Section 3.4 Exercises

CLARIFYING THE CONCEPTS

Question 3.256

1. What does it mean for a -score to be positive? Negative? Zero? (p. 155)

3.4.1

Positive : the data value is above the mean. Negative : the data value is below the mean. of zero: the data value is equal to the mean.

Question 3.257

2. Explain in your own words what the 95th percentile of a data set means. (p. 160)

Question 3.258

3. Why doesn't it make sense for there to be a 120th percentile of a data set? (p. 159)

3.4.3

No more than 100% of the data values can be less than or equal to any value in the data set.

Question 3.259

4. Is it possible for the 1st percentile of a data set to equal the 99th percentile? Explain when this would happen. (p. 159)

Question 3.260

5. Explain the difference between a percentile and a percentile rank. (p. 161)

3.4.5

A percentile is a data value while a percentile rank is a percentage.

Question 3.261

6. True or false: The IQR is sensitive to the presence of outliers. (p. 165)

PRACTICING THE TECHNIQUES

image CHECK IT OUT!

To do Check out Topic
Exercises 7–18 Example 21 Calculate z-score, given data
Exercises 19–30 Example 22 Find data value, given z-score
Exercises 31–32 Example 23 Use z-scores to compare
different data sets
Exercises 33–44 Example 24 Identify outliers using
z-scores
Exercises 45–56 Example 26 Percentiles
Exercises 57–68 Example 27 Percentile rank
Exercises 69–76 Example 28 Quartiles
Exercises 77–78 Example 30 Interquartile range

Use the following information for Exercises 7–10. Facebook reports that the average number of friends per Facebook user is 130. Assume the standard deviation is 30. Calculate the -score for the indicated number of Facebook friends.

Question 3.262

7. 190 Facebook friends

3.4.7

2

Question 3.263

8. 145 Facebook friends

Question 3.264

9. 100 Facebook friends

3.4.9

–1.0

Question 3.265

10. Zero Facebook friends

For Exercises 11–14, use the following information. Social Strand Media reports that the mean amount of video uploaded to YouTube every minute by users around the world is 100 hours. Assume the standard deviation is 25 hours. Calculate the -score for the indicated number of hours of video uploaded to YouTube.

Question 3.266

11. 125 hours

3.4.11

1

Question 3.267

12. 50 hours

Question 3.268

13. 200 hours

3.4.13

4

Question 3.269

14. 87.5 hours

Use the following information for Exercises 15–18. Suppose the mean blood sugar level is 100 mg/dl (milligrams per deciliter), with a standard deviation of 10 mg/dl.

Question 3.270

15. Alyssa has a blood sugar level of 90 mg/dl. How many standard deviations is Alyssa's blood sugar level below the mean?

3.4.15

1

Question 3.271

16. Benjamin has a blood sugar level of 135 mg/dl. How many standard deviations is Benjamin's blood sugar level above the mean?

Question 3.272

17. Chelsea has a blood sugar level of 125 mg/dl.

  1. If we calculate Chelsea's -score, what is the scale?
  2. Calculate Chelsea's -score.
  3. Interpret her -score.

3.4.17

(a) The standard deviation, which is 10 mg/dl. (b) 2.5 (c) Chelsea's blood sugar level lies 2.5 standard deviations above the mean blood sugar level of 100 mg/dl.

Question 3.273

18. David has a blood sugar level of 85 mg/dl.

  1. Calculate David's -score.
  2. Interpret his -score.

For Exercises 19–22, use the following information. Facebook reports that the average number of friends per Facebook user is 130. Assume the standard deviation is 30. Find the number of Facebook friends represented by the following -scores.

Question 3.274

19.

3.4.19

100

Question 3.275

20.

Question 3.276

21.

3.4.21

130

Question 3.277

22.

Use the following information for Exercises 23–26. Social Strand Media reports that the mean amount of video uploaded to YouTube every minute by users around the world is 100 hours. Assume the standard deviation is 25 hours. Find the number of hours of YouTube video uploaded per minute for the following -scores.

Question 3.278

23.

3.4.23

150

Question 3.279

24.

Question 3.280

25.

