Basic Business Statistics, 10/e

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1 Chapter Basic Business Statistics 11 th Edition Chapter 1 Chi-Square Tests and Nonparametric Tests Basic Business Statistics, 11e 009 Prentice-Hall, Inc. Chap 1-1 Learning Objectives In this chapter, you learn: How and when to use the chi-square test for contingency tables How to use the Marascuilo procedure for determining pairwise differences when evaluating more than two proportions How and when to use the McNemar test How to use the chi-square test for a variance or standard deviation How and when to use nonparametric tests Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap Prentice Hall, Inc.

2 Chapter 1 1- Contingency Tables Contingency Tables Useful in situations involving multiple population proportions Used to classify sample observations according to two or more characteristics Also called a cross-classification table. Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap 1-3 Contingency Table Example Left-Handed vs. Gender Dominant Hand: Left vs. Right Gender: Male vs. Female categories for each variable, so called a x table Suppose we examine a sample of 300 children Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap Prentice Hall, Inc.

3 Chapter Contingency Table Example Sample results organized in a contingency table: sample size = n = 300: 10 Females, 1 were left handed 180 Males, 4 were left handed Hand Preference Gender Left Right Female Male Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap 1-5 Test for the Difference Between Two Proportions H 0 : π 1 = π (Proportion of females who are left handed is equal to the proportion of males who are left handed) H 1 : π 1 π (The two proportions are not the same hand preference is not independent of gender) If H 0 is true, then the proportion of left-handed females should be the same as the proportion of left-handed males The two proportions above should be the same as the proportion of left-handed people overall Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap Prentice Hall, Inc.

4 Chapter The Chi-Square Test Statistic The Chi-square test statistic is: ( f f ) o e χ f all cells e where: f o = observed frequency in a particular cell f e = expected frequency in a particular cell if H 0 is true χ for the x casehas1degreeof freedom (Assumed: each cell in the contingency table has expected frequency of at least 5) Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap 1-7 Decision Rule The χ test statistic approximately follows a chisquared distribution with one degree of freedom Decision Rule: If χ χ α, reject H 0, otherwise, do not reject H 0 0 Do not reject H 0 α Reject H 0 Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap Prentice Hall, Inc.

5 Chapter Computing the Average Proportion The average proportion is: p X n 1 X n 1 X n 10 Females, 1 were left handed 180 Males, 4 were left handed Here: 1 4 p i.e., of all the children the proportion of left handers is 0.1, that is, 1% Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap 1-9 Finding Expected Frequencies To obtain the expected frequency for left handed females, multiply the average proportion left handed (p) by the total number of females To obtain the expected frequency for left handed males, multiply the average proportion left handed (p) by the total number of males If the two proportions are equal, then P(Left Handed Female) = P(Left Handed Male) =.1 i.e., we would expect (.1)(10) = 14.4 females to be left handed (.1)(180) = 1.6 males to be left handed Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap Prentice Hall, Inc.

6 Chapter Observed vs. Expected Frequencies Gender Female Male Left Observed = 1 Expected = 14.4 Observed = 4 Expected = 1.6 Hand Preference Right Observed = 108 Expected = Observed = 156 Expected = Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap 1-11 Gender Female Male The test statistic is: The Chi-Square Test Statistic Left Observed = 1 Expected = 14.4 Observed = 4 Expected = 1.6 Hand Preference Right Observed = 108 Expected = Observed = 156 Expected = χ all cells (f o f f e (114.4) 14.4 e ) ( ) (4 1.6) 1.6 ( ) Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap Prentice Hall, Inc.

7 Chapter Decision Rule Thetest statisticis χ ; χ 0.05 with 1 d.f Decision Rule: If χ > 3.841, reject H 0, otherwise, do not reject H 0 0 Do not reject H Reject H = Here, χ = < χ 0.05 = 3.841, so we do not reject H 0 and conclude that there is not sufficient evidence that the two proportions are different at = 0.05 Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap 1-13 Test for Differences Among More Than Two Proportions Extend the test to the case with more than two independent populations: H 0 : π 1 = π = = π c H 1 : Not all of the π j are equal (j = 1,,, c) Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap Prentice Hall, Inc.

