Chapter 4: More about Relationships between Two-Variables Review Sheet

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1 Review Sheet 4. Which of the following is true? A) log(ab) = log A log B. D) log(a/b) = log A log B. B) log(a + B) = log A + log B. C) log A B = log A log B. 5. Suppose we measure a response variable Y at each of several times. A scatterplot of log Y versus time of measurement looks approximately like a positively sloping straight line. We may conclude that A) the correlation between time of measurement and Y is negative, since logarithms of positive fractions (such as correlations) are negative. B) the rate of growth of Y is positive, but slowing down over time. C) a logarithmic growth model would approximately describe the relationship between Y and the time of measurement. D) a mistake has been made. It would have been better to plot Y versus the logarithm of the time of measurement. E) an exponential growth model would approximately describe the relationship between Y and time of measurement. 6. Using least-squares regression, I determine that the (base 10) logarithm of the population of a country is approximately described by the equation [y-hat]log(population)[y-hat] = (year) Based on this equation, the population of the country in the year 2010 should be about A) 6.6. B) 735. C) 2,000,000. D) 3,981,072. E) 33,000, Which of the following would provide evidence that a power law model describes the relationship between a response variable y and an explanatory variable x? A) A scatterplot of y versus x looks approximately linear. B) A scatterplot of log y versus x looks approximately linear. C) A scatterplot of y versus log x looks approximately linear. D) A scatterplot of log y versus log x looks approximately linear. E) A scatterplot of the square root of y versus x looks approximately linear. 10. Suppose the relationship between a response variable y and a predictor variable x is approximately y = x Which of the following plots would approximately follow a straight line? A) A plot of y against x. D) A plot of 10 y against x. B) A plot of y against log x. E) A plot of log y against log x. C) A plot of log y against x.

2 12. Researchers studied a sample of 100 adults between the ages of 25 and 35 and found a strong negative correlation between the amount of vitamin C an individual consumed and the number of pounds the individual was overweight. Which of the following may we conclude? A) This is strong, but not conclusive, evidence that large amounts of vitamin C inhibit weight gain. B) If the amount of vitamin C consumed and the number of pounds overweight for each individual in this study were plotted on a scatterplot, the points would lie close to a negatively sloping straight line. C) If a larger sample of adults between the ages of 25 and 35 had been studied, the correlation would have been even stronger. D) If people consumed more vitamin C, they would likely lose more weight. 13. The owner of a chain of supermarkets notices that there is a positive correlation between the sales of beer and the sales of ice cream over the course of the previous year. During seasons when sales of beer were above average, sales of ice cream also tended to be above average. Likewise, during seasons when sales of beer were below average, sales of ice cream also tended to be below average. Which of the following would be a valid conclusion from these facts? A) The sales records must be in error. There should be no association between beer and ice cream sales. B) Temperature is clearly a lurking variable when considering sales of beer and ice cream. C) A scatterplot of monthly ice cream sales versus monthly beer sales would show that a straight line describes the pattern in the plot, but it would have to be a horizontal line. D) Evidently, for a significant proportion of customers of these supermarkets, drinking beer causes a desire for ice cream or eating ice cream causes a thirst for beer. E) None of the above. 47

3 15. When exploring very large sets of data involving many variables, which of the following is true? A) The correlation coefficient will be close to 1 due to the large sample size. B) Associations will be stronger than would be seen in a much smaller subset of the data. C) A strong association is good evidence for causation because it is based on a large quantity of information. D) Extrapolation is safe because it is based on a greater quantity of evidence. E) None of the above. 18. Two variables, an explanatory variable x and a response variable y, are measured on each of several individuals. The correlation between these variables is found to be To help us interpret this correlation, we should do which of the following? A) Compute the least-squares regression line of y on x and consider whether the slope is positive or negative. B) Interchange the roles of x and y (i.e., treat x as the response variable and y as the explanatory variable) and recompute the correlation. C) Plot the data. D) Determine whether x or y has larger values before computing the residuals. 23. Which of the following would be necessary to establish a cause-and-effect relation between two variables? A) Strong association between the variables. B) A well-designed experiment. C) Plausibility of the alleged cause. D) An association between the variables observed in many different settings. Use the following to answer questions 26-27: An article in the student newspaper of a large university with the headline A s Swapped for Evaluations? included the following: According to a new study, teachers may be more inclined to give higher grades to students, hoping to gain favor with the university administrators who grant tenure. The study examined the average grade and teaching evaluation in a large number of courses given in 1997 in order to investigate the effects of grade inflation on evaluations. I am concerned with student evaluations because instruction has become a popularity contest for some teachers, said Professor Smith, who recently completed the study. Results showed higher grades directly corresponded to a more positive evaluation. 48

