Chisquare.indep.sol1

Multiple Choice Questions Chi-square tests for independence

The next set of questions refer to the following situation: A survey was conducted to investigate the severity of rodent problems in egg and poultry operations. A random sample of operators was selected, and the operators were classified according to the type of operation and the extent of the rodent population. A total of 78 egg operators and 53 turkey operators were classified and the summary information is:

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1. Which of the following is not correct? (a) Operators who had both operations could not be used because this type of analysis requires each unit to be counted in one and only one cell. (b) The null hypothesis is that the severity of the rodent problem is independent of the type of operator. (c) The alternate hypothesis is that the proportion of turkey operators with mild, moderate, and severe rodent problems is different from the proportion of egg operators with mild, moderate, and severe rodent problems. (d) A Type I error would be to conclude that the severity of rodent problems is dependent upon the type of operator while, in fact, the proportion of turkey operators with mild, moderate, and severe rodent problems is the same as the proportion of egg operators with mild, moderate, and severe rodent problems. (e) A Type II error would be to conclude that the proportion of egg operators with mild, moderate, or severe rodent problems is the same as the proportion of turkey operators with mild, moderate, or severe rodent problems when in fact they are independent. Solution: e Past performance 1993 Apr - 52% (a-10%; b-10%; c-14%; d-14%) Past performance 1996 Dec - 61% (a-10%, d-12%) Past performance 1998 Dec - 72% 2. The value of the test statistic is: (a) about 5.99

c

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(b) about 9.71 (c) about 6.81 (d) about 5.64 (e) about 8.60 Solution: d Past performance 1993 Apr - 65% (a-14%; c-10%) Past performance 1998 Dec - 99% 3. The expected count in the (egg, mild infestation) cell is: (a) about 26.00 (b) about 33.33 (c) about 53.00 (d) about 31.55 (e) about 78.00 Solution: d Past performance 1996 Dec - 71% (a-16%) Past performance 1998 Dec - 87% 4. The approximate p-value is found to be: (a) about .060 (b) about .014 (c) about .032 (d) about .008 (e) about .05 Solution: a Past performance 1993 Apr - 48% (b-14%; c-16%; e-13%) Past performance 1996 Dec - 89% Past performance 1998 Dec - 96% 5. One reviewer of the study suggested that there may be a problem with the study because results from small operators were pooled with the results from large operators. Which of the following is NOT CORRECT? (a) Simpson’s paradox occurs when conclusions from a pooled table differ from the individual tables. (b) Tables can be pooled when the underlying rates are equal among tables. c

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(c) Simpson’s paradox occurs when tables with unequal row totals are pooled. (d) Inspection of the row or column percents will give a good clue if Simpson’s paradox is likely to occur. (e) Simpson’s paradox occurs when the pooled table gives no evidence of an effect but the individual tables show evidence of an effect. Solution: c Past performance Past performance Past performance Past performance

1990 1993 1996 1998

Dec - 68% Apr - 32% (b-16%; d-22%; e-25%) Dec - 65% (b-10%, d-10%) Dec - 73% ( d-10%)

The next set of questions refer to the following situation In the paper “Color Association of Male and Female Fourth-Grade School Children” (J. Psych., 1988, 383-8), children were asked to indicate what emotion they associated with the color red. The response and the sex of the child are noted and summarized below. The first number in each cell is the count, the second number is the row percent.

Frequency| Row Pct |anger |happy |love |pain | ---------+--------+--------+--------+--------+ f | 27 | 19 | 39 | 17 | | 26.47 | 18.63 | 38.24 | 16.67 | ---------+--------+--------+--------+--------+ m | 34 | 12 | 38 | 28 | | 30.36 | 10.71 | 33.93 | 25.00 | ---------+--------+--------+--------+--------+ Total 61 31 77 45

Total 102

112

214

Statistic DF Value Prob -----------------------------------------------------Pearson Chi-Square * 4.629 ***** Likelihood Ratio Chi-Square * 4.661 ***** Mantel-Haenszel Chi-Square 1 0.307 ***** 6. Under a suitable null hypothesis, the expected frequency for the cell corresponding to Anger and Males is: (a) 15.9 (b) 55.7 (c) 30.4 (d) 31.9 c

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(e) 29.1 Solution: d Past performance 1991 Apr - 63% (C-17%, E-15%) Past performance 1991 Dec - 84% (e-11%) Past performance 1997 Aug - 87% 7. The null hypothesis will be rejected at α=0.05 if the test statistic exceeds: (a) 3.84 (b) 5.99 (c) 7.81 (d) 9.49 (e) 14.07 Solution: c Past performance 1991 Apr - 86% 8. The approximate p-value is: (a) Between .100 and .900 (b) Between .050 and .100 (c) Between .025 and .050 (d) Between .010 and .025 (e) Between .005 and .010 Solution: a Past performance 1991 Dec - 77% (e-11%) 9. Which of the following is NOT CORRECT? (a) The children were cross-classified by sex and emotion associated with red. Each child was counted in one and only one cell. (b) The null hypothesis is that the type of emotion associated with red is independent of the sex of the child. (c) The null hypothesis is that the proportion of emotions associated with red is the same for both sexes. (d) All expected cell counts should be greater than five in order that the distribution of the test statistic is an approximate chi-square distribution. (e) If we reject the null hypothesis than we have proven that the two sexes associate red with emotions in different ways. c

