Chapter 22—Bivariate Statistical Analysis: Differences Between Two Variables TRUE/FALSE Business researchers frequently use cross-tabulation techniques in data analysis ANS: T PTS: NAT: AACSB: Reflective Thinking REF: p 530 When the difference between two groups is measured, this is a form of bivariate statistics ANS: T PTS: NAT: AACSB: Reflective Thinking REF: p 530 A statistical test that is used often in business research tests the difference between two groups in terms of some behavior (e.g attitude) ANS: T PTS: NAT: AACSB: Reflective Thinking REF: p 530 If a researcher is interested in whether males file work grievances more frequently than females, this is an example of univariate statistics ANS: F This is an example of bivariate statistics PTS: REF: p 530 NAT: AACSB: Reflective Thinking A cross-tabulation is a simple way to describe the relationships between two groups ANS: T PTS: NAT: AACSB: Reflective Thinking REF: p 530 One way to test the significance of contingency tables is by means of the chi-square test ANS: T PTS: NAT: AACSB: Reflective Thinking REF: p 530 To use the chi-square test, both variables in a x contingency table must be measured on a ratio scale ANS: F A frequency count of data that nominally identify or categorically rank groups is acceptable PTS: REF: p 530 NAT: AACSB: Reflective Thinking The chi-square test requires the researcher to compare the observed frequencies of the groups with the expected frequencies of the groups ANS: T PTS: NAT: AACSB: Reflective Thinking REF: p 532 © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part 9 The chi-square test tests the significance of the data in an R X C contingency table in which R stands for row and C stands for column ANS: T PTS: NAT: AACSB: Reflective Thinking REF: p 532 10 The chi-square test requires that the expected frequency in each cell of the contingency table be at least 30 ANS: F The expected frequency in each cell should be at least PTS: REF: p 534 NAT: AACSB: Reflective Thinking 11 The independent samples t-test tests the differences between means taken from two independent samples or groups ANS: T PTS: NAT: AACSB: Reflective Thinking REF: p 534 12 The t-test for comparing the means of two groups assumes that the data are in nominal scales ANS: F It tests a dichotomous (two-level) independent variable associated with changes in a continuous dependent variable PTS: REF: p 534 NAT: AACSB: Reflective Thinking 13 A pooled estimate of the standard error is a poorer estimate of the standard error than one based on the variance from either sample ANS: F A pooled estimate of the standard error is a better estimate PTS: REF: p 535 NAT: AACSB: Reflective Thinking 14 The degrees of freedom are calculated as d.f = n − when using the t-test for comparing two means ANS: F In a test of two means, degrees of freedom are calculated as d.f = n − k, where n = n1 + n2 and k = number of groups PTS: REF: p 536 NAT: AACSB: Reflective Thinking 15 In practice, t-test results are commonly calculated by hand because they are so simple ANS: F In practice, computer software is used to compute the t-test results PTS: REF: p 536 NAT: AACSB: Reflective Thinking 16 When no direction of a relationship is stated in the hypothesis, a one-tailed test is appropriate © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part ANS: F A two-tailed test is appropriate PTS: REF: p 536 NAT: AACSB: Reflective Thinking 17 The first basic step in interpreting t-test results is to find the p-value associated with a particular t and the corresponding degrees of freedom ANS: F This is the third step The first step is to examine the difference in means to find the “direction” of any difference PTS: REF: pp 536-537 NAT: AACSB: Reflective Thinking 18 The t-test assumes that the two population variances are equal ANS: T PTS: NAT: AACSB: Reflective Thinking REF: p 538 19 Means that appear to be not so close to each other are usually statistically concluded that they are different ANS: T Even though mans appear to be not so close to each other, the statistical conclusion could be that they are the same PTS: REF: p 538 NAT: AACSB: Reflective Thinking 20 A t-test is not appropriate and should not be used when the sample size is greater than 30 ANS: T Researchers often employ a t-test even with large samples As samples get larger, the t-test and Z-test will tend to yield the same results PTS: REF: p 538 NAT: AACSB: Reflective Thinking 21 A redundant-sample t-test is an appropriate test for comparing the scores of two interval drawn from related populations ANS: F This is referred to as a paired-sample t-test PTS: REF: p 538 NAT: AACSB: Reflective Thinking 22 A Z-test for differences of proportions requires a sample size greater than 100 ANS: F It requires a sample size greater than 30 PTS: REF: p 540 NAT: AACSB: Reflective Thinking 23 The null hypothesis for an ANOVA test comparing the means of three groups is: à1 = à2 = à3 â 