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Chapter 13: Inference for Tables: Chi-Square Procedures

Use the following to answer questions 1-3: Using computer software, I generate 1000 random numbers that are supposed to follow a standard normal distribution. I classify these 1000 numbers according to whether their values are less than ­2 (value < ­2), between ­2 and 0 (­2 value < 0), between 0 and 2 (0 value < 2), or greater than or equal to 2 (2 value). The results are given in the following table. The expected counts are computed using the 68­95­99.7 rule. Between Between < ­2 ­2 and 0 0 and 2 2 Observed 18 492 468 22 Expected 25 475 475 25 ___ 1. To test if the distribution of observed counts differ significantly from the expected distribution of counts we use the X2 statistic. We know that this X2 statistic has approximately a 2 distribution. How many degrees of freedom does this distribution have? a) 3. b) 4. c) 999. d) 1000. ___ 2. To test if the observed counts differ significantly from the expected distribution of counts we use the X2 statistic. The component of this X2 statistic corresponding to the "less than 2" category is a) (O ­ E)2/E = 0.28. c) (O ­ E)2/E = 2.72. 2 b) (O ­ E) /E = 1.96. d) (O ­ E)2/E = 49. ___ 3. To test if the observed counts differ significantly from the expected distribution of counts we use the X2 statistic. The value of this X2 statistic is found to be 3.03. The P-value of our test is a) greater than 0.20. c) between 0.10 and 0.05. b) between 0.20 and 0.10. d) less than 0.05. ___ 4. Which of the following is true of chi-square distributions? a) They take on only positive values. b) They are skewed to the left. c) As the number of degrees of freedom increases, they look less and less like a normal curve. d) All of the above. Use the following to answer questions 5-8: A random sample of 100 traffic tickets given to motorists in a large city is examined. The tickets were classified according to the race of the driver. The results are summarized in the following table. White 46 Black 37 Hispanic 11 Other 6

Number of tickets

The proportion of the population of the city in each of the race categories above is the following. White 0.65 Black 0.30 Hispanic 0.03 Other 0.02

Proportion

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___ 5. We wish to test whether the racial distribution of traffic tickets in the city is the same as the racial distribution of the population of the city. To do so we use the X2 statistic. The component of this X2 statistic corresponding to the Hispanic category is a) (O ­ E)2/E = 2.67. c) (O ­ E)2/E = 21.33. 2 b) (O ­ E) /E = 5.82. d) (O ­ E)2/E = 4011.36. ___ 6. We wish to test whether the racial distribution of traffic tickets in the city is the same as the racial distribution of the population of the city. To do so we use the X2 statistic and we compute its value to be 36.51. Assuming that this X2 statistic has approximately a 2 distribution, the P-value of our test is a) greater than 0.10. c) between 0.05 and 0.01. b) between 0.10 and 0.05. d) less than 0.01. ___ 7. We wish to test whether the racial distribution of traffic tickets in the city is the same as the racial distribution of the population of the city. To do so we plan to use the X2 statistic. We may assume the X2 statistic has an approximate chi-square distribution because of which of the following? a) The number of tickets given in each race category is greater than 5. b) The sample size is 100, which is large enough for the chi-square approximation to be valid. c) The number of categories is small relative to the number of observations. d) We may not assume the X2 statistic has an approximate chi-square distribution. ___ 8. We wish to test whether the racial distribution of traffic tickets in the city is the same as the racial distribution of the population of the city. To do so we plan to use the X2 statistic. The category that contributes the largest component to the X2 statistic is a) White. b) Black. c) Hispanic. d) Other. Use the following to answer questions 9-13: I teach a large introductory statistics course. In the past, the proportion of students that receive a grade of A is 0.20. The proportion that receives a B is 0.30. The proportion that receives a C is 0.30. The proportion that receives a D is 0.10. The proportion that receives an F is 0.10. This year, there were 200 students in the class and I gave the following grades. A 56 B 74 C 60 D 9 F 1

Number

___ 9. I wish to test whether the distribution of grades this year is the same as in the past. To do so I plan to use the X2 statistic. We know that this X2 statistic has approximately a 2 distribution. How many degrees of freedom does this distribution have? a) 200. b) 199. c) 5. d) 4. ___ 10. I wish to test whether the distribution of grades this year is the same as in the past. To do so I plan to use the X2 statistic. The component of this X2 statistic corresponding to a grade of C is a) (O ­ E)2/E = 0. c) (O ­ E)2/E = 30. 2 b) (O ­ E) /E = 1. d) (O ­ E)2/E = 11,880.3. ___ 11. I wish to test whether the distribution of grades this year is the same as in the past. To do so I use the X2 statistic and I compute its value to be 33.77. The P-value of our test is a) greater than 0.10. c) between 0.05 and 0.01. b) between 0.10 and 0.05. d) less than 0.01.

