 # Differences Between Group Means

## Presentation on theme: "Differences Between Group Means"— Presentation transcript:

Differences Between Group Means

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

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

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

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

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

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

Null Hypothesis About Mean Differences Between Groups

t-Test for Difference of Means

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.

t-Test for Difference of Means

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.

Pooled Estimate of the Standard Error

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

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

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

t-Test for Difference of Means Example

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

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

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

Z-Test for Differences of Proportions

Z-Test for Differences of Proportions

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

Z-Test for Differences of Proportions

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

Z-Test for Differences of Proportions

Z-Test for Differences of Proportions

A Z-Test for Differences of Proportions

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

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

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 =

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