1. Statistics for the Social Sciences
Psychology 340
Spring 2010
Using t-tests
Basic introduction and 1-sample t-tests

2. Outline (for content up to Exam 2)
Review t-tests (note: next exam is March 4, right before spring break)
One sample, related samples, independent samples
Effect sizes with t-tests
Confidence intervals in t-test type designs

3. Statistical analysis follows design
Population mean (μ) and standard deviation (σ)are known
One score per subject
1 sample
The one-sample z-test can be used when:

4. Statistical analysis follows design
Population mean (μ) is known
but Population standard deviation (σ) is NOT known
One score per subject
1 sample
The one-sample t-test can be used when:

5. 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

6. 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
μ is known
Memory
treatment
patients
Test X
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:

7. Performing your statistical test
What are we doing when we test the hypotheses?
Real world (‘truth’)
H0: is true (no treatment effect)
One population XA
the memory treatment sample are the same as those in the population of memory patients.
H0: is false (is a treatment effect)
Two populations XA
they aren’t the same as those in the population of memory patients

8. Performing your statistical test
What are we doing when we test the hypotheses?
Computing a test statistic: Generic test
Could be difference between a sample and a population, or between different samples
Based on standard error or an estimate of the standard error

9. Performing your statistical test
One sample z
One sample t
identical
Test statistic

10. Performing your statistical test
One sample z
One sample t
Test statistic
different
Diff. Expected by chance
Standard error
don’t know this,
so need to estimate it

11. Performing your statistical test
“Unbiased estimate of the population standard deviation”
This is the same as the “sample standard deviation”
One sample z
One sample t
Test statistic
different
Diff. Expected by chance
Standard error
Estimated standard error

12. Performing your statistical test
One sample z
One sample t
Test statistic
different
Diff. Expected by chance
Standard error
don’t know this,
so need to estimate it
Estimated standard error
Degrees of freedom

13. 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

14. One sample t-test
The t-statistic distribution (a transformation of the distribution of sample means transformed)
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
Distribution of the t-statistic
If test statistic is here Reject H0
If test statistic is here Fail to reject H0

15. One sample t-test α - level New table: the t-table One tailed - or -
Two-tailed
Degrees of freedom df
Critical values of t
tcrit

16. One sample t-test
What is the tcrit for a two-tailed hypothesis test with a sample size of n = 6 and an α-level of 0.05?
Distribution of the t-statistic
α = 0.05
Two-tailed
n = 6
df = n - 1 = 5
tcrit =

17. One sample t-test
What is the tcrit for a one-tailed hypothesis test with a sample size of n = 6 and an α-level of 0.05?
Distribution of the t-statistic
α = 0.05
n = 6
One-tailed
df = n - 1 = 5
tcrit =

18. One sample t-test Memory experiment example:
An example: One sample t-test
Memory experiment example:
We give a n = 16 memory patients a memory improvement treatment.
After the treatment they have an average score of = 55, s = 8 memory errors.
Do know s
How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, μ = 60?
Don’t know σ

19. One sample t-test Memory experiment example:
An example: One sample t-test
Memory experiment example:
Step 1: State your hypotheses
H0:
the memory treatment sample are the same (or worse) as those in the population of memory patients.
We give a n = 16 memory patients a memory improvement treatment.
After the treatment they have an average score of = 55, s = 8 memory errors.
μTreatment > μpop = 60
How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, μ = 60?
HA:
they perform better than those in the population of memory patients
μTreatment < μpop = 60

20. One sample t-test An example: One sample t-test
Memory experiment example:
H0: μTreatment > μpop = 60
HA: μTreatment < μpop = 60
We give a n = 16 memory patients a memory improvement treatment.
Step 2: Set your decision criteria
One -tailed
After the treatment they have an average score of = 55, s = 8 memory errors.
α = 0.05
How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, μ = 60?

21. One sample t-test An example: One sample t-test
Memory experiment example:
H0: μTreatment > μpop = 60
HA: μTreatment < μpop = 60
We give a n = 16 memory patients a memory improvement treatment.
Step 2: Set your decision criteria
After the treatment they have an average score of = 55, s = 8 memory errors.
One -tailed
α = 0.05
How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, μ = 60?

22. One sample t-test An example: One sample t-test
Memory experiment example:
H0: μTreatment > μpop = 60
HA: μTreatment < μpop = 60
We give a n = 16 memory patients a memory improvement treatment.
One -tailed
α = 0.05
Step 3: Collect your data
After the treatment they have an average score of = 55, s = 8 memory errors.
How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, μ = 60?

23. One sample t-test An example: One sample t-test
Memory experiment example:
H0: μTreatment > μpop = 60
HA: μTreatment < μpop = 60
We give a n = 16 memory patients a memory improvement treatment.
One -tailed
α = 0.05
After the treatment they have an average score of = 55, s = 8 memory errors.
Step 4: Compute your test statistics
How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, μ = 60?
= -2.5

24. One sample t-test An example: One sample t-test
Memory experiment example:
H0: μTreatment > μpop = 60
HA: μTreatment < μpop = 60
We give a n = 16 memory patients a memory improvement treatment.
One -tailed
α = 0.05
t = -2.5
After the treatment they have an average score of = 55, s = 8 memory errors.
Step 4: Compute your test statistics
How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, μ = 60?

25. One sample t-test An example: One sample t-test
Memory experiment example:
H0: μTreatment > μpop = 60
HA: μTreatment < μpop = 60
We give a n = 16 memory patients a memory improvement treatment.
One -tailed
α = 0.05
After the treatment they have an average score of = 55, s = 8 memory errors.
Step 5: Make a decision about your null hypothesis
How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, μ = 60?
tcrit =

26. One sample t-test An example: One sample t-test
Memory experiment example:
H0: μTreatment > μpop = 60
HA: μTreatment < μpop = 60
We give a n = 16 memory patients a memory improvement treatment.
One -tailed
α = 0.05
After the treatment they have an average score of = 55, s = 8 memory errors.
Step 5: Make a decision about your null hypothesis
How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, μ = 60?
tobs=-2.5
- Reject H0
= tcrit

27. Assumptions of t-statistics
Values in the sample must be independent observations
Each observation is independent of all of the other observations (see pg 253 of your textbook for examples of non-independence)
The population that is sampled must be normally distributed
It turns out that t-tests are pretty robust against violations of this assumption, especially with large samples

28. Effect Sizes & Power for 1-sample t Test
Recall for the 1-sample z-test:
Remember we don’t know these with the t-test design
So for the 1-sample t-test we estimate the Cohen’s d:
Pop mean from null hypothesis
sample mean
sample standard deviation

29. Using SPSS: 1 sample t Entering the data Performing the analysis
Observations go into one column
e.g., 2, 6, 5, 9
Performing the analysis
Analyze -> Compare means -> one sample t-test
Identify which column to do your test on
Enter the ‘Test value’ – this is the population mean in the H0
Be careful, the default is 0, this may not be what you want
Reading the output
Mean of the sample, the computed-t, degrees of freedom, p-value (Sig.) (also reminds you of your test value)

30. Next time
Related samples t-tests
Independent samples t-tests