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part ANS: T PTS: NAT: AACSB: Reflective Thinking REF: p 541 24 The total mean is the mean of a variable over all observations ANS: F This is called the grand mean PTS: REF: p 544 NAT: AACSB: Reflective Thinking 25 The F-test partitions the total variance into within-group variance and between-group variance ANS: T PTS: NAT: AACSB: Reflective Thinking REF: p 546 MULTIPLE CHOICE A researcher hypothesizes that males and females differ with respect to attitude toward sports sponsorships To investigate this hypothesis that these two groups’ attitudes differ, he will use a: a bivariate test of differences b univariate test of differences c multivariate test of differences d cluster analysis ANS: A PTS: NAT: AACSB: Reflective Thinking REF: p 530 Another name for a cross-tabulation is: a ANOVA b Z-test c contingency table d between-group variance ANS: C PTS: NAT: AACSB: Reflective Thinking REF: p 530 The formula given below represents the method for calculating the: a b c d Z-test F-test χ2 test α ANS: C PTS: NAT: AACSB: Reflective Thinking REF: p 532 The formula for the chi-square test uses: a observed and expected frequencies b observed and expected percentages c the two sample means © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part d the two sample standard deviations ANS: A PTS: NAT: AACSB: Reflective Thinking REF: p 532 How many degrees of freedom are there in a four-cell chi-square test? a (R + 1) b (R - 1) c (R - 1)(C - 1) d R(C - 1) ANS: C PTS: NAT: AACSB: Reflective Thinking REF: p 533 If 25 of the 35 females in a research study agree with a statement, and 15 of the 35 males agree with this statement, the expected value for males-agree is: a 15 b 20 c 25 d 35 ANS: B PTS: REF: p 533 NAT: AACSB: Reflective Thinking|AACSB: Analytic In a brand awareness study, if 25 of a group of 35 males identify the brand correctly and 15 of a group of 35 females identify this brand correctly, the chi-square value for this study is approximately: a 3.26 b 4.15 c 5.84 d 2.92 ANS: C PTS: REF: p 533 NAT: AACSB: Reflective Thinking| AACSB: Analytic If the critical value of chi-square in the chi-square table is 3.84, and the obtained value of chi-square in the study is 3.45, the researcher should: a reject the null hypothesis b accept the null hypothesis c accept the research hypothesis d reject both the null and the research hypotheses ANS: B PTS: NAT: AACSB: Reflective Thinking REF: p 533 In order to use the chi-square test, the expected frequency in each of the cells of the contingency table should be at least: a b c 30 d 40 ANS: B PTS: REF: p 535 NAT: AACSB: Reflective Thinking| AACSB: Analytic 10 In using the t-test to compare the means of two groups, the alternative hypothesis is typically stated as: a à1 à2 â 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part b µ1 = µ2 c µ1 - µ2 = d µ1 + µ2 = ANS: A PTS: NAT: AACSB: Reflective Thinking REF: p 535 11 In comparing the difference of the means between two groups, the null hypothesis can also be stated as: a µ1 - µ2 = b µ1 + µ2 = c µ1 x µ2 = d µ1 / µ2 = ANS: A PTS: NAT: AACSB: Reflective Thinking REF: p 535 12 Another form of the alternative hypothesis when testing the difference between the means of the two groups is: a µ1 - µ2 = b µ1 - µ2 ≠ c µ1 - µ2 = d µ1 - µ2 ≠ ANS: B PTS: NAT: AACSB: Reflective Thinking REF: p 535 13 Supposed you used a 10-point rating scale to measure intention-to-buy (1 = definitely would not buy and 10 = definitely would buy) If a group of 40 males had a mean of and a standard deviation of 2.5, while a group of 35 females had a mean of and a standard deviation of 1.4, the standard error of the difference between the means would be approximately: a 0.48 b 1.36 c 2.45 d not enough information to determine ANS: A PTS: REF: p 535 NAT: AACSB: Reflective Thinking| AACSB: Analytic 14 In using the t-test to compare the means of two groups, the degrees of freedom are: a n1 - n2 - b n1 + n2 - c n1 x n2 -2 d n1 / n2 - ANS: B PTS: NAT: AACSB: Reflective Thinking REF: p 536 15 Suppose that you used a 9-point rating scale and that you wanted to compare men who had an annual income over $50,000 (Group 1) with men who had an annual income less than or equal to $50,000 (Group 2) on their liking of a new product If you studied 40 men in Group and they have a mean of and a standard deviation of 2.5, while the 35 men in Group have a mean of and a standard deviation of 1.4, what is the approximate value of t using the t-test? a 3.43 © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part b 4.19 c 2.64 d not enough information to determine ANS: B PTS: REF: p 536 NAT: AACSB: Reflective Thinking| AACSB: Analytic 16 In a study comparing the means of two groups in which there are 45 males in Group and 37 females in Group 2, the degrees of freedom for this study when using the t-test for the difference between means is: a 84 b 80 c 82 d 160 ANS: B PTS: REF: p 536 NAT: AACSB: Reflective Thinking| AACSB: Analytic 17 In practice, what is the first step in interpreting the t-test when comparing two means? a compute the t-test value b find the p-value associated with the t and the corresponding degrees of freedom c examine the difference in means to find the “direction” of any difference d examine if there is a difference at the 50 percent confidence interval before examining the 95 percent confidence interval ANS: C PTS: NAT: AACSB: Reflective Thinking REF: p 536 18 Which test is appropriate for comparing the scores of two interval variables drawn from related populations? a ANOVA b relative t-test c relative χ2 test d pair-samples t-test ANS: D PTS: NAT: AACSB: Reflective Thinking REF: p 538 19 In using the Z-test for comparing two proportions, the null hypothesis is typically stated as: a π1 = π2 b π1 ≠ π2 c π1 x π2 = d π1 - π2 = ANS: A PTS: NAT: AACSB: Reflective Thinking REF: p 540 20 The alternative hypothesis when using the Z-test for the difference between the proportions for two groups is: a π1 = π2 b π1 ≠ π2 c π1 x π2 = d π1 / π2 = ANS: B PTS: REF: p 540 © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part NAT: AACSB: Reflective Thinking 21 In using the Z-test for comparing two proportions, if the pooled estimate of the proportion of successes is 0.36, then the pooled estimate of proportion of failures is: a b 0.36 c 0.64 d 0.13 ANS: C PTS: REF: p 540 NAT: AACSB: Reflective Thinking| AACSB: Analytic 22 Suppose that you wanted to determine if the males in your sales force differed from the females in your sales force in the proportion of sales that they made to mass merchandiser chains Your data show that the 80 males had 40% of their sales to these chains, while the 60 females had 25% of their sales to these chains For these data, the standard error of the difference in proportions is approximately: a 0.08 b 0.15 c 0.02 d 1.67 ANS: A PTS: REF: p 541 NAT: AACSB: Reflective Thinking| AACSB: Analytic 23 In an ANOVA test of the differences between the means of three groups, the null hypothesis is: a µ1 = µ2 = µ3 b µ1 ≠ µ2 ≠ µ3 c µ1 - µ2 - µ3 = d µ1 + µ2 + µ3 = ANS: A PTS: NAT: AACSB: Reflective Thinking REF: p 541 24 Which test should be used when comparing the means of three groups to see if they are significantly different from one another? a One-group t-test b ANOVA test c Two-group t-test d Chi-square test ANS: B PTS: NAT: AACSB: Reflective Thinking REF: p 541 25 The mean of a variable over all observations is called the: a master mean b average mean c grand mean d ANOVA mean ANS: C PTS: NAT: AACSB: Reflective Thinking REF: p 544 26 Which of the following is the sum of differences between the group mean and the grand mean summed over all groups for a given set of observations? © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part a b c d between-groups variance total error variance F-statistic χ2 value ANS: A PTS: NAT: AACSB: Reflective Thinking REF: p 544 27 The sum of the differences between observed values and the group mean for a given set of observations is known as the: a within-group error b between-groups variance c F-ratio d γ statistic ANS: A PTS: NAT: AACSB: Reflective Thinking REF: p 545 28 The key statistical test for an ANOVA model is the: a χ2 test b t-test c F-test d Z-test ANS: C PTS: NAT: AACSB: Reflective Thinking REF: p 545 29 The F-distribution is a function of: a SSB - SSE b SSE - SSB c SSB/SSE d SSE/SSB ANS: C PTS: NAT: AACSB: Reflective Thinking REF: p 546 30 Practically speaking, what is the first thing a researcher should when interpreting ANOVA results? a examine the actual means for each group b determine the between-groups variance c examine the total variance d check whether or not the overall model F is significant ANS: D PTS: NAT: AACSB: Reflective Thinking REF: p 546 COMPLETION A test of a hypothesis to determine if two groups differ with respect to the scores on one variable is called a(n) ANS: test of differences PTS: REF: p 530 NAT: AACSB: Reflective Thinking © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part 2 When two variables are studied simultaneously in a research study, this is called statistical analysis ANS: bivariate PTS: REF: p 530 NAT: AACSB: Reflective Thinking The test studies the significance of a joint frequency distribution ANS: chi-square PTS: REF: p 530 NAT: AACSB: Reflective Thinking The formula for determining the degrees of freedom for a chi-square test is ANS: (R - 1)(C - 1) PTS: REF: p 533 NAT: AACSB: Reflective Thinking When a researcher needs to compare means for a variable grouped into two categories based on some less-than interval variable, a(n) test is appropriate ANS: t PTS: REF: p 534 NAT: AACSB: Reflective Thinking In using the t-test to compare the difference between the means of two groups, the null hypothesis is normally stated as ANS: µ1 = µ2 PTS: REF: p 535 NAT: AACSB: Reflective Thinking In using the t-test to compare the difference between the means of two groups, the