 5.5
€
t =D − μ
D
sD
25 = SSD
(D - D)2
2.9 = sD
45.1
5.5 Dμ−=
1.45 = sD
H0: Memory performance at the post-test are equal to memory performance at the pre-test.
€
μD
= 0
-3.5
0.5-0.5
3.5
D - DDifferencescores
12.25
0.250.25
12.25
€
nD = 4
Statistics for the Social Sciences
Performing your statistical test
€
test statistic =observed difference
difference expected by chance
What are all of these “D’s” referring to?
Person
1
23
4
Pre-testPost-test
45
5540
60
43
4935
51
2
65
9
22D = 5.5
€
t =D − μ
D
sD
25 = SSD
(D - D)2
2.9 = sD
45.1
05.5 −=
1.45 = sD
8.3=This is our tobs
-3.5
0.5-0.5
3.5
D - DDifferencescores
12.25
0.250.25
12.25
€
nD = 4
Statistics for the Social Sciences
Performing your statistical test
€
test statistic =observed difference
difference expected by chance
What are all of these “D’s” referring to?
Person
1
23
4
Pre-testPost-test
45
5540
60
43
4935
51
2
65
9
22D = 5.5
€
t =D − μ
D
sD
25 = SSD
(D - D)2
2.9 = sD
45.1
05.5 −=
1.45 = sD
8.3=tobs
Proportion in one tail0.10 0.05 0.025 0.01 0.005
Proportion in two tailsdf 0.20 0.10 0.05 0.02 0.011 3.078 6.314 12.706 31.821 63.6572 1.886 2.920 4.303 6.965 9.9253 1.638 2.353 3.182 4.541 5.8414 1.533 2.132 2.776 3.747 4.604
= 0.05Two-tailedtcrit
€
df = nD −1
=±3.18
-3.5
0.5-0.5
3.5
D - DDifferencescores
12.25
0.250.25
12.25
€
nD = 4
Statistics for the Social Sciences
+3.18 = tcrit
- Reject H0
Performing your statistical test
€
test statistic =observed difference
difference expected by chance
What are all of these “D’s” referring to?
Person
1
23
4
Pre-testPost-test
45
5540
60
43
4935
51
2
65
9
22D = 5.5
€
t =D − μ
D
sD
25 = SSD
(D - D)2
2.9 = sD
45.1
05.5 −=
1.45 = sD
8.3=tobs= 0.05Two-tailedtcrit
€
df = nD −1
=±3.18
-3.5
0.5-0.5
3.5
D - DDifferencescores
12.25
0.250.25
12.25
€
nD = 4 tobs=3.8
Statistics for the Social Sciences
Performing your statistical test
€
test statistic =observed difference
difference expected by chance
What are all of these “D’s” referring to?
Person
1
23
4
Pre-testPost-test
45
5540
60
43
4935
51
2
65
9
22D = 5.5
€
t =D − μ
D
sD
25 = SSD
(D - D)2
2.9 = sD
45.1
05.5 −=
1.45 = sD
8.3=tobs= 0.05Two-tailedtcrit
€
df = nD −1
=±3.18
Tobs > tcrit
so we reject the H0
-3.5
0.5-0.5
3.5
D - DDifferencescores
12.25
0.250.25
12.25
€
nD = 4
Statistics for the Social Sciences
• Using SPSSPersonPre-testPost-test
1
23
4
45
5540
60
43
4935
51
Statistics for the Social Sciences
Next time
• Independent samples t-tests