3.4.25

87.5

Question 3.281

26.

Use the following information for Exercises 27–30. Suppose the mean blood sugar level is 100 mg/dl (milligrams per deciliter), with a standard deviation of 10 mg/dl. Find the blood sugar levels associated with the following -scores.

Question 3.282

27.

3.4.27

119.6

Question 3.283

28.

Question 3.284

29.

3.4.29

80.4

Question 3.285

30.

Question 3.286

31. Elizabeth's statistics class had a mean quiz score of 70 with a standard deviation of 15. Fiona's statistics class had a mean quiz score of 75 with a standard deviation of 5. Both Elizabeth and Fiona got an 85 on the quiz. Who did better relative to her class?

3.4.31

Elizabeth: ; Fiona: ; Fiona did better.

Question 3.287

32. Juan's business class had a mean quiz score of 60 with a standard deviation of 15. Luis's business class had a mean quiz score of 70 with a standard deviation of 5. Both Juan and Luis got a 75 on the quiz. Who did better relative to his class?

For Exercises 33–44, determine whether the data value represents an outlier, using the -score method.

Question 3.288

33. The 190 Facebook friends from Exercise 7

3.4.33

Moderately unusual

Question 3.289

34. The 145 Facebook friends from Exercise 8

169

Question 3.290

35. The 100 Facebook friends from Exercise 9.

3.4.35

Not unusual

Question 3.291

36. The zero Facebook friends from Exercise 10.

Question 3.292

37. The 125 hours of YouTube video from Exercise 11.

3.4.37

Not unusual

Question 3.293

38. The 50 hours of YouTube video from Exercise 12.

Question 3.294

39. The 200 hours of YouTube video from Exercise 13.

3.4.39

Outlier

Question 3.295

40. The 87.5 hours of YouTube video from Exercise 14.

Question 3.296

41. Alyssa's blood sugar level from Exercise 15.

3.4.41

Not unusual

Question 3.297

42. Benjamin's blood sugar level from Exercise 16.

Question 3.298

43. Chelsea's blood sugar level from Exercise 17.

3.4.43

Moderately unusual

Question 3.299

44. David's blood sugar level from Exercise 18.

Use the following data for Exercises 45–50. The variable is Highway MPG, which is the number of miles a vehicle can travel on a highway on one gallon of gas. The sample is taken from the Chapter 8 Case Study, Motor Vehicle Fuel Efficiency. Find the highway MPG represented by the indicated percentiles.

Vehicle Highway
MPG		 Vehicle Highway
MPG
Honda CR-V 30 Subaru Impreza 25
Nissan Pathfinder 26 Ford Mustang 26
Chevrolet
Chevy SS		 21 Cadillac ATS 31
Dodge Charger 27 Chevrolet Camaro 24
Jeep Compass 23 Ford Taurus 29
Lincoln MKT 25 Ford Expedition 20
Table 3.89: Source: www.fueleconomy.gov.

Question 3.300

45. 75th

3.4.45

28

Question 3.301

46. 5th

Question 3.302

47. 95th

3.4.47

31

Question 3.303

48. 90th

Question 3.304

49. 10th

3.4.49

21

Question 3.305

50. 99th

Use the following data for Exercises 51–56. Research has shown that the amount of sodium consumed in food has been associated with hypertension (high blood pressure). The table provides a list of 16 breakfast cereals, along with their sodium content, in milligrams per serving. Find the amount of sodium represented by the indicated percentiles.

Cereal Sodium Cereal Sodium
Apple Jacks 125 Grape Nuts Flakes 140
Cap'n Crunch 220 Kix 260
Cinnamon Toast
Crunch		 210 Life 150
Corn Flakes 290 Lucky Charms 180
Count Chocula 180 Raisin Bran 210
Cream of Wheat 80 Rice Chex 240
Fruit Loops 125 Special K 230
Fruity Pebbles 135 Total Whole Grain 200

Question 3.306

cereals

51. 75th

3.4.51

225

Question 3.307

cereals

52. 10th

Question 3.308

cereals

53. 90th

3.4.53

260

Question 3.309

cereals

54. 30th

Question 3.310

cereals

55. 5th

3.4.55

80

Question 3.311

cereals

56. 95th

Using the highway MPG data above, calculate the percentile rank for the indicated highway MPG in Exercises 57–62.

