Describing Data: Displaying and Exploring Data
Chapter 4
Describing Data: Displaying and Exploring Data
True/False
1. A dot plot and a scatter diagram are different names for the same graph.
Answer: False
2. A dot plot is an easy way to represent the relationship between two variables.
Answer: False
3. A dot plot is useful for quickly graphing frequencies in a small data set.
Answer: True
4. A stem and leaf diagram shows the actual data values.
Answer: True
5. There is some loss of information when raw data is tallied into a stem-and-leaf display.
Answer: False
6. For a stem-and-leaf display, the leaf for the value 98 is 9.
Answer: False
7. The stem in a stem-and-leaf display is the leading digit.
Answer: True
8. In a stem-and-leaf display, the leaf represents a class of a frequency distribution.
Answer: False
9. In a stem-and-leaf display, the leaf represents a member of a class in a frequency distribution.
Answer: True
10. In a stem-and-leaf display, for each class, the leaves are arranged or sorted from smallest to largest.
Answer: True
11. In a stem-and-leaf display, it is easy to find the range for a data set.
Answer: True
12. Quartiles divide a distribution into four equal parts.
Answer: True
13. A percentile divides a distribution into one hundred equal parts.
Answer: True
14. A student scored in the 85th percentile on a standardized test. This means that the student scored lower than 85% of all students who took the test.
Answer: False
15. Quartiles are another way to describe the central location of a distribution.
Answer: False
16. Quartiles are another way to describe the dispersion of a distribution.
Answer: True
17. The 50th percentile of a distribution is the same as the distribution mean.
Answer: False
18. A percentile can be a decile, but a decile can not be a quartile.
Answer: True
19. A quartile can be a decile, but a decile can not be a percentile.
Answer: False
20. For a distribution, the 2nd quartile, the 5th decile, and the 50th percentile, are the same as the median.
Answer: True
21. The interquartile range is the difference between the values of the first and third quartile, indicating the range of the middle fifty percent of the observations.
Answer: True
22. A box plot graphically shows data that are in percentiles.
Answer: False
23. The "box" in a box plot shows the interquartile range.
Answer: True
24. An outlier is a data point that occurs in the first quartile.
Answer: False
25. An outlier is a value in a data set that is inconsistent with the rest of the data.
Answer: True
26. A box plot shows the relative symmetry of a distribution.
Answer: True
27. A box plot shows a distribution's mean and mode.
Answer: False
28. A box plot shows the range of values that correspond to the upper 25% of the distribution.
Answer: True
29. In a box plot, if a value is more than 1.5 times the standard deviation from the first or third quartile, the value is an outlier.
Answer: False
30. In a box plot, if a value is more than 1.5 times the interquartile range from the first or third quartile, the value is an outlier.
Answer: True
31. The coefficient of variation is a measure of relative dispersionthat expresses the standard deviation as a percent of the mean.
Answer: True
32. The Pearson's coefficient of skewness is a measure of distribution's symmetry.
Answer: True
33. The coefficient of variation is useful for comparing distributions with different units.
Answer: True
34. The coefficient of variation is computed by dividing the standard deviation by the median and multiplying the quotient by 100.
Answer: False
35. Negatively skewed indicates that a distribution is not symmetrical. The long tail is to the left or in the negative direction.
Answer: True
36. In a negatively skewed distribution, the mean is smaller than the median or mode and the mode occurs at the peak of the curve.
Answer: True
37. If Pearson's coefficient of skewness is equal to 0, then the mean and median are equal.
Answer: True
38. If Pearson's coefficient of skewness is negative, then the mean is greater than the median.
Answer: False
39. If Pearson's coefficient of skewness is negative, then the distribution is skewed to the left.
Answer: True
40. If Pearson's coefficient of skewness is negative, then the distribution is skewed to the right.
Answer: False
41. A scatter diagram of sales versus production may be constructed by plotting the data on a graph labeled with sales on the Y-axis and production on the X-axis.
Answer: True
42. A relationship between gender and preference for Coke or Pepsi can be best represented by a scatter diagram.
Answer: False
43. A relationship between gender and preference for Coke or Pepsi can be best represented by a contingency table.
Answer: True
Multiple Choice
44. A dot plot shows
A) The general shape of a distribution
B) The mean, median, and mode
C) The relationship between two variables
D) The interquartile range.
Answer: A
45. A row of a stem-and-leaf chart appears as follows: 3 | 0 1 3 5 7 9. Assume that the data is rounded to the nearest unit.
A) The frequency of the class is seven.
B) The minimum value in the class is 0.
C) The maximum value in the class is 39.
D) The class interval is 5.
Answer: C
46. The test scores for a class of 147 students are computed. What is the location of the test score associated with the third quartile?
A) 111
B) 37
C) 74
D) 75%
Answer: A
47. What statistics are needed to draw a box plot?
A) Minimum, maximum, median, first and third quartiles
B) Median, mean and standard deviation
C) A median and an interquartile range
D) A mean and a standard deviation.
Answer: A
48. A box plot shows
A) The mean and variance
B) The relative symmetry of a distribution for a set of data
C) The percentiles of a distribution
D) The deciles of a distribution
Answer: B
49. What does the interquartile range describe?
A) The lower 50% of the observations
B) The middle 50% of the observations
C) The upper 50% of the observations
D) The lower 25% and the upper 25% of the observations
E) None of the above
Answer: B
50. The coefficient of variation for a set of annual incomes is 18%; the coefficient of variation for the length of service with the company is 29%. What does this indicate?
