statistics for the social sciences psychology 340 spring 2005 using t-tests
TRANSCRIPT
Statistics for the Social Sciences
Psychology 340Spring 2005
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
Statistical analysis follows design
• The one-sample z-test can be used when:
– 1 sample
€
zX
=X − μ
X
σX
– One score per subject
– Population mean () and standard deviation ()are known
Statistics for the Social Sciences
Statistical analysis follows design
• The one-sample t-test can be used when:
– 1 sample– One score per subject
– Population mean () is known
– but standard deviation () is NOT known
€
t =X − μ
X
sX
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
Performing your statistical test
• What are we doing when we test the hypotheses?– Consider a variation of our memory experiment example
Population of memory patients
MemoryTest is known
Memory treatment
Memory patients
MemoryTestX
Compare these two means
Conclusions: • the memory treatment sample are the same as those in the population of memory patients.• they aren’t the same as those in the population of memory patients
H0
:HA:
Statistics for the Social Sciences
Performing your statistical test
• What are we doing when we test the hypotheses?
Real world (‘truth’)
H0: is false (is a treatment effect)
Two populations
XA
they aren’t the same as those in the population of memory patients
H0: is true (no treatment effect)
One population
XA
the memory treatment sample are the same as those in the population of memory patients.
Statistics for the Social Sciences
Performing your statistical test
• What are we doing when we test the hypotheses?– Computing a test statistic: Generic test
€
test statistic =observed difference
difference expected by chance
Could be difference between a sample and a population, or between different samples
Based on standard error or an estimate
of the standard error
Statistics for the Social Sciences
Performing your statistical test
€
test statistic =observed difference
difference expected by chance
€
t =X − μ
X
sX
€
zX
=X − μ
X
σX
Test statistic
One sample z One sample tidentical
Statistics for the Social Sciences
Performing your statistical test
€
test statistic =observed difference
difference expected by chance
€
t =X − μ
X
sX
€
zX
=X − μ
X
σX
Test statistic
Diff. Expected by chance
€
X
=σ
nStandard error
One sample z One sample t
different
don’t know this,so need to estimate it
Statistics for the Social Sciences
Performing your statistical test
€
test statistic =observed difference
difference expected by chance
€
t =X − μ
X
sX
€
zX
=X − μ
X
σX
€
sX
=s
n
Test statistic
Diff. Expected by chance
€
X
=σ
nStandard error
Estimated standard error
One sample z One sample t
different
don’t know this,so need to estimate it
€
df = n −1Degrees of freedom
Statistics for the Social Sciences
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 4.6045 1,476 2.015 2.571 3.365 4.0326 1.440 1.943 2.447 3.143 3.707:
15:
:1.341
:
:1.753
:
:2.131
:
:2.602
:
:2.947
:
One sample t-test
• The t-statistic distribution (a transformation of the distribution of sample means transformed)– Varies in shape according to the degrees of freedom
• New table: the t-table
Statistics for the Social Sciences
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 4.6045 1,476 2.015 2.571 3.365 4.0326 1.440 1.943 2.447 3.143 3.707:
15:
:1.341
:
:1.753
:
:2.131
:
:2.602
:
:2.947
:
One sample t-test
If test statistic is here Fail to reject H0
Distribution of the t-statistic If test
statistic is here Reject H0
– To reject the H0, you want a computed test statistics that is large• The alpha level gives us the decision criterion• New table: the t-table
• The t-statistic distribution (a transformation of the distribution of sample means transformed)
Statistics for the Social Sciences
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 4.6045 1,476 2.015 2.571 3.365 4.0326 1.440 1.943 2.447 3.143 3.707:
15:
:1.341
:
:1.753
:
:2.131
:
:2.602
:
:2.947
:
One sample t-test
• New table: the t-table
One tailed - or -
Two-tailed
levels
Critical values of ttcrit
Degrees of freedomdf
Statistics for the Social Sciences
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 4.6045 1,476 2.015 2.571 3.365 4.0326 1.440 1.943 2.447 3.143 3.707:
15:
:1.341
:
:1.753
:
:2.131
:
:2.602
:
:2.947
:
One sample t-test
• What is the tcrit for a two-tailed hypothesis test with a sample size of n = 6 and an -level of 0.05?