8 Chapter The Chi-Square Test Statistic The Chi-square test statistic is: χ all cells ( f Where: f o = observed frequency in a particular cell of the x c table f e = expected frequency in a particular cell if H 0 is true χ (Assumed: each cell in the contingency table has expected frequency of at least 1) o f for the x c casehas(- 1)(c- 1) c - 1degreesof freedom e f e ) Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap 1-15 Computing the Overall Proportion The overall proportion is: p X X n n 1 X n 1 c c X n Expected cell frequencies for the c categories are calculated as in the x case, and the decision rule is the same: Decision Rule: If χ χ, reject H 0, α otherwise, do not reject H 0 χ α Where is from the chisquared distribution with c 1 degrees of freedom Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap Prentice Hall, Inc.

9 Chapter The Marascuilo Procedure Used when the null hypothesis of equal proportions is rejected Enables you to make comparisons between all pairs Start with the observed differences, p j p j, for all pairs (for j j )......then compare the absolute difference to a calculated critical range Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap 1-17 The Marascuilo Procedure Critical Range for the Marascuilo Procedure: Criticalrange χ α p j(1 p j) n j p j' (1 p n j' j' ) (Note: the critical range is different for each pairwise comparison) A particular pair of proportions is significantly different if p j p j > critical range for j and j Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap Prentice Hall, Inc.

10 Chapter Marascuilo Procedure Example A University is thinking of switching to a trimester academic calendar. A random sample of 100 administrators, 50 students, and 50 faculty members were surveyed Opinion Administrators Students Faculty Favor Oppose Totals Using a 1% level of significance, which groups have a different attitude? Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap 1-19 Chi-Square Test Results H 0 : π 1 = π = π 3 H 1 : Not all of the π j are equal (j = 1,, 3) Expected Chi-Square Test: Administrators, Students, Faculty Admin Students Faculty Total Favor Oppose Total Observed χ χ 9.103so reject H 0 Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap Prentice Hall, Inc.

11 Chapter Marascuilo Procedure: Solution Excel Output: Marascuilo Procedure compare Sample Sample Absolute Std. Error Critical Group Proportion Size ComparisonDifference of Difference Range Results to Means are not different to Means are not different to Means are different Level of significance d.f Q Statistic Other Data 0.01 Chi-sq Critical Value At 1% level of significance, there is evidence of a difference in attitude between students and faculty Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap 1-1 Test of Independence Similar to the test for equality of more than two proportions, but extends the concept to contingency tables with r rows and c columns H 0 : The two categorical variables are independent (i.e., there is no relationship between them) H 1 : The two categorical variables are dependent (i.e., there is a relationship between them) Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap Prentice Hall, Inc.

12 Chapter Test of Independence The Chi-square test statistic is: all cells where: f o = observed frequency in a particular cell of the r x c table f e = expected frequency in a particular cell if H 0 is true χ χ ( f for the r x c case has (r -1)(c-1)degrees of (Assumed: each cell in the contingency table has expected frequency of at least 1) o f e f e ) freedom Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap 1-3 Expected Cell Frequencies Expected cell frequencies: f e row total columntotal n Where: row total = sum of all frequencies in the row column total = sum of all frequencies in the column n = overall sample size Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap Prentice Hall, Inc.

13 Chapter Decision Rule The decision rule is If χ χ, reject H 0, α otherwise, do not reject H 0 χ α Where is from the chi-squared distribution with (r 1)(c 1) degrees of freedom Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap 1-5 Example The meal plan selected by 00 students is shown below: Class Standing Number of meals per week 0/week 10/week none Total Fresh Soph Junior Senior Total Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap Prentice Hall, Inc.

14 Chapter Example The hypothesis to be tested is: H 0 : Meal plan and class standing are independent (i.e., there is no relationship between them) H 1 : Meal plan and class standing are dependent (i.e., there is a relationship between them) Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap 1-7 Class Standing Observed: Number of meals per week 0/wk 10/wk none Total Fresh Soph Junior Senior Total Example: Expected Cell Frequencies Example for one cell: row total columntotal f e n Class Standing Expected cell frequencies if H 0 is true: Number of meals per week 0/wk 10/wk none Total Fresh Soph Junior Senior Total Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap Prentice Hall, Inc.

