13
CHAPTER 2
GRAPHICAL AND TABULAR DESCRIPTIVE TECHNIQUES
SECTIONS 1
MULTIPLE CHOICE QUESTIONS In the following multiple-choice questions, please circle the correct answer.
1. Which of the following statements is false?
a. All calculations are permitted on interval data b. All calculations are permitted on nominal data
c. The most important aspect of ordinal data is the order of the data values
d. The only permissible calculations on ordinal data are ones involving a ranking process
ANSWER: b
2. The average number of units earned per semester by college students is suspected to be rising. A researcher at Boston College wishes to estimate the number of units earned by students during the spring semester of 2004 at Boston. To do so, he randomly selects 250 student transcripts and records the number of units each student earned in the spring term of 2004. The variable of interest to the researcher is the
a. number of students enrolled at Boston College during the spring term of 2004 b. average indebtedness of Boston College students enrolled in the spring c. age of Boston College students enrolled in the spring
d. number of units earned by Boston College students during the spring term of 2004 ANSWER: d
3. The classification of student major (accounting, economics, management, marketing, other) is an example of
a. a categorical random variable.
b. a discrete random variable c. a continuous random variable d. a parameter.
ANSWER: a
4. A study is under way in national forest to determine the adult height of pine trees.
Specifically, the study is attempting to determine what factors aid a tree in reaching heights greater than 50 feet tall. It is estimated that the forest contains 32,000 adult pines.
The study involves collecting heights from 500 randomly selected adult pine trees and analyzing the results. The variable of interest in the study is the
a. age of a pine tree in the national forest.
b. height of a pine tree in the national forest.
c. number of pine trees in the national forest.
d. species of trees in the national forest.
ANSWER: b
5. The classification of student class designation (freshman, sophomore, junior, senior) is an example of
a. a categorical random variable.
b. a discrete random variable.
c. a continuous random variable.
d. a parameter.
ANSWER: a
6. Most analysts focus on the cost of tuition as the way to measure the cost of a college education. But incidentals, such as textbook costs, are rarely considered. A researcher at Ferris State University wishes to estimate the textbook costs of first-year students at Ferris. To do so, he monitored the textbook cost of 200 first-year students and found that their average textbook cost was $275 per semester. The variable of interest to the researcher is the
a. textbook cost of first-year Ferris State University students.
b. year in school of Ferris State University students.
c. age of Ferris State University students.
d. cost of incidental expenses of Ferris State University students.
ANSWER: a
7. The manager of the customer service division of a major consumer electronics company is interested in determining whether the customers who have purchased a videocassette recorder made by the company over the past 12 months are satisfied with their products.
The possible responses to the question “Are you happy, indifferent, or unhappy with the performance per dollar spent on the videocassette recorder?” are values from a
a. discrete numerical random variable.
b. continuous numerical random variable c. categorical random variable.
d. parameter.
ANSWER: c
TRUE / FALSE QUESTIONS
8. There are actually four types of data: nominal, ordinal, interval, and ratio. However, for statistical purposes, there is no difference between interval and ratio data, and the authors of your book combine the two types.
ANSWER: T
9. Quantitative variables usually represent membership in groups or categories.
ANSWER: F
10. Interval data, such as heights, weights, and incomes, are also referred to as quantitative or numerical data.
ANSWER: T
11. Nominal data are also called qualitative or categorical data.
ANSWER: T
12. ATP singles rankings for tennis players is an example of an interval scale.
ANSWER: F
13. Interval data may be treated as ordinal or nominal.
ANSWER: T
14. Nominal data may be treated as ordinal or interval ANSWER: F
15. Professor Hogg graduated from the University of Iowa with a code value = 2 while Professor Maas graduated from Michigan State University with a code value = 1. The scale of measurement likely represented by this information is interval.
ANSWER: F
16. Ordinal data may be treated as interval but not as nominal.
ANSWER: F
17. A variable is some characteristic of a population, while data are the observed values of a variable based on a sample.
ANSWER: F
18. An automobile insurance agent believes that company “A” is more reliable than company
“B”. The scale of measurement that this information represents is the ordinal scale.
ANSWER: T
STATISTICAL CONCEPTS & APPLIED QUESTIONS
19. The Dean of Students conducted a survey on campus. SAT score in mathematics is an example of a __________, or __________ variable.
ANSWER:
quantitative, numerical
20. Provide one example for nominal, ordinal, and interval data.
ANSWER:
Nominal data example: Political party affiliation for voters recorded using the code: 1
= Democrat, 2 = Republican, and 3 = Independent.
Ordinal data example: Response to market research survey measured on the Likert scale using the code: 1 = Strongly agree, 2 = Agree, 3 = Neutral, 4 = Disagree, and 5 = Strongly disagree.
Interval data example: Temperature in tennis courts during the US Open
21. The dean of students conducted a survey on campus. The gender of the student is an example of a __________, or __________ variable.
ANSWER:
categorical, qualitative
22. For each of the following examples, identify the data type as nominal, ordinal, or interval.
a. The letter grades received by students in a computer science class b. The number of students in a statistics course
c. The starting salaries of newly Ph.D. graduates from a statistics program
d. The size of fries (small, medium, large) ordered by a sample of Burger King customers
e. The college you are enrolled in (Arts and science, Business, Education, etc.)
