Exam 12: Chi-Square and Nonparametric

The squared difference between the observed and theoretical frequencies should be large if there is no significant difference between the proportions.

SCENARIO 12-5 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: $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { 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 Scenario 12-5, what is the value of the critical range for the Marascuilo procedure to test for the difference in proportions between procedure B and procedure C using a 0.05 level of significance?

The procedure for the Wilcoxon rank sum test requires that you rank each group separately rather than together.

SCENARIO 12-2 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 Scenario 12-2, the same decision would be made with this test if the level of significance had been 0.10 rather than 0.05.

If the assumptions of the one-way ANOVA F test are not met, which of the following tests could be used?

SCENARIO 12-3 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 Scenario 12-3, the critical value of the test is _________.

SCENARIO 12-11 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 Scenario 12-11, the alternative hypothesis claims that "there is some connection between admission status at the college and the type of community in which an applicant resides."

The Kruskal-Wallis test is an extension of which of the following for two independent samples?

SCENARIO 12-5 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: $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { 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 Scenario 12-5, the value of the test statistic is _________.

SCENARIO 12-10 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 5-9 Years Defect Rate: High 6 9 9 Average 9 19 23 Low 7 8 10 -Referring to Scenario 12-10, find the rejection region necessary for testing at the 0.05 level of significance whether there is a relationship between defect rate and years of experience. a) Reject $H _ { 0 }$ if $\chi ^ { 2 } > 16.919$ b) Reject $H _ { 0 }$ if $\chi ^ { 2 } > 15.507$ c) Reject $H _ { 0 }$ if $\chi ^ { 2 } > 11.143$ d) Reject $H _ { 0 }$ if $\chi ^ { 2 } > 9.488$

The Kruskal-Wallis Rank Test for differences in more than two medians is a nonparametric alternative to

SCENARIO 12-17 Three new different models of compact SUVs have just arrived at the market. You are interested in comparing the gas mileage performance of all three models to see if they are the same. A partial computer output for twelve compact SUVs of each model is given below: -Referring to Scenario 12-17, what is the p-value of the Kruskal-Wallis test statistic?

SCENARIO 12-16 As part of an evaluation program, a sporting goods retailer wanted to compare the downhill coasting speeds of 4 brands of bicycles. She took 3 of each brand and determined their maximum downhill speeds. The results are presented in miles per hour in the table below. 1 43 37 41 43 2 46 38 45 45 3 43 39 42 46 -Referring to Scenario 12-16, the sporting goods retailer decided to perform a Kruskal-Wallis test even though the sample size in each of the 4 groups is not larger than 5. The null hypothesis of the test is ________.

SCENARIO 12-6 According to an article in Marketing News, fewer checks are being written at the grocery store checkout than in the past. To determine whether there is a difference in the proportion of shoppers who pay by check among three consecutive years at a 0.05 level of significance, the results of a survey of 500 shoppers in three consecutive years are obtained and presented below. -Referring to Scenario 12-6, what is the critical value?

SCENARIO 12-10 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 5-9 Years Defect Rate: High 6 9 9 Average 9 19 23 Low 7 8 10 -Referring to Scenario 12-10, for the cell with 1 to 4 years of training time and a high defect rate, what is the contribution to the overall $\chi ^ { 2 }$ statistic for the independence test?

SCENARIO 12-13 Recent studies have found that American children are more obese than in the past. The amount of time children spent watching television has received much of the blame. A survey of 100 ten-year- olds revealed the following with regards to weights and average number of hours a day spent watching television. We are interested in testing whether the mean number of hours spent watching TV and weights are independent at 1% level of significance. -Referring to Scenario 12-13, how many children in the survey spend more than 6 hours watching TV and are more than 10 lbs. overweight?

SCENARIO 12-14 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 the more expensive layout to 4 groups and determines the approval rating for each group. He presents the "scratch-and-sniff" layout to 5 groups and again determines the approval rating of the perfume for each group. The data are given below. Use this to test the appropriate hypotheses with the Wilcoxon Rank Sum Test with a level of significance of 0.05. 52 37 68 43 43 53 48 39 47 -Referring to Scenario 12-14, the rank given to the second observation in the "scratch-and-sniff" group is ________.

SCENARIO 12-15 Two new different models of compact SUVs have just arrived at the market. You are interested in comparing the gas mileage performance of both models to see if they are the same. A partial computer output for twelve compact SUVs of model 1 and thirteen of model 2 is given below: You are told that the gas mileage population distributions for both models are not normally distributed. -Referring to Scenario 12-15, what is(are) the critical value(s) using a 5% level of significance?

If the sample sizes in each group is larger than 5, the Kruskal-Wallis rank test statistic can be approximated by a standardized normal distribution.

Suppose there is interest in comparing the median response time for three independent groups learning a specific task. The appropriate nonparametric procedure is