15 Chapter Example: The Test Statistic The test statistic value is: χ ( f f ) o e f all cells e ( ) ( ) ( ) χ 0.05 = 1.59 from the chi-squared distribution with (4 1)(3 1) = 6 degrees of freedom Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap 1-9 Example: Decision and Interpretation The test statisticis χ ; χ with 6 d.f Decision Rule: If χ > 1.59, reject H 0, otherwise, do not reject H 0 0 Do not reject H Reject H =1.59 Here, χ = < χ 0.05 = 1.59, so do not reject H 0 Conclusion: there is not sufficient evidence that meal plan and class standing are related at = 0.05 Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap Prentice Hall, Inc.

16 Chapter McNemar Test (Related Samples) Used to determine if there is a difference between proportions of two related samples Uses a test statistic the follows the normal distribution Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap 1-31 McNemar Test (Related Samples) Consider a X contingency table: Condition Condition 1 Yes No Totals Yes A B A+B No C D C+D Totals A+C B+D n Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap Prentice Hall, Inc.

17 Chapter McNemar Test (Related Samples) The sample proportions of interest are A B p1 proportionof respondents who answeryes to condition1 n p A C proportionof n respondents who answeryes to condition Test H 0 : π 1 = π (the two population proportions are equal) H 1 : π 1 π (the two population proportions are not equal) Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap 1-33 McNemar Test (Related Samples) The test statistic for the McNemar test: Z B C B C where the test statistic Z is approximately normally distributed Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap Prentice Hall, Inc.

18 Chapter McNemar Test Example Suppose you survey 300 homeowners and ask them if they are interested in refinancing their home. In an effort to generate business, a mortgage company improved their loan terms and reduced closing costs. The same homeowners were again surveyed. Determine if change in loan terms was effective in generating business for the mortgage company. The data are summarized as follows: Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap 1-35 McNemar Test Example Survey response before change Survey response after change Yes No Totals Yes No Totals Test the hypothesis (at the 0.05 level of significance): H 0 : π 1 π : The change in loan terms was ineffective H 1 : π 1 < π : The change in loan terms increased business Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap Prentice Hall, Inc.

19 Chapter McNemar Test Example Survey response before change Survey response after change Yes No Totals Yes No Totals The critical value (0.05 significance) is Z 0.05 = The test statistic is: Z B C B C 4.08 Since Z = < , you reject H 0 and conclude that the change in loan terms significantly increase business for the mortgage company. Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap 1-37 Chi-Square Test for a Variance or Standard Deviation A χ test statistic is used to test whether or not the population variance or standard deviation is equal to a specified value: χ (n - 1)S σ Where n = sample size S = sample variance σ = hypothesized population variance χ follows a chi-square distribution with n 1 d.f. Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap Prentice Hall, Inc.

20 Chapter Wilcoxon Rank-Sum Test for Differences in Medians Test two independent population medians Populations need not be normally distributed Distribution free procedure Used when only rank data are available Must use normal approximation if either of the sample sizes is larger than 10 Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap 1-39 Wilcoxon Rank-Sum Test: Small Samples Can use when both n 1, n 10 Assign ranks to the combined n 1 + n sample observations If unequal sample sizes, let n 1 refer to smaller-sized sample Smallest value rank = 1, largest value rank = n 1 + n Assign average rank for ties Sum the ranks for each sample: T 1 and T Obtain test statistic, T 1 (from smaller sample) Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap Prentice Hall, Inc.

21 Chapter Checking the Rankings The sum of the rankings must satisfy the formula below Can use this to verify the sums T 1 and T T1 T n(n 1) where n = n 1 + n Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap 1-41 Wilcoxon Rank-Sum Test: Hypothesis and Decision Rule M 1 = median of population 1; M = median of population Test statistic = T 1 (Sum of ranks from smaller sample) Two-Tail Test Left-Tail Test Right-Tail Test H 0 : M 1 = M H 0 : M 1 M H 0 : M 1 M H 1 : M 1 M H 1 : M 1 < M H 1 : M 1 M Reject Do Not Reject Reject Reject Do Not Reject Do Not Reject Reject T 1L T 1U T 1L T 1U Reject H 0 if T 1 T 1L Reject H 0 if T 1 T 1L Reject H 0 if T 1 T 1U or if T 1 T 1U Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap Prentice Hall, Inc.

