Business Research Methods William G. Zikmund Chapter 22: Bivariate Analysis - Tests of Differences

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Cop yright © 2000 by Harcourt, Inc. All rights reserved. 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

Cop yright © 2000 by Harcourt, Inc. All rights reserved. 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

Cop yright © 2000 by Harcourt, Inc. All rights reserved. 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 tst Common Bivariate Tests

Cop yright © 2000 by Harcourt, Inc. All rights reserved. Type of Measurement Differences between two independent groups Nominal Chi-square test

Cop yright © 2000 by Harcourt, Inc. All rights reserved. DIFFERENCES BETWEEN GROUPS CONTINGENCY TABLES CROSS-TABULATION CHI-SQUARE ALLOWS TESTING FOR SIGNIFICANT DIFFERENCES BETWEEN GROUPS “GOODNESS OF FIT”

Cop yright © 2000 by Harcourt, Inc. All rights reserved. Chi-Square Test x² = chi-square statistics O i = observed frequency in the i th cell E i = expected frequency on the i th cell

Cop yright © 2000 by Harcourt, Inc. All rights reserved. Chi-Square Test R i = total observed frequency in the i th row C j = total observed frequency in the j th column n = sample size

DEGREES OF FREEDOM (R-1)(C-1)=(2-1)(2-1)=1 Copyright © 2000 by Harcourt, Inc. All rights reserved.

DEGREES OF FREEDOM D.f.=(R-1)(C-1) Copyright © 2000 by Harcourt, Inc. All rights reserved.

Awareness of Tire Manufacturer’s Brand MenWomenTotal Aware Unaware

Cop yright © 2000 by Harcourt, Inc. All rights reserved. Chi-Square test: Differences among groups example

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X 2 =3.84 with 1 d.f.

Cop yright © 2000 by Harcourt, Inc. All rights reserved. Type of Measurement Differences between two independent groups Interval and ratio t-test or Z-test

Cop yright © 2000 by Harcourt, Inc. All rights reserved. DIFFERENCE 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

Cop yright © 2000 by Harcourt, Inc. All rights reserved. NULL HYPOTHESIS ABOUT MEAN DIFFERENCES BETWEEN GROUPS

Cop yright © 2000 by Harcourt, Inc. All rights reserved. t-Test for difference of means

Cop yright © 2000 by Harcourt, Inc. All rights reserved. t-Test for the difference of Means X 1 = mean for Group 1 X 2 = mean for Group 2 S X 1 -X 2 = the pooled or combined standard error of difference between means.

Cop yright © 2000 by Harcourt, Inc. All rights reserved. t-Test for the difference of Means

Cop yright © 2000 by Harcourt, Inc. All rights reserved. X 1 = mean for Group 1 X 2 = mean for Group 2 S X 1 -X 2 = the pooled or combined standard error of difference between means. t-Test for the difference of Means

Cop yright © 2000 by Harcourt, Inc. All rights reserved. Pooled estimate of the standard error

Cop yright © 2000 by Harcourt, Inc. All rights reserved. Pooled estimate of the standard error S 1 2 = the variance of Group 1 S 2 2 = the variance of Group 2 n 1 = the sample size of Group 1 n 2 = the sample size of Group 2

Cop yright © 2000 by Harcourt, Inc. All rights reserved. P ooled estimate of the standard error t-Test for the difference of Means S 1 2 = the variance of Group 1 S 2 2 = the variance of Group 2 n 1 = the sample size of Group 1 n 2 = the sample size of Group 2

Cop yright © 2000 by Harcourt, Inc. All rights reserved. DEGREES OF FREEDOM d.f. = n - k where: –n = n 1 + n 2 –k = number of groups

Cop yright © 2000 by Harcourt, Inc. All rights reserved. t-Test for difference of means -example

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

Cop yright © 2000 by Harcourt, Inc. All rights reserved. COMPARING TWO GROUPS WHEN COMPARING PROPORTIONS PERCENTAGE COMPARISONS SAMPLE PROPORTION - P POPULATION PROPORTION -

Cop yright © 2000 by Harcourt, Inc. All rights reserved. DIFFERENCES BETWEEN TWO GROUPS WHEN COMPARING PROPORTIONS The hypothesis is H o :  1   may be restated as H o :  1   

Cop yright © 2000 by Harcourt, Inc. All rights reserved. A Z-test for differences of proportions or

Cop yright © 2000 by Harcourt, Inc. All rights reserved. Z-Test for differences of proportions

