1. Independent Sample T-test Formula

2. Independent-samples t-tests
But what if you have more than two groups?
One suggestion: pairwise comparisons (t-tests)

3. Multiple independent-samples t-tests
That’s a lot of tests!
# groups # tests
2 groups = 1 t-test
3 groups = 3 t-tests
4 groups = 6 t-tests
5 groups = 10 t-tests
...
10 groups = 45 t-tests

4. Inflation of familywise error rate
Familywise error rate – the probability of making at least one Type I error (rejecting the Null Hypothesis when the null is true)
Every hypothesis test has a probability of making a Type I error (a).
For example, if two t-tests are each conducted using a = .05, there is a probability of committing at least one Type I error.

5. Inflation of familywise error rate
The formula for familywise error rate:
# groups # tests nominal alpha familywise alpha
2 groups t-test
3 groups t-tests
4 groups t-tests
5 groups t-tests
...
10 groups 45 t-tests

6. Analysis of Variance: Purpose
Are there differences in the central tendency (mean) of groups?
Inferential: Could the observed differences be due to chance?

7. Assumptions of ANOVA
Normality – scores should be normally distributed within each group.
Homogeneity of variance – scores should have the same variance within each group.
Independence of observations – observations are randomly selected.

8. Logic of Analysis of Variance
Null hypothesis (Ho): Population means from different conditions are equal
m1 = m2 = m3 = m4
Alternative hypothesis: H1
Not all population means equal.

9. Lets visualize total amount of variance in an experiment
Total Variance = Mean Square Total
Between Group Differences
(Mean Square Group)
Error Variance
(Individual Differences + Random Variance)
Mean Square Error
F ratio is a proportion of the MS group/MS Error.
The larger the group differences, the bigger the F
The larger the error variance, the smaller the F

10. Logic--cont. Create a measure of variability among group means
MSgroup
Create a measure of variability within groups
MSerror

11. Example: Test Scores and Attitudes on Statistics
Loves Statistics
Hates Statistics
Indifferent 9 4 8 7 6 5 11 12 3

12. Find the sum of squares between groups
Find the sum of squares within groups
Total sum of squares = sum of between group and within group sums of squares.

13. To find the mean squares: divide each sum of squares by the degrees of freedom (2 different dfs)
Degrees of freedom between groups =
k-1, where k = # of groups
Degrees of freedom within groups = n-k
MSbetween= SSbetween/dfbetween
MSwithin= SSwithin/dfwithin
F = MSbetween / MSwithin
Compare your F with the F in Table D