Question 3.312

57. 30

3.4.57

92nd percentile

Question 3.313

58. 31

Question 3.314

59. 20

3.4.59

8th percentile

Question 3.315

60. 25

Question 3.316

61. 27

3.4.61

75th percentile

Question 3.317

62. 29

Use the cereal sodium data above to calculate the percentile rank for the indicated amount of sodium (in mg) in Exercises 63–68.

Question 3.318

63. 80

3.4.63

6th percentile

Question 3.319

64. 290

Question 3.320

65. 260

3.4.65

94th percentile

Question 3.321

66. 125

Question 3.322

67. 230

3.4.67

81st percentile

Question 3.323

68. 220

Use the highway MPG data above for Exercises 69–72.

Question 3.324

69. Find Q1, the first quartile.

3.4.69

23.5

Question 3.325

70. Calculate Q2, the second quartile.

Question 3.326

71. Compute Q3, the third quartile.

3.4.71

28

Question 3.327

72. Find the median, and compare it to Q2.

For Exercises 73–76, use the cereal sodium data above.

Question 3.328

73. Find Q1, the first quartile.

3.4.73

137.5

Question 3.329

74. Calculate Q2, the second quartile.

Question 3.330

75. Compute Q3, the third quartile.

3.4.75

225

Question 3.331

76. Find the median, and compare it to Q2.

Question 3.332

77. Use your work in Exercises 69 and 71 to compute the IQR for the highway MPG data. What does this number mean?

3.4.77

4.5. The middle 50%, or middle half, of the highway MPG data ranged over 4.5 miles per gallon.

Question 3.333

78. Use your work in Exercises 73 and 75 to compute the IQR for the cereal sodium data. What does this number mean?

APPLYING THE CONCEPTS

Breakfast Calories. Refer to Table 23 for Exercises 79–86.

Table 3.91: Table 23 Calories in 12 breakfast cereals
Cereal Calories
Apple Jacks 110
Basic 4 130
Bran Chex 90
Bran Flakes 90
Cap'n Crunch 120
Cheerios 110
Cinnamon Toast Crunch 120
Cocoa Puffs 110
Corn Chex 110
Corn Flakes 100
Corn Pops 110
Count Chocula 110

170

Question 3.334

breakfastcal

79. Find the -scores for the calories for the following cereals:

  1. Corn Flakes
  2. Basic 4
  3. Bran Flakes
  4. Cap'n Crunch

3.4.79

(a) −0.79 (b) 1.79 (c) −1.65 (d) 0.93

Question 3.335

breakfastcal

80. Find the number of calories associated with the following -scores:

  1. 0
  2. 1
  3. −1
  4. 0.5

Question 3.336

breakfastcal

81. Determine whether any of the cereals is an outlier.

3.4.81

No outliers

Question 3.337

breakfastcal

82. Find the following percentiles:

  1. 25th
  2. 50th
  3. 75th
  4. 95th

Question 3.338

breakfastcal

83. Find the percentile rank for each of the following:

  1. 90 calories
  2. 120 calories
  3. 110 calories
  4. 100 calories

3.4.83

(a) 17% (b) 92% (c) 75% (d) 25%

Question 3.339

breakfastcal

84. Find the following:

  1. Q1
  2. Q2
  3. Q3
  4. IQR

Question 3.340

breakfastcal

85. Explain what the IQR value from Exercise 84 (d) means.

3.4.85

The middle 50%, or half, of the number of calories in 12 breakfast cereals ranges over 10 calories.

Question 3.341

breakfastcal

86. Suppose that a weight-control organization recommended eating breakfast cereals with the lowest 10% of calories.

  1. How many calories does this cutoff represent?
  2. Which cereals are recommended?

Dietary Supplements. Refer to Table 24 for Exercises 87–94. The table gives the number of American adults who have used the indicated “nonvitamin, nonmineral, natural products.”

