# tests of goodness of fit and independencec

Statistics for Business and Economics (13e) Statistics for Business and Economics (13e) Anderson, Sweeney, Williams, Camm, Cochran 2017 Cengage Learning Slides by John Loucks St. Edwards University 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 11 Statistics for Business and Economics (13e) Chapter 12 Comparing Multiple Proportions, Test of Independence and Goodness of Fit Testing For Equality of Three or More Population Proportions Test of Independence Goodness of Fit Test 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 2

Statistics for Business and Economics (13e) Tests of Goodness of Fit, Independence, and Multiple Proportions In this chapter we introduce three additional hypothesis-testing procedures. The test statistic and the distribution used are based on the chi-square (c2) distribution. In all cases, the data are categorical. 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 3 Statistics for Business and Economics (13e) Testing the Equality of Population Proportions for Three or More Populations Using the notation p1 = population proportion for population 1 p2 = population proportion for population 2 pk = population proportion for population k The hypotheses for the equality of population proportions for k > 3 populations are as follows: H0: p1 = p2 = . . . = pk

Ha: Not all population proportions are equal 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 4 Statistics for Business and Economics (13e) Testing the Equality of Population Proportions for Three or More Populations If H0 cannot be rejected, we cannot detect a difference among the k population proportions. If H0 can be rejected, we can conclude that not all k population proportions are equal. Further analyses can be done to conclude which population proportions are significantly different from others. 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 5 Statistics for Business and Economics (13e) Testing the Equality of Population Proportions

for Three or More Populations Example: Finger Lakes Homes Finger Lakes Homes manufactures three models of prefabricated homes, a two-story colonial, a log cabin, and an A-frame. To help in product-line planning, management would like to compare the customer satisfaction with the three home styles. p1 = proportion likely to repurchase a Colonial for the population of Colonial owners p2 = proportion likely to repurchase a Log Cabin for the population of Log Cabin owners p3 = proportion likely to repurchase an A-Frame for the population of A-Frame owners 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 6 Statistics for Business and Economics (13e) Testing the Equality of Population Proportions for Three or More Populations We begin by taking a sample of owners from each of the three populations. Each sample contains categorical data indicating whether the respondents are likely or not likely to repurchase the home. 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or

otherwise on a password-protected website or school-approved learning management system for classroom use. 7 Statistics for Business and Economics (13e) Testing the Equality of Population Proportions for Three or More Populations Observed Frequencies (sample results) Home Owner Colonial Log A-Frame Total Likely to Yes 97 83 80 260 Repurchase No 38 18 44 100 Total 135 101 124

360 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 8 Statistics for Business and Economics (13e) Testing the Equality of Population Proportions for Three or More Populations Next, we determine the expected frequencies under the assumption H0 is correct. Expected Frequencies Under the Assumption H0 is True ( Row Total )(Column Total ) = Total Sample If a significant difference exists between the observed and expected frequencies, H0 can be rejected. 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 9

Statistics for Business and Economics (13e) Testing the Equality of Population Proportions for Three or More Populations Expected Frequencies (computed) Home Owner Colonial Log A-Frame Total Likely to Yes 97.50 72.94 89.56 260 Repurchase No 37.50 28.06 34.44 100 Total 135 101 124 360 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 10 Statistics for Business and Economics (13e) Testing the Equality of Population Proportions

for Three or More Populations Next, compute the value of the chi-square test statistic. 2 2 = where: ( ) fij = observed frequency for the cell in row i and column j eij = expected frequency for the cell in row i and column j under the assumption H0 is true Note: The test statistic has a chi-square distribution with k 1 degrees of freedom, provided the expected frequency is 5 or more for each cell. 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.

11 Statistics for Business and Economics (13e) Testing the Equality of Population Proportions for Three or More Populations Computation of the Chi-Square Test Statistic. Obs. Exp. Sqd. Sqd. Diff. / Likely to Home Freq. Freq. Diff. Diff.

Exp. Freq. Repurchase Owner fij eij (fij - eij) (fij - eij)2 (fij - eij)2/eij Yes Colonial 97 97.50 -0.50

0.2500 0.0026 Yes Log Cab. 83 72.94 10.06 101.1142 1.3862 Yes A-Frame 80 89.56

-9.56 91.3086 1.0196 No Colonial 38 37.50 0.50 0.2500 0.0067 No Log Cab. 18

28.06 -10.06 101.1142 3.6041 No A-Frame 44 34.44 9.56 91.3086 2.6509 Total 360

360 c2 = 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 8.6700 12 Statistics for Business and Economics (13e) Testing the Equality of Population Proportions for Three or More Populations Rejection Rule p-value approach: Critical value approach: Reject H0 if p-value < Reject H0 if > where is the significance level and there are k - 1 degrees of freedom 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.

13 Statistics for Business and Economics (13e) Testing the Equality of Population Proportions for Three or More Populations Rejection Rule (using = .05) Reject H0 if p-value < .05 or c2 > 5.991 With = .05 and k-1=3-1=2 degrees of freedom Do Not Reject H0 Reject H0 c2 5.991 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 14 Statistics for Business and Economics (13e)

Testing the Equality of Population Proportions for Three or More Populations Conclusion Using the p-Value Approach Area in Upper Tail c2 Value (df = 2) .10 .05 .025 .01 .005 4.605 5.991 7.378 9.210 10.597 Because c2 = 8.670 is between 9.210 and 7.378, the area in the upper tail of the distribution is between .01 and .025. The p-value < . We can reject the null hypothesis. (Actual p-value is .0131) 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.

15 Statistics for Business and Economics (13e) Testing the Equality of Population Proportions for Three or More Populations We have concluded that the population proportions for the three populations of home owners are not equal. To identify where the differences between population proportions exist, we will rely on a multiple comparisons procedure. 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 16 Statistics for Business and Economics (13e) Multiple Comparisons Procedure We begin by computing the three sample proportions. Colonial: Log Cabin: A-Frame: We will use a multiple comparison procedure known as the Marascuilo procedure.

2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 17 Statistics for Business and Economics (13e) Multiple Comparisons Procedure Marascuilo Procedure We compute the absolute value of the pairwise difference between sample proportions. Colonial and Log Cabin: = = .1033 Colonial and A-Frame: = = .0733 Log Cabin and A-Frame: = = .1766 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.

18 Statistics for Business and Economics (13e) Multiple Comparisons Procedure Critical Values for the Marascuilo Pairwise Comparison For each pairwise comparison compute a critical value as follows: = 2 , 1 (1 ) (1 ) + For = .05 and k = 3: c2 = 5.991 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.

19 Statistics for Business and Economics (13e) Multiple Comparisons Procedure Pairwise Comparison Tests Significant if > CVij Pairwise Comparison | | CVij Colonial vs. Log Cabin .1033 .1329 Not Significant Colonial vs. A-Frame .0733

.1415 Not Significant Log Cabin vs. A-Frame .1766 .1405 Significant | | 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 20 Statistics for Business and Economics (13e) Goodness of Fit Test: Multinomial Probability Distribution

1. State the null and alternative hypotheses. H0: The population follows a multinomial distribution with specified probabilities for each of the k categories Ha: The population does not follow a multinomial distribution with specified probabilities for each of the k categories 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 21 Statistics for Business and Economics (13e) Goodness of Fit Test: Multinomial Probability Distribution 2. Select a random sample and record the observed frequency, fi , for each of the k categories. 3. Assuming H0 is true, compute the expected frequency, ei , in each category by multiplying the category probability by the sample size. 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 22 Statistics for Business and Economics (13e)

Goodness of Fit Test: Multinomial Probability Distribution 4. Compute the value of the test statistic. 2 = =1 where: ( ) 2 fi = observed frequency for category i ei = expected frequency for category i k = number of categories Note: The test statistic has a chi-square distribution with k 1 df provided that the expected frequencies are 5 or more for all categories. 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or

otherwise on a password-protected website or school-approved learning management system for classroom use. Statistics for Business and Economics (13e) Goodness of Fit Test: Multinomial Probability Distribution 5. Rejection rule: p-value approach: Critical value approach: Reject H0 if p-value < a Reject H0 if > where is the significance level and there are k - 1 degrees of freedom 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 24 Statistics for Business and Economics (13e) Multinomial Distribution Goodness of Fit Test Example: Finger Lakes Homes (A)

Finger Lakes Homes manufactures four models of prefabricated homes, a two-story colonial, a log cabin, a split-level, and an A-frame. To help in production planning, management would like to determine if previous customer purchases indicate that there is a preference in the style selected. 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 25 Statistics for Business and Economics (13e) Multinomial Distribution Goodness of Fit Test Example: Finger Lakes Homes (A) The number of homes sold of each model for 100 sales over the past two years is shown below. Split- AModel Colonial Log Level Frame # Sold 30 20 35 15 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.

26 Statistics for Business and Economics (13e) Multinomial Distribution Goodness of Fit Test Hypotheses H0: pC = pL = pS = pA = .25 Ha: The population proportions are not pC = .25, pL = .25, pS = .25, and pA = .25 where: pC = population proportion that purchase a colonial pL = population proportion that purchase a log cabin pS = population proportion that purchase a split-level pA = population proportion that purchase an A-frame 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 27 Statistics for Business and Economics (13e) Multinomial Distribution Goodness of Fit Test Rejection Rule Reject H0 if p-value < .05 or c2 > 7.815. With = .05 and

k-1=4-1=3 degrees of freedom Do Not Reject H0 Reject H0 c2 7.815 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 28 Statistics for Business and Economics (13e) Multinomial Distribution Goodness of Fit Test Expected Frequencies e1 = .25(100) = 25 e2 = .25(100) = 25 e3 = .25(100) = 25 e4 = .25(100) = 25

Test Statistic + =1+1+4+4 = 10 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 29 Statistics for Business and Economics (13e) Multinomial Distribution Goodness of Fit Test Conclusion Using the p-Value Approach Area in Upper Tail .10 .05 .025 .01 .005 c2 Value (df = 3) 6.251 7.815 9.348 11.345 12.838 Because c2 = 10 is between 9.348 and 11.345, the area in the upper tail of the distribution is between .025 and .01. The p-value < . We can reject the null hypothesis. 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or

otherwise on a password-protected website or school-approved learning management system for classroom use. 30 Statistics for Business and Economics (13e) Multinomial Distribution Goodness of Fit Test Conclusion Using the Critical Value Approach c2 = 10 > 7.815 We reject, at the .05 level of significance, the assumption that there is no home style preference. 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 31 Statistics for Business and Economics (13e) Test of Independence 1. Set up the null and alternative hypotheses. H0: The column variable is independent of the row variable Ha: The column variable is not independent of the row variable 2. Select a random sample and record the observed frequency, fij , for each cell of the contingency table. 3. Compute the expected frequency, eij , for each cell.

( Row Total )(Column Total ) = Sample 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 32 Statistics for Business and Economics (13e) Test of Independence 4. Compute the test statistic. 2 2 = ( )

5. Determine the rejection rule. Reject H0 if p -value < a or > . where is the significance level and, with n rows and m columns, there are (n - 1)(m - 1) degrees of freedom. 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 33 Statistics for Business and Economics (13e) Test of Independence Example: Finger Lakes Homes (B) Each home sold by Finger Lakes Homes can be classified according to price and to style. Finger Lakes manager would like to determine if the price of the home and the style of the home are independent variables. 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 34

Statistics for Business and Economics (13e) Test of Independence Example: Finger Lakes Homes (B) The number of homes sold for each model and price for the past two years is shown below. For convenience, the price of the home is listed as either less than \$200,000 or more than or equal to \$200,000. Price Colonial < \$200,000 18 > \$200,000 12 Log 6 14 Split-Level 19 16 A-Frame 12 3

2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 35 Statistics for Business and Economics (13e) Test of Independence Hypotheses H0: Price of the home is independent of the style of the home that is purchased Ha: Price of the home is not independent of the style of the home that is purchased 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 36 Statistics for Business and Economics (13e) Test of Independence Expected Frequencies Price Colonial

Log Split-Level A-Frame Total < \$200K 18 6 19 12 55 > \$200K 12 14

16 3 45 Total 30 20 35 15 100 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 37 Statistics for Business and Economics (13e)

Test of Independence Rejection Rule With = .05 and (2 - 1)(4 - 1) = 3 d.f., Reject H0 if p-value < .05 or c2 > 7.815 Test Statistic 2 2 2 (18 16.5) (6 11) (3 6.75) 2= + ++ 16.5 11 6.75 = .1364 + 2.2727 + . . . + 2.0833 = 9.149

2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 38 Statistics for Business and Economics (13e) Test of Independence Conclusion Using the p-Value Approach Area in Upper Tail .10 .05 .025 .01 .005 c2 Value (df = 3) 6.251 7.815 9.348 11.345 12.838 Because c2 = 9.145 is between 7.815 and 9.348, the area in the upper tail of the distribution is between .05 and .025. The p-value < . We can reject the null hypothesis. (Actual p-value is .0274) 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 39 Statistics for Business and Economics (13e)

Test of Independence Conclusion Using the Critical Value Approach c2 = 9.145 > 7.815 We reject at the .05 level of significance, the assumption that the price of the home is independent of the style of home that is purchased. 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 40 Statistics for Business and Economics (13e) Testing the Equality of Population Proportions for Three or More Populations Using the notation p1 = population proportion for population 1 p2 = population proportion for population 2 pk = population proportion for population k The hypotheses for the equality of population proportions for k > 3 populations are as follows: H0: p1 = p2 = . . . = pk Ha: Not all population proportions are equal 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or

otherwise on a password-protected website or school-approved learning management system for classroom use. 41 Statistics for Business and Economics (13e) Testing the Equality of Population Proportions for Three or More Populations If H0 cannot be rejected, we cannot detect a difference among the k population proportions. If H0 can be rejected, we can conclude that not all k population proportions are equal. Further analyses can be done to conclude which population proportions are significantly different from others. 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 42 Statistics for Business and Economics (13e) Testing the Equality of Population Proportions for Three or More Populations Example: Finger Lakes Homes Finger Lakes Homes manufactures three models of prefabricated homes, a

two-story colonial, a log cabin, and an A-frame. To help in product-line planning, management would like to compare the customer satisfaction with the three home styles. p1 = proportion likely to repurchase a Colonial for the population of Colonial owners p2 = proportion likely to repurchase a Log Cabin for the population of Log Cabin owners p3 = proportion likely to repurchase an A-Frame for the population of A-Frame owners 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 43 Statistics for Business and Economics (13e) Testing the Equality of Population Proportions for Three or More Populations We begin by taking a sample of owners from each of the three populations. Each sample contains categorical data indicating whether the respondents are likely or not likely to repurchase the home. 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 44

Statistics for Business and Economics (13e) Testing the Equality of Population Proportions for Three or More Populations Observed Frequencies (sample results) Home Owner Colonial Log A-Frame Total Likely to Yes 97 83 80 260 Repurchase No 38 18 44 100 Total 135 101 124 360 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or

otherwise on a password-protected website or school-approved learning management system for classroom use. 45 Statistics for Business and Economics (13e) Testing the Equality of Population Proportions for Three or More Populations Next, we determine the expected frequencies under the assumption H0 is correct. Expected Frequencies Under the Assumption H0 is True ( Row Total )(Column Total ) = Total Sample If a significant difference exists between the observed and expected frequencies, H0 can be rejected. 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 46 Statistics for Business and Economics (13e)

Testing the Equality of Population Proportions for Three or More Populations Expected Frequencies (computed) Home Owner Colonial Log A-Frame Total Likely to Yes 97.50 72.94 89.56 260 Repurchase No 37.50 28.06 34.44 100 Total 135 101 124 360 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 47 Statistics for Business and Economics (13e) Testing the Equality of Population Proportions for Three or More Populations Next, compute the value of the chi-square test statistic. 2

2 = where: ( ) fij = observed frequency for the cell in row i and column j eij = expected frequency for the cell in row i and column j under the assumption H0 is true Note: The test statistic has a chi-square distribution with k 1 degrees of freedom, provided the expected frequency is 5 or more for each cell. 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 48 Statistics for Business and Economics (13e)

Testing the Equality of Population Proportions for Three or More Populations Computation of the Chi-Square Test Statistic. Obs. Exp. Sqd. Sqd. Diff. / Likely to Home Freq. Freq. Diff. Diff. Exp. Freq.

Repurchase Owner fij eij (fij - eij) (fij - eij)2 (fij - eij)2/eij Yes Colonial 97 97.50 -0.50 0.2500

0.0026 Yes Log Cab. 83 72.94 10.06 101.1142 1.3862 Yes A-Frame 80 89.56 -9.56

91.3086 1.0196 No Colonial 38 37.50 0.50 0.2500 0.0067 No Log Cab. 18 28.06

-10.06 101.1142 3.6041 No A-Frame 44 34.44 9.56 91.3086 2.6509 Total 360 360

c2 = 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 8.6700 49 Statistics for Business and Economics (13e) Testing the Equality of Population Proportions for Three or More Populations Rejection Rule p-value approach: Critical value approach: Reject H0 if p-value < Reject H0 if > where is the significance level and there are k - 1 degrees of freedom 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 50

Statistics for Business and Economics (13e) Testing the Equality of Population Proportions for Three or More Populations Rejection Rule (using = .05) Reject H0 if p-value < .05 or c2 > 5.991 With = .05 and k-1=3-1=2 degrees of freedom Do Not Reject H0 Reject H0 c2 5.991 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 51 Statistics for Business and Economics (13e) Testing the Equality of Population Proportions for Three or More Populations Conclusion Using the p-Value Approach

Area in Upper Tail c2 Value (df = 2) .10 .05 .025 .01 .005 4.605 5.991 7.378 9.210 10.597 Because c2 = 8.670 is between 9.210 and 7.378, the area in the upper tail of the distribution is between .01 and .025. The p-value < . We can reject the null hypothesis. (Actual p-value is .0131) 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 52

Statistics for Business and Economics (13e) Testing the Equality of Population Proportions for Three or More Populations We have concluded that the population proportions for the three populations of home owners are not equal. To identify where the differences between population proportions exist, we will rely on a multiple comparisons procedure. 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 53 Statistics for Business and Economics (13e) Multiple Comparisons Procedure We begin by computing the three sample proportions. Colonial: Log Cabin: A-Frame: We will use a multiple comparison procedure known as the Marascuillo procedure. 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.

54 Statistics for Business and Economics (13e) Multiple Comparisons Procedure Marascuillo Procedure We compute the absolute value of the pairwise difference between sample proportions. Colonial and Log Cabin: = = .1033 Colonial and A-Frame: = = .0733 Log Cabin and A-Frame: = = .1766 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 55

Statistics for Business and Economics (13e) Multiple Comparisons Procedure Critical Values for the Marascuillo Pairwise Comparison For each pairwise comparison compute a critical value as follows: = 2 , 1 (1 ) (1 ) + For = .05 and k = 3: c2 = 5.991 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 56 Statistics for Business and Economics (13e)

Multiple Comparisons Procedure Pairwise Comparison Tests Significant if > CVij Pairwise Comparison | | CVij Colonial vs. Log Cabin .1033 .1329 Not Significant Colonial vs. A-Frame .0733 .1415

Not Significant Log Cabin vs. A-Frame .1766 .1405 Significant | | 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 57 Statistics for Business and Economics (13e) End of Chapter 12 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.

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