1. T-tests Part 1 PS1006 Lecture 2
Sam Cromie

2. What has this got to do with statistics?

3. Overview Review: hypothesis testing The need for the t-test
The logic of the t-test
Application to:
Single sample designs
Two group designs
Within group designs (Related samples)
Between group designs (Independent samples)
Assumptions
Advantages and disadvantages

4. Generic Hypothesis testing steps
State hypotheses – Null and alternative
State alpha value
Calculate the statistic (z score, t score, etc.)
Look the statistic up on the appropriate probability table
Accept or reject null hypothesis

5. Generic form of a statistic
Data – Hypothesis
Error
What you got – what you expected (null)
The unreliability of your data
Z = Individual score – Population mean
Population standard deviation

6. Hypothesis testing with an individual data point…
State null hypothesis
State α value
Convert the score to a z score
Look z score up on z score tables

7. Standard deviation known
In this case we have…
Sample
Population
Data of interest
Individual score, n=1
Mean known
Error
Standard deviation known

8. Hypothesis testing with one sample (n>1) …
100 participants saw video containing violence
Then they free associated to 26 homonyms with aggressive & non-aggressive forms - e.g., pound, mug,
Mean number of aggressive free associates = 7.10
Suppose we know that without an aggressive video the mean ()=5.65 and the standard deviation () = 4.5
Is 7.10 significantly larger than 5.65?

9. Standard deviation known
In this case we have…
Sample
Population
Data of interest
Mean, n=100
Mean known
Error
Standard deviation known

10. Hypothesis testing with one sample (n>1) …
Use the sample mean instead of x in the z score formula
Use standard error of sample instead of the population standard deviation
becomes
where
n = the number of scores in the sample

11. Standard error:
If we know  then can be calculated using the formula

12. Sampling distribution
Will always be narrower than the parent population
The more samples that are taken the more normal the distribution
As sample size increases standard error decreases

13. Back to video violence If z > + 1.96, reject H0
H1:  5.65(two-tailed)
Calculate p for sample mean of 7.10 assuming =5.65
Use z from normal distribution as sampling distribution can be assumed to be normal
Calculate z = = =
If z > , reject H0
3.22 > 1.96  the difference is significant

14. But mostly we do not know σ
E.g. do penalty-takers show a preference for right or left?
16 penalty takers; 60 penalties each; null hypothesis = 50% or 30 each way
Result mean of 39 penalties to the left; is this significantly different?
µ = 30, but how do we calculate the standard error without the σ?

15. In this case we have… Sample Population Data of interest Mean n=100
Hypothesis= 30
Error
Variance ?

16. Using s to estimate σ
Can’t substitute s for  in a z score because s likely to be too small
So we need:
a different type of score – a t-score
a different type of distribution – Student’s t distribution

17. T distribution First published in 1908 by William Sealy Gosset,
Worked at a Guinness Brewery in Dublin on best yielding varieties of barley
Prohibited from publishing under his own name
so the paper was written under the pseudonym Student.

18. Allows us to calculate precisely what small samples tell us
T-test in a nut-shell…
Allows us to calculate precisely what small samples tell us
Uses three critical bits of information – mean, standard deviation and sample size

19. t test for one mean
Calculated the same way as z except  is replaced by s.
For the video example we gave before, s = 4.40 = = =

20. Degrees of freedom
t distribution is dependent on the sample size and this must be taken into account when calculating p
Skewness of sampling distribution decreases as n increases
t will differ from z less as sample size increases
t based on df where df = n - 1

21. t table

22. Statistical inference made
With n = 100, t = 1.98
Because t = 3.30 > 1.98, reject H0
Conclude that viewing violent video leads to more aggressive free associates than normal

23. Factors affecting t Difference between sample & population means
As value increases so t increases
Magnitude of sample variance
As sample variance decreases t increases
Sample size - as it increases
The value of t required to be significant decreases
The distribution becomes more like a normal distribution

24. Application of t-test to Within group designs

25. t for repeated measures scores
Same participants give data on two measures
Someone high on one measure probably high on other
Calculate difference between first and second score
Base subsequent analysis on these difference scores. Before and after data are ignored

26. Example - Therapy for PTSD
Therapy for victims of psychological trauma-Foa et al (1991)
9 Individuals received Supportive Counselling
Measured post-traumatic stress disorder symptoms before and after therapy

27. In this case we have… Sample Population Data of interest
Mean Difference
(n=9)
Hypothesis: = 0
Error
S of the Difference ?

28. Results
The Supportive Counselling group decreased number of symptoms - was difference significant?
If no change, mean of differences should be zero
So, test the obtained mean of difference scores against  = 0.
We don’t know , so use s and solve for t

29. Repeated measures t test
and = mean and standard deviation of differences respectively
df = n - 1 = = 8

30. Inference made With 8 df, t.025 = +2.306 We calculated t = 6.85
Since 6.85 > 2.306, reject H0
Conclude that the mean number of symptoms after therapy was less than mean number before therapy.
Infer that supportive counselling seems to work

31. + & - of Repeated measures design
Advantages
Eliminate subject-to-subject variability
Control for extraneous variables
Need fewer subjects
Disadvantages
Order effects
Carry-over effects
Subjects no longer naïve
Change may just be a function of time

32. t test is robust Test assumes that variances are the same
Even if the variances are not the same, the test still works pretty well
Test assumes data are drawn from a normally distributed population
Even if the population is not normally distributed, the test still works pretty well