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Solution: e Past performance 1991 Apr - 76% (C-12%) Past performance 1991 Dec - 77% (c-9%, d-12%) Past performance 1993 Feb - 67% (d-16%) 10. Which of the following is not correct? (a) A lower percentage of female students associate the emotion “anger” with the color red than do male students. (b) More students associate the color red with the emotion “love” than with the emotion “anger”. (c) Each student was classified by gender and by emotion association. Each student was counted in one and only one cell. (d) We will be unable to compute a correlation for this data because the variables are not both interval or ratio in scale. (e) We compute row or column percentages by dividing the cell count by the table total (214). Solution: e Past performance 1993 Feb - 67% (d-16%) Past performance 1996 Oct - 92% 11. A Type I error would be committed if: (a) We conclude that the sex of the child and the emotion associated with red are independent when in fact they are not independent. (b) We conclude that the sex of the child and the emotion associated with red are not independent when in fact they are not independent. (c) We conclude that the proportion of emotions associated with red differs between males and female when in fact they are the same. (d) We conclude that the proportion of emotions associated with red is the same for male and female when in fact they are the same. (e) We fail to find any association between the color red and emotions for either sex. Solution: c Past performance 1991 Apr - 76% (E-20%) Past performance 1991 Dec - 84% Past performance 1997 Aug - 76% 12. The null hypothesis is: (a) emotional association with red is independent of gender c

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(b) gender is dependent upon the emotional association with red (c) the probability of selecting an emotion with red is related to gender (d) the number of children in each cell does not depend upon gender nor upon emotion (e) the color red is independent of the emotion associated with it and with gender. Solution: c Past performance 1997 Aug - 74% 13. The test statistic and approximate p-value is: (a) 4.661 .1983 (b) 4.661 .3966 (c) 4.629 .2011 (d) 4.629 .4022 (e) 4.629 .1006 Solution: b Past performance 1997 Aug - 76% 14. Each person in a random sample of 50 was asked to state his/her sex and preferred colour. The resulting frequencies are shown below.

Sex

Male Female

Red 5 15

Colour Blue 14 6

Green 6 4

A chi-square test is used to test the null hypothesis that sex and preferred colour are independent. Which of the following statements is a correct decision about the null hypothesis? (a) Reject at the 0.005 level. (b) Reject at the 0.01 level but not at the 0.005 level. (c) Reject at the 0.025 level but not at the 0.01 level. (d) Reject at the 0.05 level but not at the 0.025 level. (e) Accept at the 0.05 level. Solution: not available

c

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15. The following data were obtained from a company which manufactures special plastic containers which are to hold a specified volume of hazardous material. On each of the three 8 hour shifts workers are able to make 500 of the containers. Some containers do not meet specifications as required by the company’s customer because they are too small, others because they are too large. Conformance to Specification Shift Too Small Within Spec. 8am 36 452 4pm 24 443 midnight 12 438

Too Large 12 33 50

If conformance to specifications is independent of shift, the expected number of containers that meet specification on the 4pm shift is (a) 166.7 (b) 443 (c) 33 (d) 444.3 (e) 500 Solution: not available 16. Are all employees equally prone to having accidents? To investigate this hypothesis, Parry (1985) looked at a light manufacturing plant and classified the accidents by type and by age of the employee.

Age Under 25 25 or over

Accident Type Sprain Burn | 9 17 | 61 13

Cut 5 12

A chi-square test gave a test-statistic of 20.78. If we test at α =.05: (a) There appears to be no association between accident type and age. (b) Age seems to be independent of accident type. (c) Accident type does not seem to be independent of age. (d) There appears to be a 20.78% correlation between accident type and age. (e) The proportion of sprain, cuts and burns seems to be similar for both age classes.

c

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Solution: c Past performance 1989 Apr - 64% 17. A random sample of 100 members of a union are asked to respond to two questions: Question 1. Are you happy with your financial situation? Question 2. Do you approve of the Federal government’s economic policies? The responses are:

Question 2

Yes 22 12 34

Yes No Total

Question 1. No 48 18 66

| | | |

Total 70 30 100

To test the null hypothesis that response to Question 1 is independent of response to Question 2 at 5% level, the expected frequency for the cell (Yes,Yes) and the critical value of the associated test statistic are: (a) 23.8 and 1.96 respectively (b) 10.2 and 3.84 respectively (c) 23.8 and 3.84 respectively (d) 23.8 and 7.81 respectively (e) 10.2 and 7.81 respectively Solution: c 18. A survey was conducted to investigate whether alcohol consumption and smoking are related. The following information was compiled for 600 individuals:

