1. Differences Between Group Means

7. Common Bivariate Tests
Type of Measurement
Differences between
two independent groups
Differences among
three or more
independent groups
Interval and ratio
Independent groups:
t-test or Z-test
One-way
ANOVA

8. Common Bivariate Tests
Type of Measurement
Differences between
two independent groups
Differences among
three or more
independent groups
Ordinal
Mann-Whitney U-test
Wilcoxon test
Kruskal-Wallis test

9. Common Bivariate Tests
Type of Measurement
Differences between
two independent groups
Differences among
three or more
independent groups
Nominal
Z-test (two proportions)
Chi-square test
Chi-square test

10. Differences Between Groups
Contingency Tables
Cross-Tabulation
Chi-Square allows testing for significant differences between groups
“Goodness of Fit”

11. two independent groups
Type of
Measurement
Differences between
two independent groups
Interval and
ratio
t-test or
Z-test

12. Differences Between Groups when Comparing Means
Ratio scaled dependent variables
t-test
When groups are small
When population standard deviation is unknown
z-test
When groups are large

13. Null Hypothesis About Mean Differences Between Groups

14. t-Test for Difference of Means

15. t-Test for Difference of Means
X1 = mean for Group 1
X2 = mean for Group 2
SX1-X2 = the pooled or combined standard error
of difference between means.

16. t-Test for Difference of Means

17. t-Test for Difference of Means
X1 = mean for Group 1
X2 = mean for Group 2
SX1-X2 = the pooled or combined standard error
of difference between means.

18. Pooled Estimate of the Standard Error

19. Pooled Estimate of the Standard Error
S12 = the variance of Group 1
S22 = the variance of Group 2
n1 = the sample size of Group 1
n2 = the sample size of Group 2

20. Pooled Estimate of the Standard Error t-test for the Difference of Means
S12 = the variance of Group 1
S22 = the variance of Group 2
n1 = the sample size of Group 1
n2 = the sample size of Group 2

21. Degrees of Freedom d.f. = n - k n = n1 + n2 k = number of groups
where:
n = n1 + n2
k = number of groups

22. t-Test for Difference of Means Example

24. two independent groups
Type of
Measurement
Differences between
two independent groups
Nominal
Z-test (two proportions)

25. Comparing Two Groups when Comparing Proportions
Percentage Comparisons
Sample Proportion - P
Population Proportion -

26. Differences Between Two Groups when Comparing Proportions
The hypothesis is:
Ho: P1 = P2
may be restated as:
Ho: P1 - P2 = 0

27. Z-Test for Differences of Proportions

28. Z-Test for Differences of Proportions

29. Z-Test for Differences of Proportions
p1 = sample portion of successes in Group 1
p2 = sample portion of successes in Group 2
(p1 - p1) = hypothesized population proportion 1
minus hypothesized population
proportion 1 minus
Sp1-p2 = pooled estimate of the standard errors of
difference of proportions

30. Z-Test for Differences of Proportions

31. Z-Test for Differences of Proportions
p = pooled estimate of proportion of success in a
sample of both groups
p = (1- p) or a pooled estimate of proportion of
failures in a sample of both groups
n1= sample size for group 1
n2= sample size for group 2

32. Z-Test for Differences of Proportions

33. Z-Test for Differences of Proportions

34. A Z-Test for Differences of Proportions

35. Hypothesis Test of a Proportion
p is the population proportion
p is the sample proportion
p is estimated with p

36. Hypothesis Test of a Proportion= H : . 5 p H : . 5 1

39. Hypothesis Test of a Proportion: Another Example
200 , 1 n = 20 . p = n pq S p =
1200 ) 8
)(. 2 (. S p =
1200 16 . S p =
000133 . S p =
0115 . S p =

40. Hypothesis Test of a Proportion: Another Example- p Z = S p . 20 - . 15 Z = .
0115 . 05 Z = .
0115 Z = 4 .
348
.001
the
beyond t
significant is it
level.
.05 at
rejected be
should
hypothesis
null so
1.96,
exceeds
value Z
The
Indeed