4 27. Which of the following would be a valid conclusion to draw from the study cited in the article? A) Teachers who give higher grades are more likely to gain tenure. B) A good teacher, as measured by teaching evaluations, helps students learn better, resulting in higher grades. C) Teachers of courses in which the mean grade is above average apparently tend to have above-average teaching evaluations. D) A teacher can improve his or her teaching evaluations by giving good grades. 28. A researcher computed the average SAT math score of all high school seniors who took the SAT exam for each of the 50 states. The researcher also computed the average salary of high school teachers in each of these states and plotted these average salaries against the average SAT math scores for each state. The plot showed a distinct negative association between average SAT math scores and teacher salaries. The researcher may legitimately conclude which of the following? A) Increasing the average salary of teachers will cause the average of SAT math scores to decrease, but it is not correct to conclude that increasing the salaries of individual teachers causes the SAT math scores of individual students to increase. B) States that pay teachers high salaries tend to do a poor job of teaching mathematics, on average. C) As the pay for an individual teacher increases, the teacher s students are more likely to do poorly on the SAT math. D) The data used by the researcher do not provide evidence that increasing the salaries of teachers will cause the performance of students on the SAT math to get worse. E) States in which students tend to perform poorly in mathematics probably have a higher proportion of problem students and thus need to pay teachers higher salaries in order to attract them to teach in those states. 30. A researcher notices that in a sample of adults, those that take larger amounts of vitamin C have fewer illnesses. However, those that take larger amounts of vitamin C also tend to exercise more. As explanations for having fewer illnesses, the variables amount of vitamin C taken and amount of exercise are A) skewed. B) confounded. C) common responses. D) symmetric. E) linked. 49

5 31. In 1982 Kennesaw, Georgia, passed a law requiring all citizens to own at least one gun. Although the law was never enforced, six months after the law was passed the number of burglaries in that month was less than in the month prior to passage of the law. We may conclude which of the following? A) Gun ownership and burglary rates are negatively associated. B) Gun ownership causes a reduction in crime. This is because there is a negative association between gun ownership and burglary rates and because there is a plausible explanation for this association (gun ownership acts as a deterrent to crime). C) Criminals are more likely to avoid homes in towns where guns are more prevalent. D) All of the above. E) None of the above. 32. A study of the salaries of full professors at Upper Wabash Tech shows that the median salary for female professors is considerably less than the median male salary. However, further investigation shows that the median salaries for male and female full professors are about the same in every department (English, physics, etc.) of the university. This apparent contradiction is an example of A) extrapolation. B) Simpson's paradox. C) confounded responses. D) correlation. E) causation. 34. The two-way table below categorizes suicides committed in 1983 by the sex of the victim and the method used. Method Male Female Firearms 13,959 2,641 Poison 3,148 2,469 Hanging 3, Other 1, Which of the following statements is consistent with the table? A) There is absolutely no evidence of a relation between the sex of the victim and the method of suicide used. B) More women commit suicide than men. C) Men display a greater tendency to use firearms to commit suicide than do women. D) The correlation between method of suicide and sex of the victim is clearly positive. E) The proportion of men who use poison to commit suicide is higher than the proportion of women who use poison to commit suicide. 50

6 36. X and Y are two categorical variables. The best way to determine whether there is a relationship between them is to A) compute the least-squares regression line between X and Y. B) draw a scatterplot of the X and Y values. C) make a two-way table of the X and Y values. D) calculate the correlation between X and Y. E) do all of the above. Use the following to answer questions 37 through 39: A business has two types of employees, managers and workers. Managers earn either $100,000 or $200,000 per year. Workers earn either $10,000 or $20,000 per year. The number of male and female managers at each salary level and the number of male and female workers at each salary level are given in the tables below. Managers Workers Male Female Male Female $100, $10, $200, $20, The proportion of male managers who make $200,000 per year is A) B) C) D) E) Use the following to answer questions 40 through 43: A review of voter registration records in a small town yielded the following table of the number of males and females registered as Democrat, Republican, or some other affiliation. Male Female Democrat Republican Other The proportion of males that are registered as Democrats is A) 300. B) 30. C) D) E)

7 Answer Key 1. B 2. C 3. A 4. D 5. E 6. D 7. D 8. D 9. A 10. C 11. E 12. B 13. E 14. E 15. E 16. B 17. C 18. C 19. D 20. B 21. C 22. D 23. E 24. C 25. C 26. A 27. C 28. D 29. A 30. B 31. E 32. B 33. A 34. C 35. A 36. C 37. C 38. E 39. E 40. C 41. D 42. C 43. E 52