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___ 12. I wish to test whether the distribution of grades this year is the same as in the past. To do so I use the X2 statistic. I may assume the X2 statistic has an approximate chi-square distribution because of which of the following? a) The expected number of people in each grade category is greater than 5. b) The sample size is 200, which is large enough for the chi-square approximation to be valid. c) The number of categories is small relative to the number of observations. d) I may not assume the X2 statistic has an approximate chi-square distribution because there is only one person in the F grade category. ___ 13. I wish to test whether the distribution of grades this year is the same as in the past. To do so I use the X2 statistic. The grade category that contributes the largest component to the X2 statistic is a) A. b) B. c) D. d) F. Use the following to answer questions 14-15: Using computer software, I generate 1000 random numbers that are supposed to follow a standard normal distribution. I classify these 1000 numbers according to whether their values are less than 0 or greater than or equal to 0. The results are given in the table below. Less Than 0 512 Greater Than or Equal to 0 488

Number

Because the standard normal distribution is symmetric about 0, I would expect half of the random numbers generated to be less than 0 and half to be greater than or equal to 0. To test if the distribution of the observed number in each category differs significantly from the expected distribution of counts I use the X2 statistic. ___ 14. The value of this statistic is a) 0.024. b) 0.048. c) 0.288.

d) 0.576.

___ 15. I know that this X2 statistic has approximately a 2 distribution. How many degrees of freedom does this distribution have? a) 0. b) 1. c) 2. d) none of the above. Use the following to answer questions 16-20: Are avid readers more likely to wear glasses than those who read less frequently? Three hundred men in the Korean army were selected at random and characterized as to whether they wore glasses and whether the amount of reading they did was above average, average, or below average. The results are presented in the following table. Wear Glasses No 26 80 70

Amount of Reading Above average Average Below average

Yes 47 48 31

___ 16. This is an r × c table. The number r has value a) 2. b) 3. c) 4. d) 6. ___ 17. The proportion of men in the table who wore glasses is a) 0.24. b) .37. c) 0.42. d) 0.64.

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___ 18. The proportion of all above-average readers who wear glasses is a) 0.24. b) 0.37. c) 0.42. d) .64. ___ 19. Suppose we wish to test the null hypothesis that there is no association between the amount of reading you do and whether you wear glasses. Under the null hypothesis, the expected number of above average readers who wear glasses is a) 81.1. b) 47. c) 30.7. d) 27.2. ___ 20. Suppose we wish to display in a graphic the proportion of all above average readers that wear glasses and do not wear glasses, respectively. Which of the following graphical displays is best suited to this purpose? a) a stemplot. b) a scatterplot. c) a boxplot. d) a bar graph. Use the following to answer questions 21-23: When a police officer responds to a call for help in a case of spousal abuse, what should the officer do? A randomized controlled experiment in Charlotte, North Carolina, studied three police responses to spousal abuse: advise and possibly separate the couple, issue a citation to the offender, and arrest the offender. The effectiveness of the three responses was determined by re-arrest rates. The table below shows these rates. Assigned Treatment # of Re-arrests Arrest Citation Advise/Separate 0 175 181 187 1 36 33 24 2 2 7 1 3 1 1 0 4 0 2 0 ___ 21. This is an r × c table. The number r has value a) 2. b) 3. c) 4. d) 5. ___ 22. Suppose we wish to test the null hypothesis that the proportion of subsequent arrests is the same regardless of the treatment assigned. When the null hypothesis is true, the expected number of times no subsequent arrest would occur for the treatment "Advise/Separate" is a) 177. b) 181. c) 187. d) 543. ___ 23. Suppose we wish to test the null hypothesis that the proportion of subsequent arrests is the same regardless of the treatment assigned. Which of the following statements is true? a) We cannot test this hypothesis because the police officers did not record the expected counts. b) The test of the null hypothesis will have a very small P-value (below .0001) because the counts in each row are not identical. c) We cannot test this hypothesis because the expected cell counts are less than five in too many of the cells. d) The test of the null hypothesis will have a very small P-value (below .0001) because the there were so few cases where there were more than one re-arrest.

