Chap 11-1 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Chapter 11 Chi-Square Tests Business Statistics: A First Course 6 th Edition

Chap 11-2 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Learning Objectives In this chapter, you learn:  How and when to use the chi-square test for contingency tables The  2 test for the difference between two proportions The  2 test for differences in more than two proportions The  2 test for independence

Chap 11-3 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Contingency Tables Useful in situations comparing multiple population proportions Used to classify sample observations according to two or more characteristics Also called a cross-classification table. DCOVA

Chap 11-4 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Contingency Table Example Left-Handed vs. Gender Dominant Hand: Left vs. Right Gender: Male vs. Female  2 categories for each variable, so this is called a 2 x 2 table  Suppose we examine a sample of 300 children DCOVA

Chap 11-5 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Contingency Table Example Sample results organized in a contingency table: (continued) Gender Hand Preference LeftRight Female Male Females, 12 were left handed 180 Males, 24 were left handed sample size = n = 300: DCOVA

Chap 11-6 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall  2 Test for the Difference Between Two Proportions 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 H 0 : π 1 = π 2 (Proportion of females who are left handed is equal to the proportion of males who are left handed) H 1 : π 1 ≠ π 2 (The two proportions are not the same – hand preference is not independent of gender) DCOVA

Chap 11-7 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall The Chi-Square Test Statistic where: f o = observed frequency in a particular cell f e = expected frequency in a particular cell if H 0 is true (Assumed: each cell in the contingency table has expected frequency of at least 5) The Chi-square test statistic is: DCOVA

Chap 11-8 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Decision Rule  2α2α Decision Rule: If, reject H 0, otherwise, do not reject H 0 The test statistic approximately follows a chi- squared distribution with one degree of freedom 0  Reject H 0 Do not reject H 0 DCOVA

Chap 11-9 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Computing the Average Proportion Here: 120 Females, 12 were left handed 180 Males, 24 were left handed i.e., based on all 300 children the proportion of left handers is 0.12, that is, 12% The average proportion is: DCOVA

Chap Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 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) =.12 i.e., we would expect (.12)(120) = 14.4 females to be left handed (.12)(180) = 21.6 males to be left handed DCOVA

Chap Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Observed vs. Expected Frequencies Gender Hand Preference LeftRight Female Observed = 12 Expected = 14.4 Observed = 108 Expected = Male Observed = 24 Expected = 21.6 Observed = 156 Expected = DCOVA

Chap Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Gender Hand Preference LeftRight Female Observed = 12 Expected = 14.4 Observed = 108 Expected = Male Observed = 24 Expected = 21.6 Observed = 156 Expected = The Chi-Square Test Statistic The test statistic is: DCOVA

Chap Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Decision Rule Decision Rule: If > 3.841, reject H 0, otherwise, do not reject H 0 Here, = < = 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   = Reject H 0 Do not reject H 0 DCOVA

Chap Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Chi-Square Test In Excel Chi-Square Test Observed Frequencies Hand Preference Calculations GenderLeftRightTotalf 0 - f e Female Male Total Expected Frequencies Hand Preference GenderLeftRightTotal(f 0 - f e )^2/f e Female Male Total Data Level of Significance0.05 Number of Rows2 Number of Columnns2 Degrees of Freedom1=(B19-1)*(B20-1) Results Critical Value3.8415=CHIINV(B18,B21) Chi-Square Test Statistic0.7576=SUM(F13:G14) p-Value0.3841=CHIDIST(B25,B21) Do not reject the null hypothesis=IF(B26<B18,"Reject the null hypothesis", "Do not reject the null hypothesis") Expected Frequency assumption is met.=IF(OR(B13<5,C13<5,B14<5,C14<5), " is violated."," is met.") DCOVA

Chap Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Chi-Square Test In Minitab Tabulated statistics: Gender, Hand (Expected counts are below observed) (Chi-Square cell contribution below expected counts) Rows: Gender Columns: Hand Left Right All Female * Male * All * * * Pearson Chi-Square = 0.758, DF = 1, P-Value = DCOVA

Chap Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Extend the  2 test to the case with more than two independent populations:  2 Test for Differences Among More Than Two Proportions H 0 : π 1 = π 2 = … = π c H 1 : Not all of the π j are equal (j = 1, 2, …, c) DCOVA

Chap Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall The Chi-Square Test Statistic Where: f o = observed frequency in a particular cell of the 2 x c table f e = expected frequency in a particular cell if H 0 is true (Assumed: each cell in the contingency table has expected frequency of at least 1) The Chi-square test statistic is: DCOVA

Chap Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Computing the Overall Proportion The overall proportion is: Expected cell frequencies for the c categories are calculated as in the 2 x 2 case, and the decision rule is the same: Where is from the chi- squared distribution with c – 1 degrees of freedom Decision Rule: If, reject H 0, otherwise, do not reject H 0 DCOVA

Chap Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Example of  2 Test for Differences Among More Than Two Proportions 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? DCOVA

Chap Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Chi-Square Test Results Chi-Square Test: Administrators, Students, Faculty Admin Students FacultyTotal Favor Oppose Total Observed Expected H 0 : π 1 = π 2 = π 3 H 1 : Not all of the π j are equal (j = 1, 2, 3) DCOVA

Chap Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Excel Output For The Example DCOVA Chi-Square Test Observed Frequencies Role Calculations OpinionAdminStudentsFacultyTotalfo - fe Favor Oppose Total Expected Frequencies Role OpinionAdminStudentsFacultyTotal(fo - fe)^2/fe Favor Oppose Total Data Level of Significance0.01 Number of Rows2 Number of Columns3 Degrees of Freedom2=($B$19 - 1) * ($B$20 - 1) Results Critical Value9.2103=CHIINV(B18, B21) Chi-Square Test Statistic =SUM(G13:I14) p-Value0.0017=CHIDIST(B25, B21) Reject the null hypothesis=IF(B26<B18, "Reject the null hypothesis", Do not reject the null hypothesis") Expected frequency assumption is met.=IF(OR(B13<1, C13<1, D13<1, B14<1, C14<1, D14<1), " is violated.", " is met.")

Chap Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Minitab Output For The Example DCOVA Tabulated statistics: Position, Role Rows: Position Columns: Role Admin Faculty Students All Favor * Oppose * All * * * * Pearson Chi-Square = , DF = 2, P-Value = 0.002

Chap Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall  2 Test of Independence Similar to the  2 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) DCOVA

Chap Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall  2 Test of Independence 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 (Assumed: each cell in the contingency table has expected frequency of at least 1) The Chi-square test statistic is: (continued) DCOVA

Chap Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Expected Cell Frequencies Expected cell frequencies: Where: row total = sum of all frequencies in the row column total = sum of all frequencies in the column n = overall sample size DCOVA

Chap Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Decision Rule The decision rule is Where is from the chi-squared distribution with (r – 1)(c – 1) degrees of freedom If, reject H 0, otherwise, do not reject H 0 DCOVA

Chap Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Example The meal plan selected by 200 students is shown below: Class Standing Number of meals per week Total 20/week10/weeknone Fresh Soph Junior Senior Total DCOVA

Chap Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Example The hypothesis to be tested is: (continued) 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) DCOVA

Chap Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Class Standing Number of meals per week Total 20/wk10/wknone Fresh Soph Junior Senior Total Class Standing Number of meals per week Total 20/wk10/wknone Fresh Soph Junior Senior Total Observed: Expected cell frequencies if H 0 is true: Example for one cell: Example: Expected Cell Frequencies (continued) DCOVA

Chap Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Example: The Test Statistic The test statistic value is: (continued) = from the chi-squared distribution with (4 – 1)(3 – 1) = 6 degrees of freedom DCOVA

Chap Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Example: Decision and Interpretation (continued) Decision Rule: If > , reject H 0, otherwise, do not reject H 0 Here, = < = , so do not reject H 0 Conclusion: there is not sufficient evidence that meal plan and class standing are related at  = 0.05   = Reject H 0 Do not reject H 0 DCOVA

Chap Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Excel Output Of Example DCOVA Chi-Square Test of Independence Observed Frequencies Meals Per Week Calculations Class Standing20/wk10/wkNoneTotalfo - fe Fresh Soph Junior Senior Total Expected Frequencies Meals Per Week Class Standing20/wk10/wkNoneTotal(fo - fe)^2/fe Fresh Soph Junior Senior Total Data Level of Significance0.05 Number of Rows4 Number of Columns3 Degrees of Freedom6=($B$23 - 1) * ($B$24 - 1) Results Critical Value =CHIINV(B22, B25) Chi-Square Test Statistic0.7093=SUM($G$15:$I$18) p-Value0.9943=CHIDIST(B29, B25) Do not reject the null hypothesis=IF(B30<B22, "Reject the null hypothesis", Do not reject the null hypothesis) Expected frequency assumption is met.=IF(OR(B15<1, C15<1, D15<1, B16<1, C16<1, D16<1, B17<1, C17<1, D17<1, B18<1, C18<1, D18<1), " is violated.", " is met.")

Chap Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Minitab Output Of Example DCOVA Tabulated statistics: Class, Meals Rows: Class Columns: Meals All Fresh * Junior * Senior * Soph * All * * * * Pearson Chi-Square = 0.709, DF = 6, P-Value = 0.994

Chap Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Chapter Summary Developed and applied the  2 test for the difference between two proportions Developed and applied the  2 test for differences in more than two proportions Examined the  2 test for independence

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