1. S519: Evaluation of Information Systems
Social Statistics
Inferential Statistics
Chapter 9: t test

2. This week
What is t test
Types of t test
TTEST function
T-test ToolPak

3. Why not z-test
In most cases, the z-test requires more information than we have available
We do inferential statistics to learn about the unknown population but, ironically, we need to know characteristics of the population to make inferences about it
Enter the t-test: “estimate what you don’t know”

4. William S. Gossett
Employed by Guinness Brewery, Dublin, Ireland, from 1899 to 1935.
Developed t-test around 1905, for dealing with small samples in brewing quality control.
Published in 1908 under pseudonym “Student” (“Student’s t-test”)

5. Types of t test

6. Degrees of Freedom and t test
Degrees of freedom describes the number of scores in a sample that are free to vary.
degrees of freedom = df = n-1
The larger, the better

7. One sample t test Very similar like z test
Use sample statistics instead of population parameters (mean and standard deviation)
Evaluate the result through t test table (Table B2) instead of z test table (Table B1)

8. An example
We show 26 babies the two pictures at the same time (one w/ his/her mother, the other a scenery picture) for 60 seconds, and measure how long they look at the facial configuration.
Our null assumption is that they will not look at it for longer than half the time, μ = 30
Our alternate hypothesis is that they will look at the face stimulus longer b/c face recognition is hardwired in their brain, not learned (directional)
Our sample of n = 26 babies looks at the face stimulus for M = 35 seconds, s = 16 seconds
Test our hypotheses (α = .05, one-tailed)

9. Step 1: Hypotheses Sentence: Code Symbols:
Null: Babies look at the face stimulus for less than or equal to half the time
Alternate: Babies look at the face stimulus for more than half the time
Code Symbols:

10. Step 2: Determine Critical Region
Population variance is not known, so use sample variance to estimate
n = 26 babies; df = n-1 = 25
Look up values for t at the limits of the critical region from our critical values of t table
Set α = .05; one-tailed
tcrit =

11. Step 3: Calculate t statistic from sample
Central Limit Theorem
μ = 30
sM=s/ =16/ = 3.14

12. Step 4: Decision and Conclusion
The tobt=1.59 does not exceed tcrit=1.708
∴ We must retain the null hypothesis
Conclusion: Babies do not look at the face stimulus more often than chance, t(25) = +1.59, n.s., one-tailed. Our results do not support the hypothesis that face processing is innate.

13. Independent t test
A research design that uses a separate sample for each treatment condition is called an independent-measures (or between-subjects) research design.

14. t Statistic for Independent- Measures Design
The goal of an independent-measures research study:
To evaluate the difference of the means between two populations.
Mean of first population: μ1
Mean of second population: μ2
Difference between the means: μ1- μ2

15. t Statistic for Independent- Measures Design
Null hypothesis: “no change = no effect = no difference”
H0: μ1- μ2 = 0
Alternative hypothesis: “there is a difference”
H1: μ1- μ2 ≠ 0

16. T test formula : is the mean for Group 1 : is the mean for Group 2
: is the number of participants in Group 1
: is the number of participants in Group 2
: is the variance for Group 1
: is the variance for Group 2
Value for degrees of freedom: df = df1 + df2

17. An Example
 Group 1 
 Group 2  7 5 3 4 2 6 1 10 9 8 12 15

18. T test steps Step 1: A statement of the null and research hypotheses.
Null hypothesis: there is no difference between two groups
Research hypothesis: there is a difference between the two groups

19. T test steps
Step 2: setting the level of risk (or the level of significance or Type I error) associated with the null hypothesis
0.05

20. T test steps Step 3: Selection of the appropriate test statistic
Determine which test statistic is good for your research
Independent t test

21. T test steps
Step 4: computation of the test statistic value
t= 0.14

22. T test steps
Step 5: determination of the value needed for the rejection of the null hypothesis
Table B2 in Appendix B (S-p360)
Degrees of freedom (df): approximates the sample size
Group 1 sample size -1 + group 2 sample size -1
Our test df= 58
Two-tailed or one-tailed
Directed research hypothesis  one-tailed
Non-directed research hypothesis  two-tailed

23. T test steps
Step 6: A comparison of the obtained value and the critical value
0.14 and 2.001
If the obtained value > the critical value, reject the null hypothesis
If the obtained value < the critical value, retain the null hypothesis

24. T test steps Step 7 and 8: make a decision
What is your decision and why?

25. Interpretation
How to interpret t(58) = 0.14, p>0.05, n.s.

26. Excel: TTEST function TTEST (array1, array2, tails, type)
array1 = the cell address for the first set of data
array2 = the cell address for the second set of data
tails: 1 = one-tailed, 2 = two-tailed
type: 1 = a paired t test; 2 = a two-sample test (independent with equal variances); 3 = a two-sample test with unequal variances

27. Excel: TTEST function It does not compute the t value
It returns the likelihood that the resulting t value is due to chance (the possibility of the difference of two groups is due to chance)

28. Excel ToolPak Select t-test: two sample assuming equal variances
Variable 1
Variable 2
Mean
Variance
Observations 30
Pooled Variance
Hypothesized Mean Difference df 58
t Stat
P(T<=t) one-tail
t Critical one-tail
P(T<=t) two-tail
t Critical two-tail

29. Effect size
If two groups are different, how to measure the difference among them
Effect size
ES: effect size
: the mean for Group 1
: the mean for Group 2
SD: the standard deviation from either group

30. Effect size A small effect size ranges from 0.0 ~ 0.2
Both groups tend to be very similar and overlap a lot
A medium effect size ranges from 0.2 ~ 0.5
The two groups are different
A large effect size is any value above 0.50
The two groups are quite different
ES=0the two groups have no difference and overlap entirely
ES=1the two groups overlap about 45%