ANSWER:
a. Ordinal b. Interval c. Interval d. Ordinal e. Nominal
23. The Dean of Students conducted a survey on campus. Class designation (Freshman, Sophomore, Junior, and Senior) is an example of a __________, or __________ variable.
ANSWER:
categorical, qualitative
24. Most colleges admit students based on their achievements in a number of different areas.
The grade obtained in senior level English course (A, B, C, D, or F) is an example of a __________, or __________ variable.
ANSWER:
categorical, qualitative
25. At the end of an escorted motor coach vacation, the tour operator asks the vacationers to respond to the questions listed below. For each question, determine whether the possible responses are interval, nominal, or ordinal.
a. How many escorted vacations have you taken prior to this one?
b. Do you feel that the stay in New York was sufficiently long?
c. Which of the following features of the hotel in New York did you find most attractive: location, facilities, room size, or price?
d. What is the maximum number of hours per day that you would like to spend traveling?
e. Would your overall rating of this tour be excellent, good, fair, or poor?
ANSWER:
a. Interval b. Nominal c. Nominal d. Interval e. Ordinal
26. For each of the following, indicate whether the variable of interest would be nominal or interval.
a. Whether you are a US citizen b. Your marital status
c. Number of cars in a parking lot
d. Amount of time you spend per week on your homework
e. Lily’s travel time from her dorm to the student union at the university of Iowa f. Heidi’s favorite brand of tennis balls.
ANSWER:
a. Nominal b. Nominal c. Interval d. Interval e. Interval f. Nominal
27. In purchasing a used automobile, there are a number of variables to consider. The age of the car is an example of a __________, or __________ variable.
ANSWER:
quantitative, numerical
28. In purchasing an automobile, there are a number of variables to consider. The body style of the car (sedan, coupe, wagon, etc.) is an example of a __________, or __________
variable.
ANSWER:
categorical, qualitative
29. Before leaving a particular restaurant, customers are asked to respond to the questions listed below. For each question, determine whether the possible responses are interval, nominal, or ordinal.
a. What is the approximate distance of the restaurant from your residence?
b. Have you eaten at the restaurant previously?
c. On how many occasions have you eaten at the restaurant previously?
d. Which of the following attributes of the restaurant do you find most attractive:
service, prices, quality of the food, or varied menu?
e. Would your overall rating of the restaurant be excellent, good, fair, or poor?
ANSWER:
a. Interval b. Nominal c. Interval d. Nominal e. Ordinal
SECTION 2
MULTIPLE CHOICE QUESTIONS In the following multiple-choice questions, please circle the correct answer.
30. The best type of chart for comparing two sets of categorical data is a a. line chart
b. pie chart c. histogram d. bar chart ANSWER: d
31. Which of the following statements about pie charts is false?
a. Pie charts are graphical representations of the relative frequency distribution
b. Pie charts are usually used to display the relative sizes of categories for interval data.
c. Pie charts always have the shape of a circle
d. Area of each slice of a pie chart is the proportion of the corresponding category of the frequency distribution of a categorical variable
ANSWER: b
32. The two graphical techniques we usually use to present nominal data are a. bar chart and histogram
b. pie chart and ogive c. bar chart and pie chart d. histogram and ogive ANSWER: c
33. Which of the following statements is false?
a. A bar chart is similar to a histogram
b. A pie chart is a circle subdivided into slices whose areas are proportional to the frequencies
c. Pie charts emphasize the frequency of occurrences of each category in a frequency distribution
d. None of the above ANSWER: c
34. Which of the following statements is true?
a. Bar charts focus the attention on the frequency of the occurrences of the categories b. A bar chart is created by drawing a rectangle representing each category
c. The height of each rectangle in a bar chart represents the frequency for a particular category
d. All of the above ANSWER: d
TRUE / FALSE QUESTIONS
35. A bar chart is used to represent interval data.
ANSWER: F
36. One of the advantages of a pie chart is that it clearly shows that the total of all the categories of the pie adds to 100%.
ANSWER: T
37. The bar chart is preferred to the pie chart, because the human eye can more accurately judge length comparisons against a fixed scale (as in a bar chart) than angular measures (as in a pie chart).
ANSWER: T
38. Bar and pie charts are graphical techniques for nominal data. The former focus the attention on the frequency of the occurrences of the categories, and the later emphasize the proportion of occurrences of each category.
ANSWER: T
39. Bar and pie charts are two graphical techniques that can be used to represent nominal data.
ANSWER: T
40. A bar chart is similar to a histogram in the sense that the bases of the rectangles are arbitrary intervals whose centers are the midpoints of the intervals.
ANSWER: F
41. If we wish to emphasize the relative frequencies for nominal data, we draw a histogram instead of drawing a bar chart.
ANSWER: F
42. Pie and bar charts are used widely in newspapers, magazines, and business and government reports.
ANSWER: T
43. The size of each slice in a pie chart is proportional to the percentage corresponding to that category.
ANSWER: T
44. A category that contains 30% of the observations is represented by a slice of a pie chart that contains 100 degrees.
ANSWER: F
STATISTICAL CONCEPTS & APPLIED QUESTIONS
45. Identify the type of data for which each of the following graphs is appropriate.
a. Pie chart b. Bar chart
ANSWER:
a. Nominal b. Nominal
46. Voters participating in a recent election exit poll in Minnesota were asked to state their political party affiliation. Coding the data 1 for Republican, 2 for Democrat, and 3 for Independent, the data collected were as follows: 3, 1, 2, 3, 1, 3, 3, 2, 1, 3, 3, 2, 1, 1, 3, 2, 3, 1, 3, 2, 3, 2, 1, 1, and 3. Construct a frequency bar graph.
