Answers to Selected Exercises

Definitions of Statistics, Probability, and Key Terms – Practice

1. AIDS patients.

3. The average length of time (in months) AIDS patients live after treatment.

5. [latex]X[/latex] = the length of time (in months) AIDS patients live after treatment

Data, Sampling, and Variation in Data and Sampling – Practice

7. b

9. a

11.

  1. [latex]0.5242[/latex]
  2. [latex]0.03[/latex]%
  3. [latex]6.86[/latex]%
  4. [latex]\frac{823,088}{823,856}[/latex]
  5. quantitative discrete
  6. quantitative continuous
  7. In both years, underwater earthquakes produced massive tsunamis.

13. systematic

15. simple random

17. values for [latex]X[/latex], such as [latex]3[/latex], [latex]4[/latex], [latex]11[/latex], and so on

19. No, we do not have enough information to make such a claim.

21. Take a simple random sample from each group. One way is by assigning a number to each patient and using a random number generator to randomly select patients.

23. This would be convenience sampling and is not random.

25. Yes, the sample size of [latex]150[/latex] would be large enough to reflect a population of one school.

27. Even though the specific data support each researcher’s conclusions, the different results suggest that more data need to be collected before the researchers can reach a conclusion.

29. There is not enough information given to judge if either one is correct or incorrect.

31. The software program seems to work because the second study shows that more patients improve while using the software than not. Even though the difference is not as large as that in the first study, the results from the second study are likely more reliable and still show improvement.

33. Yes, because we cannot tell if the improvement was due to the software or the exercise; the data is confounded, and a reliable conclusion cannot be drawn. New studies should be performed.

35. No, even though the sample is large enough, the fact that the sample consists of volunteers makes it a self-selected sample, which is not reliable.

37. No, even though the sample is a large portion of the population, two responses are not enough to justify any conclusions. Because the population is so small, it would be better to include everyone in the population to get the most accurate data.

Frequency, Frequency Tables, and Levels of Measurement – Practice

39.

  1. ordinal
  2. interval
  3. nominal
  4. nominal
  5. ratio
  6. ordinal
  7. nominal
  8. interval
  9. ratio
  10. interval
  11. ratio
  12. ordinal

Experimental Design and Ethics – Practice

41.

  1. Inmates may not feel comfortable refusing participation, or may feel obligated to take advantage of the promised benefits. They may not feel truly free to refuse participation.
  2. Parents can provide consent on behalf of their children, but children are not competent to provide consent for themselves.
  3. All risks and benefits must be clearly outlined. Study participants must be informed of relevant aspects of the study in order to give appropriate consent.

Definitions of Statistics, Probability, and Key Terms – Homework

43.

  1. all children who take ski or snowboard lessons
  2. a group of these children
  3. the population mean age of children who take their first snowboard lesson
  4. the sample mean age of children who take their first snowboard lesson
  5. [latex]X[/latex] = the age of one child who takes his or her first ski or snowboard lesson
  6. values for [latex]X[/latex], such as [latex]3[/latex], [latex]7[/latex], and so on

45.

  1. the clients of the insurance companies
  2. a group of the clients
  3. the mean health costs of the clients
  4. the mean health costs of the sample
  5. [latex]X[/latex] = the health costs of one client
  6. values for [latex]X[/latex], such as [latex]34[/latex], [latex]9[/latex], [latex]82[/latex], and so on

47.

  1. all the clients of this counselor
  2. a group of clients of this marriage counselor
  3. the proportion of all her clients who stay married
  4. the proportion of the sample of the counselor’s clients who stay married
  5. [latex]X[/latex] = the number of couples who stay married
  6. yes, no

49.

  1. all people (maybe in a certain geographic area, such as the United States)
  2. a group of the people
  3. the proportion of all people who will buy the product
  4. the proportion of the sample who will buy the product
  5. [latex]X[/latex] = the number of people who will buy it
  6. buy, not buy

51. a

Data, Sampling, and Variation in Data and Sampling – Homework

53. quantitative discrete, [latex]150[/latex]

55. qualitative, Oakland A’s

57. quantitative discrete, [latex]11,234[/latex] students

59. qualitative, Crest

61. quantitative continuous, [latex]47.3[/latex] years

63. b

65.

  1. The survey was conducted using six similar flights. The survey would not be a true representation of the entire population of air travelers. Conducting the survey on a holiday weekend will not produce representative results.
  2. Conduct the survey during different times of the year. Conduct the survey using flights to and from various locations. Conduct the survey on different days of the week.

