Exam 12: Chi-Square Tests and Nonparametric Tests

TABLE 12-1 A study published in the American Journal of Public Health was conducted to determine whether the use of seat belts in motor vehicles depends on ethnic status in San Diego County. A sample of 792 children treated for injuries sustained from motor vehicle accidents was obtained, and each child was classified according to (1) ethnic status (Hispanic or non-Hispanic) and 1() seat belt usage (worn or not worn) during the accident. The number of children in each category is given in the table below. Hispanic Non- Hispanic Seat belts worn 31 148 Seat belts not worn 283 330 -Referring to Table 12-1, the calculated test statistic is

In testing for differences between the median of two independent populations, the null hypothesis is

The McNemar test is approximately chi-square distributed.

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

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 test will involve ______degrees of freedom.

TABLE 12-18 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. Trial Barth Tornado Reiser Shaw 1 43 37 41 43 2 46 38 45 45 3 43 39 42 46 -Referring to Table 12-18, the alternative hypothesis of the Kruskal-Wallis test is that_________

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. What is the value of the test statistic using a 5% level of significance?

TABLE 12-20 Data on the percentage of 200 hotels in each of the three large cities across the world on whether minibar charges are correctly posted at checkout are given below. Hong Kong New York Paris Yes 86\% 76\% 78\% No 14\% 24\% 22\% At the 0.05 level of significance, you want to know if there is evidence of a difference in the proportion of hotels that correctly post minibar charges among the three cities. -Referring to Table 12-20, the test will involve______ degrees of freedom.

TABLE 12-20 Data on the percentage of 200 hotels in each of the three large cities across the world on whether minibar charges are correctly posted at checkout are given below. Hong Kong New York Paris Yes 86\% 76\% 78\% No 14\% 24\% 22\% At the 0.05 level of significance, you want to know if there is evidence of a difference in the proportion of hotels that correctly post minibar charges among the three cities. -Referring to Table 12-20, there is sufficient evidence to conclude that the proportions between Hong Kong and Paris are different at a 0.05 level of significance.

TABLE 12-20 Data on the percentage of 200 hotels in each of the three large cities across the world on whether minibar charges are correctly posted at checkout are given below. Hong Kong New York Paris Yes 86\% 76\% 78\% No 14\% 24\% 22\% At the 0.05 level of significance, you want to know if there is evidence of a difference in the proportion of hotels that correctly post minibar charges among the three cities. -Referring to Table 12-20, the decision made suggests that the 3 cities all have different proportions of hotels that correctly post minibar charges.

When using the ?² tests for independence, one should be aware that expected frequencies that are too small will lead to too big a type I error.

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, what is the value of the critical range for the Marascuilo procedure to test for the difference in proportions between procedure A and procedure D using a 0.05 level of significance?

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 by Varieties blocked by Fields $\mathrm { s } =$$10.00$ $\quad \mathrm { DF } = 2 \quad \mathrm { P } = 0.007$ Est Sum of Varieties Median 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 there is no interaction between the 5 blocks and the 3 treatment levels.

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 decision made suggests that the 4 procedures do not all have the same proportion of clear operations.

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 value of the test statistic?

TABLE 12-20 Data on the percentage of 200 hotels in each of the three large cities across the world on whether minibar charges are correctly posted at checkout are given below. Hong Kong New York Paris Yes 86\% 76\% 78\% No 14\% 24\% 22\% At the 0.05 level of significance, you want to know if there is evidence of a difference in the proportion of hotels that correctly post minibar charges among the three cities. -Referring to Table 12-20, the null hypothesis will be rejected.

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

TABLE 12-12 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 average number of hours spent watching TV and weights are independent at 1% level of significance. TV Hours Weights 0-3 3-6 6+ Total More than 101 . overweight 1 9 20 30 Within 101 . of normal weight 20 15 15 50 More than 10 . underweight 10 5 5 20 Total 31 29 40 100 -Referring to Table 12-12, if there is no connection between weights and average number of hours spent watching TV, we should expect how many children to be spending no more than 6 hours on average watching TV and are more than 10 lbs. underweight?

TABLE 12-5 A corporation randomly selects 150 salespeople and finds that 66% who have never taken a self-improvement course would like such a course. The firm did a similar study 10 years ago in which 60% of a random sample of 160 salespeople wanted a self-improvement course. The groups are assumed to be independent random samples. Let u₁ and u₂ represent the true proportion of workers who would like to attend a self-improvement course in the recent study and the past study, respectively. -Referring to Table 12-5, what is the critical value when performing a chi-square test on whether population proportions are different if ? = 0.05?

TABLE 12-12 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 average number of hours spent watching TV and weights are independent at 1% level of significance. TV Hours Weights 0-3 3-6 6+ Total More than 101 . overweight 1 9 20 30 Within 101 . of normal weight 20 15 15 50 More than 10 . underweight 10 5 5 20 Total 31 29 40 100 -Referring to Table 12-12, how many children in the survey spend no more than 6 hours watching TV and are more than 10 lbs. underweight?