ANSWER:
FOR QUESTIONS 47 AND 48, USE THE FOLLOWING NARRATIVE:
Narrative: Car Dealers
Car buyers were asked to indicate the car dealer they believed offered the best overall service.
The four choices were Carriage Motors (C), Marco Chrysler (M), Triangle Auto (T), and University Chevrolet (U). The following data were obtained:
T C C C U C M T C U U M C M T C M M C M T C C T U M M C C T T U C U T M M C U T
0 2 4 6 8 10 12
Republican Democrat Independent
Frequency
47. {Car Dealers Narrative} Construct a frequency bar chart.
ANSWER:
48. {Car Dealers Narrative} Construct a pie chart. Which car dealer offered the best overall service?
ANSWER:
It seems that Carriage Motors offered the best overall service.
0 5 10 15
C M T U
Dealership
Frequency
C 35.0%
M 25.0%
T 22.5%
U 17.5%
49. Given the following five categories and the number of times each occurs, draw a pie chart and a bar chart.
Category 1 2 3 4 5
Frequency 15 30 40 25 20
ANSWER:
Bar Chart
0 5 10 15 20 25 30 35 40 45
1 2 3 4 5
Category
Frequency
Pie Chart
1 12%
2 23%
3 31%
4 19%
5 15%
FOR QUESTIONS 50 AND 51, USE THE FOLLOWING NARRATIVE:
Narrative: Business School Graduates
The frequency distribution for a sample of 200 business school graduates is shown in the following table.
50. {Business School Graduates Narrative} Draw a pie chart of the number of graduates.
ANSWER:
Major of Graduates Number of graduates
Accounting 58
Finance 42
Management 38
Marketing 52
Other 10
Accounting 29.0%
Finance 21.0%
Management 19.0%
Marketing 26.0%
Other 5.0%
51. {Business School Graduates Narrative} Draw a frequency bar chart.
ANSWER:
0 10 20 30 40 50 60 70
Accountin g
Finance
Management
Marketing
Other
Major
Frequency
SECTION 3
MULTIPLE CHOICE QUESTIONS In the following multiple-choice questions, please circle the correct answer.
52. The most appropriate type of chart for determining the number of observations at or below a specific value is:
a. a histogram b. a pie chart
c. a time-series chart
d. a cumulative frequency ogive ANSWER: d
53. In general, incomes of employees in large firms tend to be a. positively skewed
b. negatively skewed c. symmetric
d. All of the above ANSWER: a
54. The total area of the bars in a relative frequency histogram:
a. depends on the sample size b. depends on the number of bars c. depends on the width of each bar d. depends on the height of each bar ANSWER: c
55. Which of the following statements is false?
a. A frequency distribution counts the number of observations that fall into each of a series on intervals, called classes that cover the complete range of observations.
b. The intervals in a frequency distribution may overlap to ensure that each observation is assigned to an interval
c. Although the frequency distribution provides information about how the numbers in the data set are distributed, the information is more easily understood and imparted by drawing a histogram
d. The number of class intervals we select in a frequency distribution depends entirely on the number of observations in the data set
ANSWER: b
56. The total area of the five bars in a relative frequency histogram for which the width of each bar is four units is:
a. 5 b. 4 c. 9 d. 1
ANSWER: b
57. The relative frequency of a class is computed by
a. dividing the frequency of the class by the number of classes b. dividing the frequency of the class by the class width
c. dividing the frequency of the class by the total number of observations in the data set d. subtracting the lower limit of the class from the upper limit and multiplying the
difference by the number of classes ANSWER: c
58. A modal class is the class that includes a. the largest number of observations b. the smallest number of observations c. the largest observation in the data set d. the smallest observation in the data set ANSWER: a
59. The sum of the relative frequencies for all classes will always equal a. the number of classes
b. the class width
c. the total number of observations in the data set d. one
ANSWER: d
60. When ogives or histograms are constructed, which axis must show the true zero or
“origin”?
a. The horizontal axis.
b. The vertical axis.
c. Both the horizontal and vertical axes.
d. Neither the horizontal nor the vertical axis.
ANSWER: b
61. The width of each bar in a histogram corresponds to the a. differences between the lower and upper limits of the class.
b. number of observations in each class.
c. midpoint of each class
d. frequency of observations in each class.
ANSWER: a
62. The most important and commonly graphical presentation of interval data is a a. bar chart
b. histogram c. pie chart
d. cumulative frequency distribution ANSWER: b
63. According to Sturges’ rule, the ideal number of class intervals in a frequency distribution of n = 150 data equals about
a. 8 b. 15 c. 20 d. 28
ANSWER: a
64. According to Sturges’ rule, the ideal number of class intervals in a frequency distribution equals
a. 5 b. 15
c. 3.3 + log (n), where n is the size of the data set.
d. 1 + 3.3 log (n), where n is the size of the data set.