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Use the following to answer questions 24-26: Even though Puerto Rico is a territory of the United States there are many cultural differences between the states on the continent and the island of Puerto Rico. These differences include the way consumers handle problems with purchases. Two researchers surveyed owners of VCRs in the Northeast United States and in Puerto Rico. They asked those who had experienced problems with their VCRs whether they complained. The results are given in the table below. Complained No Yes States 94 330 Puerto Rico 33 64

___ 24. This is an r × c table. The number r has value a) 1. b) 2. c) 3. d) 4. ___ 25. The cell that contributes most to the chi-square statistic is a) Americans in the states that did not complain. b) Americans in the states that complained. c) Puerto Ricans that did not complain. d) Puerto Ricans that did complain. ___ 26. The P-value for testing the null hypothesis that the probability of complaining is the same for the States and Puerto Rico, against the alternative that the probability is different for the States and Puerto Rico, a) is between 0.025 and 0.05. b) is between 0.010 and 0.025. c) is less than 0.010. d) cannot be determined since these are not the hypotheses being tested by the chi-square test. Use the following to answer questions 27-30: All current-carrying wires produce electromagnetic (EM) radiation, including the electrical wiring running into, through, and out of our homes. High-frequency EM radiation is thought to be a cause of cancer; the lower frequencies associated with household current are generally assumed to be harmless. To investigate this, researchers visited the addresses of children in the Denver area who had died of some form of cancer (leukemia, lymphoma, or some other type) and classified the wiring configuration outside the building as either a high-current configuration (HCC) or as a low-current configuration (LCC). Here are some of the results of the study. Leukemia 52 84 Lymphoma 10 21 Other Cancers 17 31

HCC LCC

Computer software was used to analyze the data; the output is given below. It includes the cell counts, the expected cell counts, and the chi-square statistic. Expected counts are printed below observed counts. Leukemia 52 49.97 84 86.03 136 Lymphoma 10 11.39 21 19.61 31 Other Cancers 17 17.64 31 30.36 48 Total 79 136 215

HCC LCC Total

X2 = 0.082 + 0.170 + 0.023 + 0.048 + 0.099 + 0.013 = 0.435

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___ 27. The appropriate degrees of freedom for the chi-square statistic is a) 1. b) 2. c) 3. d) 4. ___ 28. The P-value for the chi-square statistic is a) larger than 0.10. b) between 0.10 and 0.05.

c) between 0.05 and 0.01. d) less than 0.01.

___ 29. The cell that contributes most to the chi-square statistic is a) the cases of leukemia that occurred in homes with an HCC. b) the cases of leukemia that occurred in homes with an LCC. c) the cases of other cancers that occurred in homes with an LCC. d) the cases of lymphoma that occurred in homes with an HCC. ___ 30. Which of the following may we conclude? a) There is strong evidence of an association between wiring configuration and the chance a child will develop some form of cancer. b) HCC either causes cancer directly or is a major contributing factor to the development of cancer in children. c) There is weak evidence that HCC causes cancer in children. d) There is not much evidence of an association between wiring configuration and the type of cancer children in the study died of. Use the following to answer questions 31-32: A study was performed to examine the personal goals of children in grades 4, 5, and 6. A random sample of students was selected for each of the grades 4, 5, and 6 from schools in Georgia. The students received a questionnaire regarding achieving personal goals. They were asked what they would most like to do at school: make good grades, be good at sports, or be popular. Results are presented in the table below by the sex of the child. Boys 96 32 94 Girls 295 45 40

Make good grades Be popular Be good in sports

___ 31. Which hypotheses are being tested by the chi-square test? a) The null hypothesis is that personal goals and gender are independent and the alternative is that they are dependent. b) The null hypothesis is that the mean personal goal is the same for boys and girls and the alternative is that the means differ. c) The distribution of personal goals is different for boys and girls. d) The distribution of gender is different for the three different personal goals. ___ 32. The numerical value of the chi-square statistic for this table is a) 3.84. b) 5.99. c) 16.105. d) 89.966.

