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**Differences Between Group Means**

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**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

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**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

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**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

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**Differences Between Groups**

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

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**two independent groups**

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

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**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

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**Null Hypothesis About Mean Differences Between Groups**

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**t-Test for Difference of Means**

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**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.

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**t-Test for Difference of Means**

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**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.

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**Pooled Estimate of the Standard Error**

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**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

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**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

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

where: n = n1 + n2 k = number of groups

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**t-Test for Difference of Means Example**

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**two independent groups**

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

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**Comparing Two Groups when Comparing Proportions**

Percentage Comparisons Sample Proportion - P Population Proportion -

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**Differences Between Two Groups when Comparing Proportions**

The hypothesis is: Ho: P1 = P2 may be restated as: Ho: P1 - P2 = 0

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**Z-Test for Differences of Proportions**

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**Z-Test for Differences of Proportions**

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**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

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**Z-Test for Differences of Proportions**

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**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

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**Z-Test for Differences of Proportions**

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**Z-Test for Differences of Proportions**

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**A Z-Test for Differences of Proportions**

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**Hypothesis Test of a Proportion**

p is the population proportion p is the sample proportion p is estimated with p

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**Hypothesis Test of a Proportion**

= H : . 5 p H : . 5 1

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**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 =

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**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

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