Distribution of the t-statistic= 0.05
Two-tailedn = 6
df = n - 1 = 5
tcrit = +2.571
Statistics for the Social Sciences
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 4.6045 1,476 2.015 2.571 3.365 4.0326 1.440 1.943 2.447 3.143 3.707:
15:
:1.341
:
:1.753
:
:2.131
:
:2.602
:
:2.947
:
One sample t-test
Distribution of the t-statistic= 0.05
One-tailedn = 6
df = n - 1 = 5
tcrit = +2.015
• What is the tcrit for a one-tailed hypothesis test with a sample size of n = 6 and an -level of 0.05?
Statistics for the Social Sciences
One sample t-test
An example: One sample t-test
Memory experiment example:• We give a n = 16 memory patients a memory improvement treatment.
• How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, = 60?
• After the treatment they have an average score of = 55, s = 8 memory errors.
€
X
• Step 1: State your hypotheses
H0
:the memory treatment sample are the same as those in the population of memory patients.HA: they aren’t the same as those in the population of memory patients
Treatment > pop = 60
Treatment < pop = 60
Statistics for the Social Sciences
One sample t-test
Memory experiment example:• We give a n = 16 memory patients a memory improvement treatment.
• Step 2: Set your decision criteria
H0: Treatment > pop = 60 HA: Treatment < pop = 60
= 0.05One -tailed
• How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, = 60?
• After the treatment they have an average score of = 55, s = 8 memory errors.
€
X
An example: One sample t-test
Statistics for the Social Sciences
One sample t-test
An example: One sample t-test
Memory experiment example:• We give a n = 16 memory patients a memory improvement treatment.
• Step 2: Set your decision criteria
H0: Treatment > pop = 60 HA: Treatment < pop = 60
= 0.05One -tailed
• How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, = 60?
• After the treatment they have an average score of = 55, s = 8 memory errors.
€
X
Statistics for the Social Sciences
One sample t-test
An example: One sample t-test
Memory experiment example:• We give a n = 16 memory patients a memory improvement treatment.
= 0.05One -tailed
• Step 3: Collect your data
H0: Treatment > pop = 60 HA: Treatment < pop = 60
• How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, = 60?
• After the treatment they have an average score of = 55, s = 8 memory errors.
€
X
Statistics for the Social Sciences
One sample t-test
An example: One sample t-test
Memory experiment example:• We give a n = 16 memory patients a memory improvement treatment.
= 0.05One -tailed
• Step 4: Compute your test statistics
€
t =X − μ
X
sX
€
=55 − 60
816
⎛ ⎝ ⎜
⎞ ⎠ ⎟
= -2.5
H0: Treatment > pop = 60 HA: Treatment < pop = 60
• How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, = 60?
• After the treatment they have an average score of = 55, s = 8 memory errors.
€
X
Statistics for the Social Sciences
One sample t-test
An example: One sample t-test
Memory experiment example:• We give a n = 16 memory patients a memory improvement treatment.
= 0.05One -tailed
• Step 4: Compute your test statistics
€
df = n −1
t = -2.5
H0: Treatment > pop = 60 HA: Treatment < pop = 60
• How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, = 60?
• After the treatment they have an average score of = 55, s = 8 memory errors.
€
X
€
=16 −1=15
Statistics for the Social Sciences
One sample t-test
An example: One sample t-test
Memory experiment example:• We give a n = 16 memory patients a memory improvement treatment.
= 0.05One -tailed
€
t(15) = −2.5
• Step 5: Make a decision about your null hypothesis
H0: Treatment > pop = 60 HA: Treatment < pop = 60
• How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, = 60?
• After the treatment they have an average score of = 55, s = 8 memory errors.
€
X
€
df =15
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 0.01:
15:
:1.341
:
:1.753
:
:2.131
:
:2.602
:
:2.947
:
tcrit = -1.753
Statistics for the Social Sciences
One sample t-test
An example: One sample t-test
Memory experiment example:• We give a n = 16 memory patients a memory improvement treatment.
= 0.05One -tailed
€
t(15) = −2.5
• Step 5: Make a decision about your null hypothesis
H0: Treatment > pop = 60 HA: Treatment < pop = 60
• How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, = 60?
• After the treatment they have an average score of = 55, s = 8 memory errors.
€
X
€
df =15
-1.753 = tcrit
tobs=-2.5
- Reject H0
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−D
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−D
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
D = 0
D ≠ 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
−=
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
−=
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
−=
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
−=
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 D−=
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 D−=
1.45 = sD
H0: Memory performance at the post-test are equal to memory performance at the pre-test.
€
D
= 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
Next time
• Finishing up related samples t-tests
• Independent samples t-tests• Using SPSS with t-tests (maybe next Monday)