ANSWER: d
65. How many classes should a histogram contain if the number of observations is 250?
a. 5, 6, or 7 b. 7, 8, or 9 c. 9 or 10 d. 10 or 11 ANSWER: c
66. How many classes should a frequency distribution contain if the number of observations is 45?
a. 5, 6, or 7 b. 7, 8, or 9 c. 9 or 10 d. 10 or 11 ANSWER: a
67. Sturge’s formula recommends that the number of class intervals to construct a frequency distribution or draw a histogram using a data set with n observations is determined by:
a. log(n) b. 3.3 log(n) c. 1 + 3.3 log(n) d. 2 – 3.3 log(n) ANSWER: c
68. Which of the following statements about number of modal classes is false?
a. A unimodal histogram is one with a single peak
b. A bimodal histogram is one with two peaks, not necessarily equal in height c. A bimodal histogram is one with two peaks equal in height
d. None of the above ANSWER: c
69. Which of the following statements about shapes of histograms is true?
a. A histogram is said to be symmetric if, when we draw a vertical line down the center of the histogram, the two sides are identical in shape and size
b. A positively skewed histogram is one with a long tail extending to the right c. A negatively skewed histogram is one with a long tail extending to the left d. All of the above
ANSWER: d
TRUE / FALSE QUESTIONS
70. A relative frequency distribution describes the proportion of data values that fall within each class, and may be presented in a histogram form.
ANSWER: T
71. A relative frequency distribution describes the proportion of data values that fall within each category.
ANSWER: T
72. The stem-and-leaf display reveals far more information relative to individual values than does the histogram.
ANSWER: F
73. Individual observations within each class may be found in a frequency distribution.
ANSWER: F
74. The following stem-and-leaf output has been generated by statistical software. The median of this data is 26.
Stem-and-leaf of C2 N = 75 Leaf Unit = 10
9 0 000112333 14 0 56899 21 1 0000123 26 1 66699 33 2 3334445 (8) 2 66677888 34 3 0023344 27 3 56669999 19 4 000122233 10 4 5556667799
ANSWER: F
75. A cumulative frequency distribution lists the number of observations that are within or below each of the classes.
ANSWER: T
76. The following stem-and-leaf output has been generated by statistical software. This data has a negative mode.
Stem-and-leaf of C2 N = 75 Leaf Unit = 0.01
1 -2 6 2 -2 0 5 -1 555 8 -1 420
22 -0 99999887777665 36 -0 44322111111000 (14) 0 01122233333344 25 0 66678889999 14 1 0022222334 4 1 56
2 2 03 ANSWER: T
77. Compared to the frequency distribution, the stem-and-leaf display provides more details, since it can describe the individual data values as well as show how many are in each group, or stem.
ANSWER: T
78. A histogram represents nominal data.
ANSWER: F
79. In the term “frequency distribution,” frequency refers to the number of data values falling within each class.
ANSWER: T
80. The class interval in a frequency distribution is the number of data values falling within each class.
ANSWER: F
81. The largest value in a set of data is 140, and the lowest value is 70. If the resulting frequency distribution is to have five classes of equal width, the class width will be 14.
ANSWER: T
82. A stem-and-leaf display describes two - digit integers between 20 and 70. For one of the classes displayed, the row appears as 4|256. The numerical values being described are 24, 54, and 64.
ANSWER: F
83. The following “character histogram” has been generated by statistical software. The median class is 150.
Histogram of C1 N = 75
Midpoint Count -150 1 * -100 1 * -50 3 ***
0 2 **
50 7 *******
100 12 ************
150 18 ******************
200 20 ********************
250 5 *****
300 5 *****
350 1 *
ANSWER: T
84. The following stem-and-leaf output has been generated by statistical software. This data set has a mean that is negative, and there is no modal class.
Stem-and-leaf of C2 N = 10 Leaf Unit = 0.10
2 - 1 53 4 - 0 97 (2) - 0 65 4 0 3 3 0 6 2 1 3 1 1 8
ANSWER: T
85. A frequency distribution is a listing of the individual observations arranged in ascending or descending order.
ANSWER: F
86. When a distribution has more values to the left and tails to the right, it is skewed negatively.
ANSWER: F
87. A histogram is said to be symmetric if, when we draw a vertical line down the center of the histogram, the two sides are identical in shape and size.
ANSWER: T
88. A skewed histogram is one with a long tail extending either to the right or left. The former is called negatively skewed, and the later is called positively skewed.
ANSWER: F
89. A bimodal histogram is one with two or more peaks equal in height.
ANSWER: F
90. A cumulative frequency distribution when presented in graphic form is called an ogive.
ANSWER: T
91. When a distribution has more values to the right and tails to the left, we say it is skewed positively.
ANSWER: F
92. The sum of relative frequencies in a distribution always equals 1.
ANSWER: T
93. The stem-and-leaf display is often superior to the frequency distribution in that is maintains the original values for the further analysis.
ANSWER: T
94. The sum of cumulative frequencies in a distribution always equals 1.
ANSWER: F
95. If the values of the sixth and seventh class in a cumulative frequency distribution are the same, we know that there are no observations in the seventh class.
ANSWER: T
96. The larger the number of observations in a numerical data set, the larger the number of class intervals needed for a frequency distribution.
ANSWER: T
97. The original data values cannot be assessed once they are grouped into a frequency distribution.
ANSWER: T
98. A research analyst was directed to arrange raw data collected on the yield of wheat, ranging from 40 to 90 bushels per acre, in a frequency distribution. He should choose 40 as the class interval width.
ANSWER: F
99. The relative frequency of a class is the frequency of that class divided by the total number of classes.
ANSWER: F
100. Ogives are plotted at the midpoints of the class intervals.
ANSWER: F
101. Sturge’s formula recommends that the number of class intervals needed to draw a histogram using a data set with 200 observations is 12.79 which we round to 13.
ANSWER: F
102. A modal class is the class with the largest number of observations.
ANSWER: T
103. Incomes of employees in large firms tend to be negatively skewed, because there is a large number of relatively low – paid workers and a small number of well – paid executives.
ANSWER: F
104. The time taken by students to write exams is frequently positively skewed because few students hand in their exams early; most prefer to reread their papers and hand them in near the end of the scheduled test period.
ANSWER: F
105. A frequency distribution counts the number of observations that fall into each of a series of intervals, called classes that cover the range of observations.
ANSWER: T
106. One of the drawbacks of the histogram is that we lose potentially useful information by classifying the observations and sacrificing whatever information was contained in the actual observations.
ANSWER: T
107. The histogram is usually preferred over the stem – and – leaf display.
ANSWER: F
108. The stem – and – leaf display’s advantage over the histogram is that we can see the actual observations rather than observations classified into different classes.
ANSWER: T
STATISTICAL CONCEPTS & APPLIED QUESTIONS
109. Identify the type of data for which a Histogram is appropriate.
ANSWER:
Interval
110. The total area under a relative frequency histogram for which the width of each class is ten units is _________.
ANSWER:
10
111. Voters participating in a recent election exit poll in Minnesota were asked to state their political party affiliation. Coding the data 1 for Republican, 2 for Democrat, and 3 for Independent, the data collected were as follows: 3, 1, 2, 3, 1, 3, 3, 2, 1, 3, 3, 2, 1, 1, 3, 2, 3, 1, 3, 2, 3, 2, 1, 1, and 3. Develop a frequency distribution and a proportion distribution for the data. What does the data suggest about the strength of the political parties in Minnesota?
ANSWER:
Party Frequency Proportion
Republican 8 0.32
Democrat 6 0.24
Independent 11 0.44
Independent in Minnesota is stronger than Republican and Democrat parties
FOR QUESTIONS 112 THROUGH 118, USE THE FOLLOWING NARRATIVE:
Narrative: Salespersons’ Ages
The ages of a sample of 25 salespersons are as follows:
47 21 37 53 28 40 30 32 34 26 34 24 24 35 45 38 35 28 43 45 30 45 31 41 56
112. {Salespersons’ Ages Narrative} Draw a histogram with four classes.
ANSWER:
Histogram
0 2 4 6 8 10 12
26 38 50 62
Age
Frequency
113. {Salespersons’ Ages Narrative} Draw a histogram with six classes.
ANSWER:
Histogram
0 1 2 3 4 5 6 7 8
23.5 30.5 37.5 44.5 51.5 58.5
Age
Frequency
114. {Salespersons’ Ages Narrative} Draw a stem and leaf display.
ANSWER:
115. {Salesperson’s Ages Narrative} Construct an ogive for the data.
ANSWER:
116. {Salesperson’s Ages Narrative} Estimate the proportion of salespersons who are less than 30 years of age.
ANSWER:
0.24
117. {Salesperson’s Ages Narrative} Estimate the proportion of salespersons who are more than 40 years of age.
ANSWER:
1-0.64 = 0.36
118. {Salesperson’s Ages Narrative} Estimate the proportion of salespersons who are between 40 and 50 years of age.
ANSWER:
0.92 - 0.64 = 0.28
STEM LEAF 2 1 4 4 6 8 8 3 0 0 1 2 4 4 5 5 7 8 4 0 1 3 5 5 5 7 5 3 6
0.00
0.24
0.64
0.92
1.00
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
20 30 40 50 60
Ages (years)
Cumulative Relative Frequency
FOR QUESTIONS 119 THROUGH 121, USE THE FOLLOWING NARRATIVE:
Narrative: Defective Items
The number of defective items produced by a machine and recorded for the last 25 days are as follows: 19, 6, 15, 20, 17, 16, 17, 12, 15, 29, 23, 17, 7, 10, 14, 14, 27, 22, 8, 5, 23, 19, 9, 28, and 5.
119. {Defective Items Narrative} What is the relationship between the total area under the histogram you have constructed and the relative frequencies of observations?
ANSWER:
Class Limits Frequency Relative Frequency
5 up to 10* 6 0.24
10 up to 15 4 0.16
15 up to 20 8 0.32
20 up to 25 4 0.16
25 up to 30 3 0.12
Total 25 1.00
*Class contains observations up to but not including 10. The other classes are defined similarly. This notion is used throughout the chapter.
120. {Defective Items Narrative} Construct a relative frequency histogram for these data.
ANSWER:
Note that the numbers that appear along the horizontal axis represent the upper limits of the class intervals even though they appear in the center of the classes
0 0.1 0.2 0.3 0.4
10 15 20 25 30
Defective Items
Relative Frequency
121. {Defective Items Narrative} Construct a frequency distribution and relative frequency distribution for these data. Use five class intervals, with the lower boundary of the first class being five items.
ANSWER:
Note that the area under the histogram between two values is five times the relative
frequency of observations that fall between those two values (since 5 is the width of each class). Hence, the total area under the histogram is therefore equal to 5.
FOR QUESTIONS 122 THROUGH 129, USE THE FOLLOWING NARRATIVE:
Narrative: Exam Grades
The grades on a calculus exam for a sample of 40 students are as follows:
63 74 42 65 51 54 36 56 68 57 62 64 76 67 79 61 81 77 59 38 84 68 71 94 71 86 69 75 91 55 48 82 83 54 79 62 68 58 41 47
122. {Exam Grades Narrative} Construct a stem and leaf display for these data.
ANSWER:
Stem Leaf
3 68
4 1278
5 14456789 6 12234578889 7 11456799
8 12346
9 14
123. {Exam Grades Narrative} Construct a frequency distribution and relative frequency distribution for these data, using seven class intervals.
ANSWER:
Class Limits Frequency Relative Frequency
30 up to 40 2 0.050
40 up to 50 4 0.100
50 up to 60 8 0.200
60 up to 70 11 0.275
70 up to 80 8 0.200
80 up to 90 5 0.125
90 up to 100 2 0.050
Total 40 1.00
124. {Exam Grade Narrative} Construct a relative frequency histogram for these data.
ANSWER:
125. {Exam Grades Narrative} Describe briefly what the histogram and the stem and leaf display tell you about the data
ANSWER:
The distribution of the data is symmetrical and bell-shaped, with 67.5% of the observations between 50 and 80.
126. {Exam Grades Narrative} Construct a cumulative frequency and a cumulative relative frequency distribution
ANSWER:
Classes Cumulative Frequency
Cumulative Relative Frequency
up to 40 2 0.050
up to 50 6 0.150
up to 60 14 0.350
up to 70 25 0.625
up to 80 33 0.825
up to 90 38 0.950
up to 100 40 1.00
127. {Exam Grades Narrative} What proportion of the grades is less than 60?
ANSWER:
0.35
0 0.05 0.1 0.15 0.2 0.25 0.3
40 50 60 70 80 90 100
Grade
Relative Frequency
128. {Exam Grades Narrative} What proportion of the grades is more than 70?
ANSWER:
1 - 0.625 = 0.375
129. {Exam Grades Narrative) Construct an ogive, and estimate the proportion of grades that are between 80 and 90.
ANSWER:
The proportion of grades that are between 80 and 90 = 0.95 - 0.825 = 0.125.
130. Car buyers were asked to indicate the car dealer they believed offered the best overall service. The four choices were A, B, C, and D as shown below:
A C C C D C B A C D D B C B A C B B C B A C C A D B B C C A A D C D T B B C D A
Construct a frequency distribution and proportion distribution.
ANSWER:
Dealer Frequency Proportion
A 9 0.225
B 10 0.250
C 14 0.350
D 7 0.175
0.000 0.050 0.150
0.350 0.625
0.825
0.950 1.000
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
30 40 50 60 70 80 90 100
Grade
Cumulative Relative Frequency
131. A grocery store’s monthly sales (in thousands of dollars) for the last year were as follows:
Month 1 2 3 4 5 6 7 8 9 10 11 12
Sales 78 74 83 87 85 93 100 105 103 89 78 94
Construct a relative frequency bar chart for these data.
ANSWER:
132. Consider the following cumulative frequency distribution.
Classes Limits Cumulative Frequency
Up to 5 8
Up to 10 15
Up to 15 21
Up to 20 30
Up to 25 42
Find the frequency for each of the following classes.
a. 0 up to 5 b. 5 up to 10 c. 10 up to 15 d. 15 up to 20 e. 20 up to 25
ANSWER:
a. 8 b. 7 c. 6 d. 9 e. 12
0.00 0.02 0.04 0.06 0.08 0.10 0.12
Jan. Feb. March April May June July Aug. Sept. Oct. Nov. Dec.
Month
Relative Frequency
133. The frequency distribution for a sample of 200 business school graduates is shown in the following table.
Construct a relative frequency distribution.
ANSWER:
FOR QUESTIONS 134 THROUGH 137, USE THE FOLLOWING NARRATIVE:
Narrative: Weights of Workers
The weights in pounds of a sample of 25 workers are given: 164, 148, 137, 157, 173, 156, 177, 172, 169, 165, 145, 168, 163, 162, 174, 152, 156, 168, 154, 151, 174, 146, 134, 140, and 171.
134. {Weights of Workers Narrative} Construct an ogive for the data.
ANSWER:
Major of Graduates Number of graduates
Accounting 58
Finance 42
Management 38
Marketing 52
Other 10
Major of Graduates Proportion of Graduates
Accounting 0.29
Finance 0.21
Management 0.19
Marketing 0.26
Other 0.05
Total 1.00
0.00
0.08
0.24
0.48
0.76
1.00
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
130 140 150 160 170 180
Weight (pounds)
Cumulative Relative Frequency
135. {Weights of Workers Narrative} What proportion of the total area under the ogive curve falls between 160 and 180?
ANSWER:
1 - 0.48 = 0.52
136. {Weights of Workers Narrative}What proportion of the total area under the ogive curve falls below or equal to 150?
ANSWER:
0.24
137. {Weights of Workers Narrative} What proportion of the total area under the ogive curve falls above or equal to 140?
ANSWER:
1 – 0.08 = 0.92
FOR QUESTIONS 138 THROUGH 142, USE THE FOLLOWING NARRATIVE:
Narrative: Years of Service
The frequency distribution of the number of years of service for 100 employees is shown below:
138. {Years of Service Narrative} Construct a relative frequency distribution for the data.
ANSWER:
Years Relative Frequency
0 up to 5 0.12
5 up to 10 0.16
10 up to 15 0.42
15 up to 20 0.20
20 up to 25 0.10
Total 1.00
Years Frequency
0 up to 5 12
5 up to 10 16
10 up to 15 42
15 up to 20 20
20 up to 25 10
Total 100
139. {Years of Service Narrative} Construct a cumulative relative frequency distribution for the data.
ANSWER:
Years Cumulative Relative Frequency
up to 5 0.12
up to 10 0.28
up to 15 0.70
up to 20 0.90
up to 25 1.00
140. {Years of Service Narrative} The proportion of employees who have less than 10 years of service is __________.
ANSWER:
0.28
141. {Years of Service Narrative} The proportion of employees who have more than 20 years of service is __________.
ANSWER:
1 – 0.90 = 0.10
142. {Years of Service Narrative} The proportion of employees who have between 10 and 20 years of service is __________.
ANSWER:
0.90 - 0.28 = 0.62
FOR QUESTIONS 143 AND 144, USE THE FOLLOWING NARRATIVE:
Narrative: Insurance Company
An insurance company evaluates many numerical variables about a person before deciding on an appropriate rate for automobile insurance. A representative from a local insurance agency selected random sample of insured drivers and recorded the number of claims made in the last three years, with the following results.
Number of claims 0 1 2 3 4 5
Frequency 10 18 16 12 3 1
143. {Insurance Company Narrative} How many drivers are represented in the sample?
ANSWER:
60
144. {Insurance Company Narrative} How many total claims are represented in the sample?
ANSWER:
103
FOR QEUSTIONS 145 AND 146, USE THE FOLLOWING NARRATIVE:
Narrative: Computer Company
At a meeting of regional offices managers of a national computer company, a survey, was taken to determine the number of employees the regional managers supervise in the operation of their departments, as shown below
145. {Computer Company Narrative} How man regional offices are represented in the survey results?
ANSWER:
50
146. {Computer Company Narrative} Across all of the regional offices, how many total employees were supervised by those surveyed?
ANSWER:
153
FOR QUESTIONS 147 THROUGH 150 USE THE FOLLOWING NARRATIVE:
Narrative: On-line Classes
A survey was conducted to determine how students rated the quality of on-line classes. Students were asked to rate the overall quality from 0 (no quality at all) to 100 (extremely good quality).
The stem- and-leaf display of the data is shown below.
Stem Leaves 3 15
4 01457889 5 0134677 6 24568 7 29 8
9 5
147. {On-line Classes Narrative} What percentage of the students rated overall quality of on- line classes with a rating of 70 or above?
ANSWER:
12%
Number of employees 1 2 3 4 5
Frequency 7 11 14 8 10
148. {On-line Classes Narrative} What percentage of the students rated overall quality of on- line classes with a rating of 60 or below?
ANSWER:
68%
149. {On-line Classes Narrative} What percentage of the students rated overall quality of on- line classes with a rating between 50 and 75, inclusive?
ANSWER:
52%
150. {On-line Classes Narrative} What percentage of the students rated overall quality of on- line classes with a rating below 40?
ANSWER:
8%
151. A __________ is a vertical bar chart in which the rectangular bars are constructed at the lower and upper limits of each class interval.
ANSWER:
histogram
152. It is desirable that the width of each class grouping or interval in a frequency distribution be __________.
ANSWER:
the same or equal
153. A __________ is a summary table in which numerical data are tallied into class intervals or categories.
ANSWER:
frequency distribution
SECTIONS 4 - 5
MULTIPLE CHOICE QUESTIONS
In the following multiple-choice questions, please circle the correct answer.
154. The relationship between two interval variables is graphically displayed by a a. scatter diagram
b. histogram c. bar chart d. pie chart ANSWER: a
155. When studying the simultaneous responses to two categorical questions, we should develop a
a. contingency table.
b. frequency distribution table.
c. cumulative percentage distribution table.
d. histogram.
ANSWER: a
156. In general, the scatter diagram of two interval variables may reveal that a. there is a positive linear relationship.
b. there is a negative linear relationship.
c. there is no relationship at all, or the relationship may be nonlinear.
d. All of the above are true statements.
ANSWER: d
157. In order to draw a scatter diagram, we need interval data for a. one variable
b. two variables c. three variables d. four variables ANSWER: b
158. A contingency table is also called a. a cross-classification table b. a cross-tabulation table c. Both (a) and (b) are true d. Neither (a) nor (b) is true.
ANSWER: c
159. In a contingency table, the number of rows and columns a. must always be the same.
b. must always be 3.
c. must add to 100.
d. None of the above.
ANSWER: d
160. Which of the following statements is false?
a. Techniques applied to single data sets are called univariate.
b. There are many situations where we wish to graphically depict the relationship between variables; in such cases multivariate methods are require.
c. The technique used to describe the relationship between two interval variables is called the scatter diagram.
d. None of the above.
ANSWER: b
161. Which of the following statements is false?
a. In applications where one variable depends to some degree on the other variable, we label the dependent variable Y and the other, called the response variable X.
b. To determine the strength of the linear relationship between two variables, we draw a straight line through the points in such a way that the line represents the relationship.
c. If most of the points in a scatter diagram appear to be scattered randomly with only a semblance of a straight line, there is no – or at best, a weak – linear relationship.
d. If, in general, when one variable increases, so does the other, we say that there is a positive linear relationship.
ANSWER: a
TRUE / FALSE QUESTIONS
162. The graphical technique used to describe the relationship between two interval variables is the scatter diagram.
ANSWER: T
163. Line charts can be used to graphically depict ordinal and interval data, but not nominal data.
ANSWER: F
164. Time series data are often graphically depicted on a line chart, which is a plot of the variable of interest over time.
ANSWER: T
165. A Line chart is created by plotting the value of the variable on the vertical axis and the time periods on the horizontal axis.
ANSWER: T
166. In order to describe how two variables are related, the two most important characteristics revealed by the scatter diagram are the strength and direction of the relationship.
ANSWER: T
167. Cross-sectional data are often graphically depicted on a line chart, which is a plot of the variable over time.
ANSWER: F
168. Techniques applied to single data sets are called univariate.
ANSWER: T
169. If we draw a straight line through the points in a scatter diagram and most of the points fall close to the line, we say that there is a positive linear relationship between the two variables.
ANSWER: F
170. Data can be classified according to whether the observations are measured at the same time or whether they represent measurements at successive points in time. The former are called cross – sectional data and the latter, time – series data.
ANSWER: T
171. When two variables are linearly related, and tend to move in opposite directions, we describe the nature of their association as a positive linear relationship.
ANSWER: F
STATISTICAL CONCEPTS & APPLIED QUESTIONS
172. To evaluate two categorical variables at the same time, a __________ also called __________ or __________ should be developed.
ANSWER:
Contingency table, cross-classification table, cross-tabulation table
173. Data can be classified according to whether the observations are measured at the same time or whether they represent measurements at successive points in time. The former are called __________, and the latter, __________.
ANSWER:
cross-sectional data, time-series data
174. A professor of economics wants to study the relationship between income and education.
A sample of 10 individuals is selected at random, and their income (in thousand of dollars) and education (in years) are shown below:
Education 12 14 10 11 13 8 10 15 13 12 Income 25 31 20 24 28 15 21 35 29 27
a. Draw a scatter diagram for these data with the income on the vertical axis.
b. Describe the relationship between income and education.
ANSWER:
a.
b. There is a very strong positive linear relationship between education and income; as years of education increase, there is a definite tendency for income to linearly increase.
0 5 10 15 20 25 30 35 40
5 7 9 11 13 15 17
Education
Income
175. The number of houses sold in Grand Rapids and the average monthly mortgage rates for 18 months randomly selected between January 1997 and April 1999 are shown in the following table.
Mortgage rate (%)
Number of houses sold
Mortgage rate (%)
Number of houses sold
7.5 90 9.5 68
9.0 72 6.5 97
7.0 89 8.0 79
10.5 62 9.0 75
10.0 58 10.5 53
9.5 70 9.5 73
8.5 74 11.0 50
10.0 65 7.5 82
11.0 51 8.5 70
a. Draw a scatter diagram with the number of houses sold on the vertical axis.
b. Describe the relationship between mortgage rate and number of houses sold.
ANSWER:
a.
b. There is a strong linear relationship between the mortgage rate and the number of houses sold.
0 20 40 60 80 100 120
6 8 10 12
Mortgage Rate (%)
Number of Houses Sold
176. A grocery store’s monthly sales (in thousands of dollars) for the last year were as follows:
Month 1 2 3 4 5 6 7 8 9 10 11 12
Sales 78 74 83 87 85 93 100 105 103 89 78 94
Construct a line chart for these data.
ANSWER:
177. A __________ is a graphical display consisting of a scatter of dots, with each dot representing one observation about a variable measured along the horizontal axis, and another observation about a different variable measured along the vertical axis.
ANSWER:
scatter diagram
178. Briefly discuss the difference between cross-sectional data and time-series data.
ANSWER:
Data can be classifies according to whether the observations are measured at the same time or whether they represent measurements at successive points in time. The former are called cross-sectional data and the later, time-series data.
0 30 60 90 120
Jan. Feb. March April May June July Aug. Sept. Oct. Nov. Dec.
Month
Sales
FOR QUESTIONS 179 THROUGH 185 USE THE FOLLOWING NARRATIVE:
Narrative: Bar Hopping
A sample of 200 students at Ohio State University was taken after the midterm to ask them whether they went bar hopping the weekend before the midterm or spent the weekend studying, and whether they did well or poorly on the midterm. The following table contains the results.
Did Well in Midterm
Did Poorly in Midterm
Studying for Exam 90 10
Went Bar Hopping 20 80
179. {Bar Hopping Narrative} Of those who went bar hopping the weekend before the midterm in the sample, what percent age of them did well on the midterm?
ANSWER:
30%
180. {Bar Hopping Narrative} Of those who did well on the midterm in the sample, what percentage of them went bar hopping the weekend before the midterm?
ANSWER:
22.22%
181. {Bar Hopping Narrative} What percentage of the students in the sample went bar hopping the weekend before the midterm and did well on the midterm?
ANSWER:
10%
182. {Bar Hopping Narrative}What percentage of the students in the sample spent the weekend studying and did well on the midterm?
ANSWER:
45%
183. {Bar Hopping Narrative} If the sample is a good representation of the population, what percentage of the students in the population can we expect to spend the weekend studying and do poorly?
ANSWER:
5%
184. {Bar Hopping Narrative} If the sample is a good representation of the population, what percentage of those who spend the weekend studying can we expect to do poorly on the midterm?
ANSWER:
10%
185. {Bar Hopping Narrative} If the sample is a good representation of the population, what percentage of those who did poorly on the midterm can we expect to have spent the weekend studying?
ANSWER:
11.11%