Table 3.92: Table 24 Use of dietary supplements
Product Usage
(in millions)		 Product Usage
(in millions)
Echinacea 14.7 Ginger 3.8
Ginseng 8.8 Soy 3.5
Ginkgo biloba 7.7 Chamomile 3.1
Garlic 7.1 Bee pollen 2.8
Glucosamine 5.2 Kava kava 2.4
St. John's wort 4.4 Valerian 2.1
Peppermint 4.3 Saw palmetto 2.0
Fish oil 4.2
Table 3.92: Source: Centers for Disease Control and Prevention, Vital and Health Statistics.

Question 3.342

dietarysupp

87. Find the -scores for usage of the following products:

  1. Echinacea
  2. Saw palmetto
  3. Valerian
  4. Ginseng

3.4.87

(a) 2.87 (b) −0.91 (c) −0.89 (d) 1.11

Question 3.343

dietarysupp

88. Find the usage associated with each of the following -scores.

  1. 0
  2. 3
  3. −3
  4. 1

Question 3.344

dietarysupp

89. Identify any outliers in the data set.

3.4.89

Echinacea with 14.7 million users is moderately unusual.

Question 3.345

dietarysupp

90. Find the following percentiles:

  1. 10th
  2. 90th
  3. 5th
  4. 95th

Question 3.346

dietarysupp

91. Find the percentile rank for each of the following usages:

  1. 14.7 million
  2. 2.0 million
  3. 8.8 million
  4. 2.1 million

3.4.91

(a) 100% (b) 7% (c) 93% (d) 13%

Question 3.347

dietarysupp

92. Find the following:

  1. Q1
  2. Q2
  3. Q3
  4. IQR

Question 3.348

dietarysupp

93. Interpret the IQR value from Exercise 92 (d) so that a nonspecialist could understand it.

3.4.93

The middle 50%, or half, of the usage of dietary supplements ranges over 4.3 million.

Question 3.349

dietarysupp

94. Suppose an advertising agency is interested in the top 15% of supplements.

  1. What usage does this represent?
  2. Which supplements would be of interest?

Question 3.350

95. Expenditure per Pupil. The 5th percentile expenditure per pupil nationwide in 2005 was $6,381, the 50th percentile was $8,998, and the 95th percentile was $17,188.5

  1. Determine whether the distribution of expenditures is symmetric, left-skewed, or right-skewed.
  2. Would we expect the mean expenditure per pupil to be less than, equal to, or greater than $8,998? Explain.
  3. Draw a distribution curve that matches this information.

3.4.95

(a) Right-skewed (b) Greater than, since the distribution is right-skewed. (c) Answers will vary.

For Exercises 96–99, consider whether the scenarios are possible. If it is possible, then clearly describe what the data set would look like. If it is not possible, explain why.

Question 3.351

96. A scenario where the first and second quartiles of a data set are equal

Question 3.352

97. A scenario where the mean of a data set is larger than Q3

3.4.97

Right-skewed with a few values much larger than the rest; median line of box plot closer to the line for Q3 than the line for Q1.

Question 3.353

98. A scenario where the median of a data set is smaller than Q1

Question 3.354

99. A scenario where the IQR is negative

3.4.99

Not possible. Q1, the 25th percentile, will always be less than or equal to Q3, the 75th percentile. Thus the IQR = Q3 – Q1 is always greater than or equal to zero.

Twitter Followers. Are you on Twitter? How many Twitter followers do you have? Jon Bruner from O'Reilly Media reported6 the information in Table 25. For selected percentiles, Table 25 shows the number of Twitter followers that each percentile represents. For example, the 50th percentile is 61 Twitter followers. Use Table 25 for Exercises 100–105. Twitter reports that there are 400 million active Twitter users worldwide who actually tweet (post messages).

Question 3.355

100. What percent of Twitter accounts have three or fewer followers?

Question 3.356

101. What percent of Twitter accounts have between three and 19 followers?

3.4.101

20%

Question 3.357

102. How many active Twitter users have between 2,991 and 24,964 followers?

Question 3.358

103. How many active Twitter users have more than 24,964 followers?

3.4.103

0.1%

Question 3.359

104. Is it possible using Table 25 to find what percent of Twitter accounts have 100 or fewer followers? How might we estimate it?