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___ 33. Are avid readers more likely to wear glasses than those who read less frequently? Three hundred men in the Korean army were selected at random and characterized as to whether they wore glasses and whether the amount of reading they did was above average, average, or below average. The results are presented in the following table. Wear Glasses No 26 78 70

Amount of Reading Above average Average Below average

Yes 47 48 31

The numerical value of the chi-square statistic for testing independence of whether you wear glasses and the amount of reading you do is a) 2. b) 8.65. c) 21. d) 30.7 Answer Key 1. a 2. b 3. a 4. a 5. c 6. d 7. d 8. c 9. d 10. a 11. d

12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

a d d b b c d c d d a

23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33.

c b c b b a d d a d c

34) Packages of mixed nuts made by a certain company contain four types of nuts. The percentage of nuts of type 1, 2, 3 and 4 are supposed to be 40%, 30%, 20% and 10% respectively. A random sample of 200 nuts is selected, and each one is categorized by type. In the sample 75 were type 1, 55 were type 2, 55 were type 3 and the rest were type 4. Is there significant evidence to conclude that these nuts do not follow the given distribution? Follow the inference toolbox using an alpha of 5%.

35) The paper "No evidence of Impaired Neurocognitive Performance in Collegiate Soccer Players" compared collegiate soccer players, athletes in sports other than soccer, and a group of students who were not involved in collegiate sports with respect to history of head injuries. The results are summarized below: 0 Concussions 1 concussion 2 concussions 3+ concussions Soccer Players 45 25 11 10 Non-Soccer Athletes 68 15 8 5 Non-Athletes 45 5 3 0 Total Is there significant evidence to conclude concussions are not evenly distributed among these groups of individuals? Follow the inference toolbox and use an alpha of 5%.

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36) The paper "Facial Expression of Pain in Elderly Adults with Dementia" examined the relationship between a nurse's assessment of a patient's facial expression and his or her self-reported level of pain. Data for 89 patients are summarized below. Self-Reported No pain Pain Nurse's Assesment No Pain 17 40 Pain 3 29 Is there significant evidence of a relationship between a facial expression that reflects pain and self-reported pain because patients with dementia do not always give a verbal indication that they are in pain? Follow the inference toolbox, use an alpha of 5%.

37) Job satisfaction of professionals was examined in the article "Psychology of the Scientist: Work-related Attitudes of U.S. Scientists." Each person in a random sample of 778 teachers was classified according to a job satisfaction variable and also by teaching level, resulting in the accompanying two-way table. Can we conclude that there is an association between job satisfaction and teaching level? Follow the inference toolbox. Job Satisfaction Satisfied Unsatisfied Teaching Level College 74 43 High School 224 171 Elementary 126 140

38) When public opinion surveys are conducted by mail, a cover letter explaining the purpose of the survey is usually included. To determine whether the wording of the cover letter influences the response rate, three different cover letters were used in a survey of students at a Midwest University ("The Effectiveness of a Cover Letter Appeals," J. of Soc. Psych. (1984): 85-91). Suppose that each of the three cover letters accompanied questionnaires sent to an equal number of randomly selected students. Returned questionnaires were then classified according to the type of cover letter (I, II, or III). Use the accompanying data to test the hypothesis that 1/3 are the true proportions of all returned questionnaires accompanied by cover letters I, II, and III. Use a .05 significance level and follow the inference toolbox. Cover Letter Type I II III Frequency 48 44 39

39) The article "Daily Weigh-ins Can Help You Keep Off Lost Pounds, Experts Say" (Associated Press, October 17, 2005) describes an experiment in which 291 people who had lost at least 10% of their body weight in a medical weight loss program were assigned at random to one of three groups for follow-up. One group met monthly in person, one group "met" online monthly in a chat room, and one group received a monthly newsletter by mail. After 18 months, participants in each group were classified according to whether or not they had regained more than 5 pounds. Amount of Weight Gained Regained 5 lbs or less Regained more than 5 lbs Row total In-Person 52 45 97 Online 44 53 97 Newsletter 27 70 97 Total 134 168 291 Does there appear to be a difference in the weight regained proportions for the three follow-up methods? Follow the inference toolbox and use an alpha of 1%.

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