formula for determining the degrees of freedom is ANS: n1 + n2 - PTS: REF: p 536 NAT: AACSB: Reflective Thinking One way to interpret the meaning of the results of the t-test is to focus on the and the group means ANS: p-value PTS: REF: p 538 NAT: AACSB: Reflective Thinking An appropriate test for comparing the scores of two interval variables drawn from related populations is the ANS: paired-samples t-test PTS: REF: p 539 NAT: AACSB: Reflective Thinking © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part 10 The null hypothesis when using the Z-test for the differences between the proportions of two groups is that ANS: π1 = π2 PTS: REF: p 540 NAT: AACSB: Reflective Thinking 11 is the appropriate test to use when comparing the means of three or more groups to see if they are significantly different from one another ANS: ANOVA PTS: REF: p 541 NAT: AACSB: Reflective Thinking 12 The is the mean of a variable over all observations ANS: grand mean PTS: REF: p 544 NAT: AACSB: Reflective Thinking 13 The sum of differences between the group mean and the grand mean summed over all groups for a given set of observations is called ANS: between-groups variance PTS: REF: p 544 NAT: AACSB: Reflective Thinking 14 The statistical test that is used to compare the sample variance with another sample variance is the ANS: F-test PTS: REF: p 545 NAT: AACSB: Reflective Thinking 15 The F-test partitions total variance into variance and variance ANS: within-group; between-group PTS: REF: p 546 NAT: AACSB: Reflective Thinking ESSAY Explain what an independent samples t-test is and the null hypothesis examined ANS: The independent samples t-test tests the differences between means taken from two independent samples or groups It is a test for hypotheses stating that the mean scores for some interval- or ratioscaled variable groups based on some less-than interval classificatory variable are not the same Thus, the null hypothesis is µ1 = µ2, which is equivalent to µ1 - µ2 = PTS: REF: pp 534-535 © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part NAT: AACSB: Reflective Thinking|AACSB: Communication List the practical steps for interpreting t-test results ANS: The basic steps for interpreting t-test results are: (1) Examine the difference in means to find the “directions” on any difference (2) Compare or locate the computed t-test value (3) Find the p-value associated with this t and the corresponding degrees of freedom (4) The difference can also be examined using the 95 percent confidence interval PTS: REF: pp 536-537 NAT: AACSB: Reflective Thinking|AACSB: Communication Discuss when a paired-samples t-test is appropriate and give an example of when it should be used ANS: A paired-samples t-test is an appropriate test for comparing the scores of two interval variables drawn from related populations This is the appropriate test when the same respondent is measured twice For example, respondents may be asked their attitudes toward corporate sponsorship in sports and their attitudes toward corporate sponsorship of the arts Since the attitude scores are both provided by the same person, the assumption that they are independent is not realistic When a paired-samples t-test is appropriate, the two numbers being compared are usually scored as separate variables PTS: REF: p 538 NAT: AACSB: Reflective Thinking|AACSB: Communication Explain what a Z-test for comparing two proportions is and the appropriate null hypothesis ANS: A Z-test for differences of proportions is a technique used to test the hypothesis that proportions are significantly different for two independent samples or groups The null hypothesis is π1 = π2, which may be restated as π1 - π2 = PTS: REF: p 540 NAT: AACSB: Reflective Thinking|AACSB: Communication What is ANOVA and when is it the appropriate statistical technique? What test is used to determine significance? ANS: ANOVA stands for analysis of variance and involves the investigation of the effects of one treatment variable on an interval-scaled dependent variable It is a hypothesis-testing technique to determine whether statistically significant differences in means occur between two or more groups The F-test is the key statistical test for an ANOVA model PTS: REF: pp 541-546 NAT: AACSB: Reflective Thinking|AACSB: Communication © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part  ... Which test should be used when comparing the means of three groups to see if they are significantly different from one another? a One-group t -test b ANOVA test c Two-group t -test d Chi-square test. .. NAT: AACSB: Reflective Thinking REF: p 545 28 The key statistical test for an ANOVA model is the: a χ2 test b t -test c F -test d Z -test ANS: C PTS: NAT: AACSB: Reflective Thinking REF: p 545 29 The... 536 18 Which test is appropriate for comparing the scores of two interval variables drawn from related populations? a ANOVA b relative t -test c relative χ2 test d pair-samples t -test ANS: D PTS: