Exam 12: Chi-Square Tests and Nonparametric Tests

TABLE 12-19 An agronomist wants to compare the crop yield of 3 varieties of chickpea seeds. She plants all 3 varieties of the seeds on each of 5 different patches of fields. She then measures the crop yield in bushels per acre. Treating this as a randomized block design, the results are presented in the table that follows. Fields Smith Walsh Trevor 1 11.1 19.0 14.6 2 13.5 18.0 15.7 3 15.3 19.8 16.8 4 14.6 19.6 16.7 5 9.8 16.6 15.2 Below is the Minitab output of the Friedman rank test: Friedman Test: Yield versus Varieties, Fields Friedman test for Yield byVarieties blocked by Fields $\mathrm{s}=$$10.00$ $\quad$$\mathrm{DF}=2 \quad \mathrm{P}=0.007$ Est Sum of Varieties N Median Ranks Smith 5 13.500 5.0 Trevor 5 15.667 10.0 Walsh 5 18.533 15.0 Grand median= 15.900 -Referring to Table 12-19, the Friedman rank test is valid only if the 5 blocks are independent so that the yields in one block have no influence on the yields in any other block.

A researcher is curious about the effect of sleep on students' test performances. He chooses 60 students and gives each 2 tests: one given after 2 hours' sleep and one after 8 hours' sleep. The test the researcher should use would be a related samples test.

TABLE 12-17 The director of the MBA program of a state university wanted to know if a one week orientation would change the proportion among potential incoming students who would perceive the program as being good. Given below is the result from 215 students' view of the program before and after the orientation. Before the Orientation Good NotGood Total Good 93 37 130 Not Good 71 14 85 Total 164 51 215 -Referring to Table 12-17, what is the value of the test statistic using a 1% level of significance?

TABLE 12-9 Four surgical procedures currently are used to install pacemakers. If the patient does not need to return for follow-up surgery the operation is called a "clear" operation. A heart center wants to compare the proportion of clear operations for the 4 procedures, and collects the following numbers of patients from their own records: Procedure A B C D Total Clear 27 41 21 7 96 Return 11 15 9 11 46 Total 38 56 30 18 142 They will use this information to test for a difference among the proportion of clear operations using a chi-square test with a level of significance of 0.05. -Referring to Table 12-9, the expected cell frequency for the Procedure A/Clear cell is______

TABLE 12-19 An agronomist wants to compare the crop yield of 3 varieties of chickpea seeds. She plants all 3 varieties of the seeds on each of 5 different patches of fields. She then measures the crop yield in bushels per acre. Treating this as a randomized block design, the results are presented in the table that follows. 1 11.1 19.0 14.6 2 13.5 18.0 15.7 3 15.3 19.8 16.8 4 14.6 19.6 16.7 5 9.8 16.6 15.2 Below is the Minitab output of the Friedman rank test: Friedman Test: Yield versus Varieties, Fields $\text { Friedman test for Yield by Varieties blocked by Fields }$ $\mathrm{S}=10.00 \quad \mathrm{DF}=2 \quad \mathrm{P}=0.007$ Est Sum of Varieties Median Ranks Smith 5 13.500 5.0 Trevor 5 15.667 10.0 Walsh 5 18.533 15.0 Grand median = 15.900 -Referring to Table 12-19, what is the null hypothesis for the Friedman rank test?

When the normality assumption is not met in a randomized block design, which of the following tests should be used?

TABLE 12-8 The director of transportation of a large company is interested in the usage of the company's van pool program. She surveyed 129 of her employees on the usage of the program before and after a campaign to convince her employees to use the service and obtained the following: $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$$\text { Before }$ Use Do Not Use Total After Use 27 44 71 Do Not Use 33 25 58 Total 60 69 129 She will use this information to perform test using a level of significance of 0.05. -Referring to Table 12-8, the director now wants to know if the proportion of employees who use the service before the campaign and the proportion of employees who use the service after the campaign are the same. She should reject the null hypothesis using a 5% level of significance.

TABLE 12-4 One criterion used to evaluate employees in the assembly section of a large factory is the number of defective pieces per 1,000 parts produced. The quality control department wants to find out whether there is a relationship between years of experience and defect rate. Since the job is repetitious, after the initial training period any improvement due to a learning effect might be offset by a loss of motivation. A defect rate is calculated for each worker in a yearly evaluation. The results for 100 workers are given in the table below. Years Since Training Period <1 Year 1-4 Years 5 - 9Years High 6 9 9 Defect Rate: Average 9 19 23 Low 7 8 10 -Referring to Table 12-4, of the cell for 1 to 4 years of training time and a high defect rate, what is the contribution to the overall ?² statistic for the independence test?

TABLE 12-16 A perfume manufacturer is trying to choose between 2 magazine advertising layouts. An expensive layout would include a small package of the perfume. A cheaper layout would include a "scratch-and-sniff" sample of the product. The manufacturer would use the more expensive layout only if there is evidence that it would lead to a higher approval rate. The manufacturer presents both layouts to 5 groups and determines the approval rating from each group on both layouts. The data are given below. Use this to test whether the median difference in approval rating is different from zero in favor of the more expensive layout with a level of significance of 0.05. Package Scratch 52 37 68 43 43 53 48 39 56 47 -Referring to Table 12-16, what is the rank of the absolute difference for the last pair of observations?

TABLE 12-9 Four surgical procedures currently are used to install pacemakers. If the patient does not need to return for follow-up surgery the operation is called a "clear" operation. A heart center wants to compare the proportion of clear operations for the 4 procedures, and collects the following numbers of patients from their own records: Procedure A B C D Total Clear 27 41 21 7 96 Return 11 15 9 11 46 Total 38 56 30 18 142 They will use this information to test for a difference among the proportion of clear operations using a chi-square test with a level of significance of 0.05. -Referring to Table 12-9, the overall or average proportion of clear operations is_____ .

TABLE 12-10 The director of admissions at a state college is interested in seeing if admissions status (admitted, waiting list, denied admission) at his college is independent of the type of community in which an applicant resides. He takes a sample of recent admissions decisions and forms the following table: Admitted Wait List Denied Total Urban 45 21 17 83 Rural 33 13 24 70 Suburban 34 12 39 85 Total 112 46 80 238 He will use this table to do a chi-square test of independence with a level of significance of 0.01. -Referring to Table 12-10, the test will involve______ degrees of freedom.

The chi-square test of independence requires that the number of expected frequency in each cell to be at least 5.

TABLE 12-17 The director of the MBA program of a state university wanted to know if a one week orientation would change the proportion among potential incoming students who would perceive the program as being good. Given below is the result from 215 students' view of the program before and after the orientation. After the Orientation Before the Orientation Good NotGood Total Good 93 37 130 Not Good 71 14 85 Total 164 51 215 -Referring to Table 12-17, which test should she use?

If we use the ?² method of analysis to test for the differences among 4 proportions, the degrees of freedom are equal to

TABLE 12-7 The director of transportation of a large company is interested in the usage of her van pool. She considers her routes to be divided into local and non-local. She is particularly interested in learning if there is a difference in the proportion of males and females who use the local routes. She takes a sample of a day's riders and finds the following: Male Female Total Local 27 44 71 Non- Local 33 25 58 Total 60 69 129 She will use this information to perform a chi-square hypothesis test using a level of significance of 0.05. -Referring to Table 12-7, the expected cell frequency in the Female/Non-Local cell is______

TABLE 12-7 The director of transportation of a large company is interested in the usage of her van pool. She considers her routes to be divided into local and non-local. She is particularly interested in learning if there is a difference in the proportion of males and females who use the local routes. She takes a sample of a day's riders and finds the following: Male Female Total Local 27 44 71 Non- Local 33 25 58 Total 60 69 129 She will use this information to perform a chi-square hypothesis test using a level of significance of 0.05. -Referring to Table 12-7, the test will involve _____degree(s) of freedom.

TABLE 12-19 An agronomist wants to compare the crop yield of 3 varieties of chickpea seeds. She plants all 3 varieties of the seeds on each of 5 different patches of fields. She then measures the crop yield in bushels per acre. Treating this as a randomized block design, the results are presented in the table that follows. Fields Smith Walsh Trevor 1 11.1 19.0 14.6 2 13.5 18.0 15.7 3 15.3 19.8 16.8 4 14.6 19.6 16.7 5 9.8 16.6 15.2 Below is the Minitab output of the Friedman rank test: Friedman Test: Yield versus Varieties, Fields Friedman test for Yield byVarieties blocked by Fields $\mathrm{s}=$$10.00$ $\quad$$\mathrm{DF}=2 \quad \mathrm{P}=0.007$ Est Sum of Varieties N Median Ranks Smith 5 13.500 5.0 Trevor 5 15.667 10.0 Walsh 5 18.533 15.0 Grand median= 15.900 -Referring to Table 12-19, the null hypothesis for the Friedman rank test for the difference in the means should be rejected at a 0.01 level of significance.

TABLE 12-10 The director of admissions at a state college is interested in seeing if admissions status (admitted, waiting list, denied admission) at his college is independent of the type of community in which an applicant resides. He takes a sample of recent admissions decisions and forms the following table: Admitted Wait List Denied Total Urban 45 21 17 83 Rural 33 13 24 70 Suburban 34 12 39 85 Total 112 46 80 238 He will use this table to do a chi-square test of independence with a level of significance of 0.01. -Referring to Table 12-10, the same decision would be made with this test if the level of significance had been 0.005.

TABLE 12-16 A perfume manufacturer is trying to choose between 2 magazine advertising layouts. An expensive layout would include a small package of the perfume. A cheaper layout would include a "scratch-and-sniff" sample of the product. The manufacturer would use the more expensive layout only if there is evidence that it would lead to a higher approval rate. The manufacturer presents both layouts to 5 groups and determines the approval rating from each group on both layouts. The data are given below. Use this to test whether the median difference in approval rating is different from zero in favor of the more expensive layout with a level of significance of 0.05. Package Scratch 52 37 68 43 43 53 48 39 56 47 -Referring to Table 12-16, the null hypothesis should be rejected.

TABLE 12-6 The dean of a college is interested in the proportion of graduates from his college who have a job offer on graduation day. He is particularly interested in seeing if there is a difference in this proportion for accounting and economics majors. In a random sample of 100 of each type of major at graduation, he found that 65 accounting majors and 52 economics majors had job offers. If the accounting majors are designated as "Group 1" and the economics majors are designated as "Group 2," perform the appropriate hypothesis test using a level of significance of 0.05. -Referring to Table 12-6, the same decision would be made with this test if the level of significance had been 0.10 rather than 0.05.