Drinker Non-drinker

Smoker 193 89

Non-smoker 165 153

Which of the following statements is true? (a) The appropriate alternative hypothesis is A: Smoking and Alcohol Consumption are independent. (b) The appropriate null hypothesis is H: Smoking and Alcohol Consumption are not independent. (c) The calculated value of the test statistic is 3.84. (d) The calculated value of the test statistic is 7.86. c

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(e) At level .01 we conclude that smoking and alcohol consumption are related. Solution: e 19. Doctors’ practices have been categorized as to being Urban, Rural, or Intermediate. The number of doctors who prescribed tetracycline to at least one patient under the age of 8 were recorded for each of these practice areas. The results are: Urban Tetracycline 95 No tetracycline 126

Intermediate 74 84

Rural 31 30

If the county type of practice and the use of tetracycline are independent, then the expected number of rural doctors who prescribe tetracycline is: (a) 31.0 (b) 27.7 (c) 1.37 (d) 51% (e) 62 Solution: b 20. For the problem outlined above, the critical value(table value) of the test statistic when the level of significance is α =0.05, is: (a) 0.1026 (b) 7.3778 (c) 5.9915 (d) 12.5916 (e) 7.8147 Solution: c The next set of questions refer to the following situation: A study was conducted to determine if the fatality rate depends on the size of the automobile. The analysis of accidents is as follows (with some values hidden):

c

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DEATH SIZE FREQUENCY| m | s | L | ---------+--------+--------+--------+ no | 63 | 128 | 46 | ---------+--------+--------+--------+ yes | 26 | 95 | 16 | ---------+--------+--------+--------+ TOTAL 89 223 62

TOTAL 237 137 374

STATISTICS FOR TABLE OF DEATH BY SIZE STATISTIC DF VALUE PROB -----------------------------------------------------CHI-SQUARE * 8.663 ***** LIKELIHOOD RATIO CHI-SQUARE * 8.838 ***** 21. Under a suitable null hypothesis, the expected frequency for the cell corresponding to fatal type of accident and small size automobile is: (a) (b) (c) (d) (e)

81.68 67.00 61.43 63.41 59.72

Solution: a Past performance 1990 Apr - 92% 22. Which of the following is NOT CORRECT? (a) The accidents were cross-classified by size of automobile and fatality status. Each accident was counted in one and only one cell. (b) The null hypothesis is that the fatality status is independent of the size of the automobile. (c) The null hypothesis is that the proportion of fatality status is the same for all three sizes of automobiles. (d) All expected cell counts should be greater than five in order that the distribution of the test statistic is an approximate chi-square distribution. (e) If we reject the null hypothesis than we have proven that the size of the automobile affects the chances of a fatality. Solution: e Past performance 1990 Apr - 39% (B-12%, C-36%) Past performance 1990 Dec - 20% ( 15% - c, 56% - d)

c

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23. The null hypothesis will be rejected at α=0.05 if the test statistic exceeds: (a) 12.59 (b) 7.81 (c) 5.99 (d) 3.84 (e) 9.49 Solution: c Past performance 1990Apr - 79% 24. The approximate p-value is: (a) less than .005 (b) between .005 and .010 (c) between .010 and .025 (d) between .025 and .050 (e) between .050 and .100 Solution: c Past performance 1990 Dec - 78% Past performance 1993 Apr - 80% 25. A controversial issue in sports is the use of the “instant replay” for making decisions on plays that are extremely close or hard to call by an official. A survey of players in each of four professional sports was conducted, asking them if they felt “instant replays” should be used to decide close or controversial calls. The results are as follows: Use of Instant Replay Favor Oppose Football 22 2 Baseball 18 6 Basketball 15 26 Soccer 3 10 In testing to see whether opinion with respect to the use of instant replays is independent of sport, a table of expected frequencies is found. In this table, the expected number of professional baseball players opposing the use of instant replays is equal to: (a) 10.4 (b) 24.1 c

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(c) 11.0 (d) 6.0 (e) 8.4 Solution: not available 26. Each person in a random sample of males and females was asked to state his/her sex and preferred colour. The resulting frequencies are shown below.

Sex

Male Female

Red 3 17

Colour Blue 11 11

Green 6 2

Which of the following is FALSE? (a) 55% of males prefer the colour blue. (b) Of those who prefer the colour green, 75% are males. (c) 44% of people surveyed prefer the colour blue. (d) A higher percentage of males prefered the colour blue than females. (e) 15% of people are males who prefer the colour red. Solution: e Past performance 2006 Oct - 76% (16%=d)

c

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