171

Question 3.360

twitterpercentile

105. What is the percentile rank of 819 Twitter followers?

Table 3.93: Table 25 Table of percentiles of Twitter followers
Percentile Number of
Twitter followers
10 3
20 9
30 19
40 36
50 61
60 98
70 154
80 246
90 458
95 819
99 2,991
99.9 24,964

3.4.105

95th percentile

WORKING WITH LARGE DATA SETS

image Financial Experts versus the Darts. This set of exercises examines measures of relative position using the Darts data set from the Chapter 3 Case Study. Open the Darts data set. Use technology to do Exercises 106–112.

Question 3.361

darts

106. Find the median for each of the professionals, the Darts, and the DJIA. To those who would say that using darts is better, what do the relative values of the medians say?

Question 3.362

darts

107. Calculate the -score for the median for each of the three groups. What does the sign of the -score for each group indicate about the relationship between the median and the mean?

3.4.107

image

image

image

For the pros and the darts data the median is below the mean. For the DJIA data the median is above the mean.

Question 3.363

darts

108. For each group, compute the stock price change represented by the following -scores.

  1. 2
  2. −2

Question 3.364

darts

109. For each group, what percentage of the data lies between the values you found in the previous exercise?

3.4.109

About 95%

Question 3.365

darts

110. For each group, calculate the first quartile and the third quartile.

Question 3.366

darts

111. Calculate and interpret the IQR for each group.

3.4.111

Pros: IQR = 32.4

For the professionals, the middle 50%, or middle half, of the change in stocks data ranged over 32.4%.

Darts: IQR = 20.60

For using darts, the middle 50%, or middle half, of the change in stocks data ranged over 20.60%.

DJIA: IQR = 11.65

For the DJIA, the middle 50%, or middle half, of the change in stocks data ranged over 11.65%.

Question 3.367

darts

112. For each group, compare the IQR with the range and standard deviation. Do all these measures of spread agree regarding which group has the least variability? The most variability?

BRINGING IT ALL TOGETHER

Pedestrian Fatalities. The Department of Transportation releases statistics on the number of pedestrians killed by vehicles in the United States. The following table contains the pedestrian fatality rate (number of fatalities per 100,000 population) for 2013 for six states. Use this information for Exercises 113–120.

State Pedestrian fatality rate
Nebraska 0.38
Ohio 0.90
Tennessee 1.25
Texas 1.64
California 1.66
Florida 2.57
Table 3.94: Source: U.S. Department of Transportation: www-fars.nhtsa.dot.gov/Main/index.aspx.

Question 3.368

pedestrians

113. Find the -scores for the pedestrian fatality rate for the following states:

  1. Ohio
  2. Texas
  3. Florida

3.4.113

(a) −0.6673 (b) 0.3203 (c) 1.5615

Question 3.369

pedestrians

114. Find the pedestrian fatality rates indicated by the following -scores:

  1. −2
  2. 1
  3. 3

Question 3.370

pedestrians

115. Determine whether the pedestrian fatality rates for any of the states represents an outlier.

3.4.115

No outliers

Question 3.371

pedestrians

116. If the pedestrian fatality rate for Nebraska and Florida do not represent outliers, explain why we need not check whether the pedestrian fatality rates for the other states are outliers.

Question 3.372

pedestrians

117. Find the following percentiles:

  1. 50th
  2. 75th
  3. 25th

3.4.117

(a) 1.445 fatalities per 100,000 people (b) 1.66 fatalities per 100,000 people (c) 0.90 fatality per 100,000 people

Question 3.373

pedestrians

118. Calculate the percentile rank for the following pedestrian fatality rates:

  1. 0.38
  2. 1.25
  3. 2.57

Question 3.374

pedestrians

119. Find the following:

  1. Q1
  2. Q2
  3. Q3
  4. IQR

3.4.119

(a) 0.90 fatality per 100,000 people (b) 1.445 fatalities per 100,000 people (c) 1.66 fatalities per 100,000 people (d) 0.76 fatality per 100,000 people

Question 3.375

pedestrians

120. Interpret the IQR value from Exercise 119 (d).