67. Answers will vary. Sample Answer: You could use a systematic sampling method. Stop the tenth person as they leave one of the buildings on campus at 9:50 in the morning. Then stop the tenth person as they leave a different building on campus at 1:50 in the afternoon.

69. Answers will vary. Sample Answer: Many people will not respond to mail surveys. If they do respond to the surveys, you can’t be sure who is responding. In addition, mailing lists can be incomplete.

71. b

73. convenience cluster stratified systematic simple random

75.

  1. qualitative
  2. quantitative discrete
  3. quantitative discrete
  4. qualitative

77. Causality: The fact that two variables are related does not guarantee that one variable is influencing the other. We cannot assume that crime rate impacts education level or that education level impacts crime rate.

Confounding: There are many factors that define a community other than education level and crime rate. Communities with high crime rates and high education levels may have other lurking variables that distinguish them from communities with lower crime rates and lower education levels. Because we cannot isolate these variables of interest, we cannot draw valid conclusions about the connection between education and crime. Possible lurking variables include police expenditures, unemployment levels, region, average age, and size.

79.

  1. Possible reasons: increased use of caller id, decreased use of landlines, increased use of private numbers, voice mail, privacy managers, hectic nature of personal schedules, decreased willingness to be interviewed
  2. When a large number of people refuse to participate, then the sample may not have the same characteristics of the population. Perhaps the majority of people willing to participate are doing so because they feel strongly about the subject of the survey.

Frequency, Frequency Tables, and Levels of Measurement

81.

# Flossing per Week Frequency Relative Frequency Cumulative Relative Frequency
[latex]0[/latex] [latex]27[/latex] [latex]0.4500[/latex] [latex]0.4500[/latex]
[latex]1[/latex] [latex]18[/latex] [latex]0.3000[/latex] [latex]0.7500[/latex]
[latex]3[/latex] [latex]11[/latex] [latex]0.1833[/latex] [latex]0.9333[/latex]
[latex]6[/latex] [latex]3[/latex] [latex]0.0500[/latex] [latex]0.9833[/latex]
[latex]7[/latex] [latex]1[/latex] [latex]0.0167[/latex] [latex]1[/latex]

b. [latex]5.00[/latex]%
c. [latex]93.33[/latex]%

83. The sum of the travel times is [latex]1,173.1[/latex]. Divide the sum by [latex]50[/latex] to calculate the mean value: [latex]23.462[/latex]. Because each state’s travel time was measured to the nearest tenth, round this calculation to the nearest hundredth: [latex]23.46[/latex].

85. b

Experimental Design and Ethics – Homework

87.
Explanatory variable: amount of sleep

Response variable: performance measured in assigned tasks

Treatments: normal sleep and [latex]27[/latex] hours of total sleep deprivation

Experimental Units: [latex]19[/latex] professional drivers

Lurking variables: none – all drivers participated in both treatments

Random assignment: treatments were assigned in random order; this eliminated the effect of any “learning” that may take place during the first experimental session

Control/Placebo: completing the experimental session under normal sleep conditions

Blinding: researchers evaluating subjects’ performance must not know which treatment is being applied at the time

89. You cannot assume that the numbers of complaints reflect the quality of the airlines. The airlines shown with the greatest number of complaints are the ones with the most passengers. You must consider the appropriateness of methods for presenting data; in this case displaying totals is misleading.

Extra Practice

91.  Answers will vary. Sample answer: The sample is not representative of the population of all college textbooks. Two reasons why it is not representative are that he only sampled seven subjects and he only investigated one textbook in each subject. There are several possible sources of bias in the study. The seven subjects that he investigated are all in mathematics and the sciences; there are many subjects in the humanities, social sciences, and other subject areas, (for example: literature, art, history, psychology, sociology, business) that he did not investigate at all. It may be that different subject areas exhibit different patterns of textbook availability, but his sample would not detect such results.

He also looked only at the most popular textbook in each of the subjects he investigated. The availability of the most popular textbooks may differ from the availability of other textbooks in one of two ways:

  • the most popular textbooks may be more readily available online, because more new copies are printed, and more students nationwide are selling back their used copies OR
  • the most popular textbooks may be harder to find available online, because more student demand exhausts the supply more quickly.

In reality, many college students do not use the most popular textbook in their subject, and this study gives no useful information about the situation for those less popular textbooks.

He could improve this study by:

  • expanding the selection of subjects he investigates so that it is more representative of all subjects studied by college students, and
  • expanding the selection of textbooks he investigates within each subject to include a mixed representation of both the most popular and less popular textbooks.