22 Chapter 1 1- Wilcoxon Rank-Sum Test: Small Sample Example Sample data are collected on the capacity rates (% of capacity) for two factories. Are the median operating rates for two factories the same? For factory A, the rates are 71, 8, 77, 94, 88 For factory B, the rates are 85, 8, 9, 97 Test for equality of the population medians at the 0.05 significance level Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap 1-43 Ranked Capacity values: Tie in 3 rd and 4 th places Wilcoxon Rank-Sum Test: Small Sample Example Capacity Rank Factory A Factory B Factory A Factory B Rank Sums: Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap Prentice Hall, Inc.

23 Chapter Wilcoxon Rank-Sum Test: Small Sample Example Factory B has the smaller sample size, so the test statistic is the sum of the Factory B ranks: T 1 = 4.5 The sample sizes are: n 1 = 4 (factory B) n = 5 (factory A) The level of significance is =.05 Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap 1-45 Wilcoxon Rank-Sum Test: Small Sample Example Lower and Upper Critical Values for T 1 from Appendix table E.8: n One- Two- Tailed Tailed 4 5 n , 8 19, , 9 17, , 30 16, , -- 15, 40 6 T 1L = 11 and T 1U = 9 Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap Prentice Hall, Inc.

24 Chapter =.05 n 1 = 4, n = 5 Reject Two-Tail Test H 0 : M 1 = M H 1 : M 1 M Do Not Reject Reject T 1L =11 T 1U =9 Reject H 0 if T 1 T 1L =11 or if T 1 T 1U =9 Wilcoxon Rank-Sum Test: Small Sample Solution Test Statistic (Sum of ranks from smaller sample): T 1 = 4.5 Decision: Do not reject at = 0.05 Conclusion: There is not enough evidence to prove that the medians are not equal. Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap 1-47 Wilcoxon Rank-Sum Test (Large Sample) For large samples, the test statistic T 1 is approximately normal with mean μ T1 and standard deviation : σ T1 μ T 1 n1( n 1) σ T 1 n1 n (n 1) 1 Must use the normal approximation if either n 1 or n > 10 Assign n 1 to be the smaller of the two sample sizes Can use the normal approximation for small samples Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap Prentice Hall, Inc.

25 Chapter Wilcoxon Rank-Sum Test (Large Sample) The Z test statistic is Z T 1 μ σ T 1 T 1 T 1 n 1 n n 1 (n 1) (n 1) 1 Where Z approximately follows a standardized normal distribution Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap 1-49 Wilcoxon Rank-Sum Test: Normal Approximation Example Use the setting of the prior example: The sample sizes were: n 1 = 4 (factory B) n = 5 (factory A) The level of significance was α =.05 The test statistic was T 1 = 4.5 Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap Prentice Hall, Inc.

26 Chapter Wilcoxon Rank-Sum Test: Normal Approximation Example n (n 1) 4(9 1) 0 1 μt 1 σ T 1 n n ( n 1) 1 1 4( 5 )( 9 1) The test statistic is T μ 1 Z σ T 1 T Z = 1.10 is not greater than the critical Z value of 1.96 (for α =.05) so we do not reject H 0 there is not sufficient evidence that the medians are not equal Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap 1-51 Wilcoxon Signed Ranks Test A nonparametric test for two related populations Steps: 1. For each of n sample items, compute the difference, D i, between two measurements. Ignore + and signs and find the absolute values, D i 3. Omit zero differences, so sample size is n 4. Assign ranks R i from 1 to n (give average rank to ties) 5. Reassign + and signs to the ranks R i 6. Compute the Wilcoxon test statistic W as the sum of the positive ranks Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap Prentice Hall, Inc.