Cop yright © 2000 by Harcourt, Inc. All rights reserved. Z-Test for differences of proportions p 1 = sample portion of successes in Group 1 p 2 = sample portion of successes in Group 2  1  1 )  = hypothesized population proportion 1 minus hypothesized population proportion 1 minus S p1-p2 = pooled estimate of the standard errors of difference of proportions

Cop yright © 2000 by Harcourt, Inc. All rights reserved. Z-Test for differences of proportions - to calculate the standard error of the difference of proportions

Cop yright © 2000 by Harcourt, Inc. All rights reserved. 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 n  = sample size for group 1 n  = sample size for group 2 Z-Test for differences of proportions - to calculate the standard error of the difference of proportions

Cop yright © 2000 by Harcourt, Inc. All rights reserved. Z-Test for differences of proportions - to calculate the standard error of the difference of proportions - weighted average

Cop yright © 2000 by Harcourt, Inc. All rights reserved. A Z-test for differences of proportions

Cop yright © 2000 by Harcourt, Inc. All rights reserved. A Z-test for differences of proportions

Cop yright © 2000 by Harcourt, Inc. All rights reserved. Type of Measurement Differences between three or more independent groups Interval or ratio One-way ANOVA

Cop yright © 2000 by Harcourt, Inc. All rights reserved. ANALYSIS OF VARIANCE HYPOTHESIS WHEN COMPARING THREE GROUPS  1    

Cop yright © 2000 by Harcourt, Inc. All rights reserved. Analysis of variance - F-Ratio

Cop yright © 2000 by Harcourt, Inc. All rights reserved. Analysis of variance - Sum of squares

Cop yright © 2000 by Harcourt, Inc. All rights reserved. Analysis of variance - Sum of squares

Cop yright © 2000 by Harcourt, Inc. All rights reserved. Analysis of variance - Sum of squares p i = individual scores, i.e., the i th observation or test unit in the j th group p i = grand mean n = number of all observations ot test units in a group c = number of j th groups (or columns)

Cop yright © 2000 by Harcourt, Inc. All rights reserved. Analysis of variance - Sum of squares

Cop yright © 2000 by Harcourt, Inc. All rights reserved. Analysis of variance - Sum of squares p i = individual scores, i.e., the i th observation or test unit in the j th group p i = grand mean n = number of all observations ot test units in a group c = number of j th groups (or columns)

Cop yright © 2000 by Harcourt, Inc. All rights reserved. Analysis of variance - Sum of squares

Cop yright © 2000 by Harcourt, Inc. All rights reserved. Analysis of variance - Sum of squares = individual scores, i.e., the i th observation or test unit in the j th group = grand mean n j = number of all observations ot test units in a group

Cop yright © 2000 by Harcourt, Inc. All rights reserved. Analysis of variance - Mean Squares Between

Cop yright © 2000 by Harcourt, Inc. All rights reserved. Analysis of variance - Mean Square Within

Cop yright © 2000 by Harcourt, Inc. All rights reserved. Analysis of variance - Calculation of F-Ratio

Cop yright © 2000 by Harcourt, Inc. All rights reserved. A Test Market Experiment on Pricing Sales in Units (thousands) Regular Price $ X 1 = X= Reduced Price $ X 2 = Cents-Off Coupon Regular Price X 1 = Test Market A, B, or C Test Market D, E, or F Test Market G, H, or I Test Market J, K, or L Mean Grand Mean

Cop yright © 2000 by Harcourt, Inc. All rights reserved. ANOVA Summary table Source of variation BETWEEN GROUPS SUM OF SQUARES –SS BETWEEN DEGREES OF FREEDOM –c-1 where c=number of groups MEAN SQUARED-MS BETWEEN –SS BETWEEN /c-1

Cop yright © 2000 by Harcourt, Inc. All rights reserved. ANOVA Summary table Source of variation WITHIN GROUPS SUM OF SQUARES –SS WITHIN DEGREES OF FREEDOM –cn-c where c=number of groups, n= number of observations in a group MEAN SQUARED-MS WITHIN –SS WITHIN /cn-c

Cop yright © 2000 by Harcourt, Inc. All rights reserved. ANOVA Summary table Source of variation TOTAL SUM OF SQUARES –SS TOTAL DEGREES OF FREEDOM –cn-1 where c=number of groups, n= number of observations in a group

Cop yright © 2000 by Harcourt, Inc. All rights reserved. Type of Measurement Differences between two independent groups Ordinal Mann-Whitney U-test Wilcocon test