27 Chapter Wilcoxon Signed Ranks Test Statistic The Wilcoxon signed ranks test statistic is the sum of the positive ranks: W n' i1 ( ) R i For small samples (n < 0), use Table E.9 for the critical value of W Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap 1-53 Wilcoxon Signed Ranks Test Statistic For samples of n > 0, W is approximately normally distributed with μ W n'(n' 1) 4 σ W n'(n' 1)(n' 1) 4 Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap Prentice Hall, Inc.

28 Chapter Wilcoxon Signed Ranks Test The large sample Wilcoxon signed ranks Z test statistic is Z n'(n' 1) W 4 n'(n' 1)(n' 1) 4 To test for no median difference in the paired values: H 0 : M D = 0 H 1 : M D 0 Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap 1-55 Kruskal-Wallis Rank Test Tests the equality of more than population medians Use when the normality assumption for oneway ANOVA is violated Assumptions: The samples are random and independent Variables have a continuous distribution The data can be ranked Populations have the same variability Populations have the same shape Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap Prentice Hall, Inc.

29 Chapter Kruskal-Wallis Test Procedure Obtain rankings for each value In event of tie, each of the tied values gets the average rank Sum the rankings for data from each of the c groups Compute the H test statistic Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap 1-57 Kruskal-Wallis Test Procedure The Kruskal-Wallis H-test statistic: (with c 1 degrees of freedom) H 1 n(n 1) c j1 T n j j 3(n 1) where: n = sum of sample sizes in all groups c = Number of groups T j = Sum of ranks in the j th group n j = Number of values in the j th group (j = 1,,, c) Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap Prentice Hall, Inc.

30 Chapter Kruskal-Wallis Test Procedure Decision rule Complete the test by comparing the calculated H value to a critical value from the chi-square distribution with c 1 degrees of freedom Reject H 0 if test statistic H > α 0 Do not reject H 0 α Reject H 0 Otherwise do not reject H 0 Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap 1-59 Kruskal-Wallis Example Do different departments have different class sizes? Class size (Math, M) Class size (English, E) Class size (Biology, B) Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap Prentice Hall, Inc.

31 Chapter Kruskal-Wallis Example Do different departments have different class sizes? Class size (Math, M) Ranking Class size (English, E) Ranking Class size (Biology, B) Ranking = 44 = 56 = Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap 1-61 Kruskal-Wallis Example H0 : Median M Median E Median B H : Not all populationmedians are equal 1 The H statistic is 1 H n(n 1) T n (151) 5 c j1 j j 3(n 1) (151) 7.1 Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap Prentice Hall, Inc.

32 Chapter Kruskal-Wallis Example Compare H = 7.1 to the critical value from the chi-square distribution for 3 1 = degrees of freedom and = 0.05:. 05 χ Since H = 7.1 > χ , reject H 0 There is sufficient evidence to reject that the population medians are all equal Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap 1-63 Friedman Rank Test Use the Friedman rank test to determine whether c groups (i.e., treatment levels) have been selected from populations having equal medians H 0 : M.1 = M. =... = M.c H 1 : Not all M.j are equal (j = 1,,, c) Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap Prentice Hall, Inc.

33 Chapter Friedman Rank Test Friedman rank test for differences among c medians: F R 1 rc(c 1) c j1 R.j 3r(c 1) where R.j = the square of the total ranks for group j r = the number of blocks c = the number of groups Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap 1-65 Friedman Rank Test The Friedman rank test statistic is approximated by a chi-square distribution with c 1 d.f. Reject H 0 if F R χ α Otherwise do not reject H 0 Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap Prentice Hall, Inc.

34 Chapter Chapter Summary Developed and applied the test for the difference between two proportions Developed and applied the test for differences in more than two proportions Applied the Marascuilo procedure for comparing all pairs of proportions after rejecting a test Examined the test for independence Applied the McNemar test for proportions from two related samples Presented the test for a variance or a standard deviation Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap 1-67 Chapter Summary Used the Wilcoxon rank sum test for two population medians Presented the Wilcoxon signed ranks test for comparing paired samples Applied the Kruskal-Wallis H-test for multiple population medians Applied the Friedman rank test for comparing multiple population medians in a randomized block design Basic Business Statistics, 11e 009 Prentice-Hall, Inc.. Chap Prentice Hall, Inc.