A) More dispersion in the distribution of the incomes compared with the dispersion of their length of service
B) More dispersion in the lengths of service compared with incomes
C) Dispersion in the two distributions (income and service) cannot be compared using percents
D) Dispersions are equal
Answer: B
51. Mr. and Mrs. Jones live in a neighborhood where the mean family income is $45,000 with a standard deviation of $9,000. Mr. and Mrs. Smith live in a neighborhood where the mean is $100,000 and the standard deviation is $30,000. What is the relative dispersion of the family incomes in the two neighborhoods?
A) Jones 40%, Smith 20%
B) Jones 20%, Smith 30%
C) Jones 30%, Smith 20%
D) Jones 50%, Smith 33%
E) None of the above
Answer: B
52. A large oil company is studying the number of gallons of gasoline purchased per customer at self-service pumps. The mean number of gallons is 10.0 with a standard deviation of 3.0 gallons. The median is 10.75 gallons. What is the Pearson's coefficient of skewness?
A) -1.00
B) -0.75
C) +0.75
D) +1.00
Answer: B
53. What is the value of the Pearson coefficient of skewness for a distribution with a mean of 17, median of 12 and standard deviation of 6?
A) +2.5
B) -2.5
C) +0.83
D) -0.83
Answer: A
54. A study of business faculty at state supported institutions in Ohio revealed that the arithmetic mean salary for nine months is $52,000 and a standard deviation of $3,000. The study also showed that the faculty had been employed an average (arithmetic mean) of 15 years with a standard deviation of 4 years. How does the relative dispersion in the distribution of salaries compare with that of the lengths of service?
A) Salaries about 100%, service about 50%
B) Salaries about 6%, service about 27%
C) Salaries about 42%, service about 81%
D) Salaries about 2%, service about 6%
Answer: B
55. What is the possible range of values for the coefficient of variation?
A) -1 and +1
B) -3 and +3
C) 0% and 100%
D) Unlimited values
Answer: C
56. A research analyst wants to compare the dispersion in the price-to-earnings ratios for a group of common stocks with their return on investment (ROI). For the price-to-earnings ratios, the mean is 10.9 and the standard deviation is 1.8. The mean return on investment is 25 percent and the standard deviation 5.2 percent. What is the relative dispersion for the price-to-earnings ratios and return on investment?
A) Price-to-earnings = 32.0 percent, ROI =19.0 percent
B) Price-to-earnings =16.5 percent, ROI = 20.8 percent
C) Price-to-earnings =132.0 percent, ROI =190.0 percent
D) Price-to-earnings = 50.0 percent, ROI =10.0 percent
Answer: B
57. A study of the scores on an in-plant course in management principles and the years of service of the employees enrolled in the course resulted in these statistics:
- Mean test score was 200 with a standard deviation of 40
- Mean number of years of service was 20 years with a standard deviation of 2 years.
In comparing the relative dispersion of the two distributions, what are the coefficients of variation?
A) Test 50%, service 60%
B) Test 100%, service 400%
C) Test 20%, service 10%
D) Test 35%, service 45%
Answer: C
58. A large group of inductees was given a mechanical aptitude and a finger dexterity test. The arithmetic mean score on the mechanical aptitude test was 200, with a standard deviation of 10. The mean and standard deviation for the finger dexterity test were 30 and 6 respectively. What is the relative dispersion in the two groups?
A) Mechanical aptitude 5 percent, finger dexterity 20 percent
B) Mechanical aptitude 20 percent, finger dexterity 10 percent
C) Mechanical aptitude 500 percent, finger dexterity 200 percent
D) Mechanical aptitude 50 percent, finger dexterity 200 percent
Answer: A
59. A sample of experienced typists revealed that their mean typing speed is 87 words per minute and the median is 73. The standard deviation is 16.9 words per minute. What is the Pearson's coefficient of skewness?
A) -2.5
B) -4.2
C) +4.2
D) +2.5
Answer: D
60. A study of the net sales of a sample of small corporations revealed that the mean net sales is $2.1 million, the median $2.4 million, the modal sales $2.6 million and the standard deviation of the distribution is $500,000. What is the Pearson's coefficient of skewness?
A) -9.1
B) +6.3
C) -3.9
D) +2.4
E) None of the above
Answer: E
61. In a scatter diagram, we describe the relationship between
A) two variables measured at the ordinal level
B) two variables, one measured as an ordinal variable and the other as a ratio variable
C) two variables measured at the interval or ratio level
D) a variable measure on the interval or ratio level and time.
Answer: C
62. In a contingency table, we describe the relationship between
A) two variables measured at the ordinal or nominal level.
B) two variables, one measured as an ordinal variable and the other as a ratio variable
C) two variables measured at the interval or ratio level
D) a variable measure on the interval or ratio level and time.
Answer: A
Fill-in-the-Blank
63. What chart or graph is useful for illustrating frequencies? ______.
Answer: dot plot
64. For a stem-and-leaf display, what is the stem for the value 67? ____.
Answer: 6
Essay
65. Construct a stem-and-leaf display for the following data:
Answer:
1| 9
2| 0 1 2 2 6 9
3| 0 1 2 3 4 5 5 7 8 8 9
4| 2 6
5| 0 1 2 4 5 7 8 9
6| 5 9
66. From the following stem-and-leaf display, find the minimum value, the 1st quartile, the median, the 3rd quartile, and the maximum value. List and interpret the interquartile range.
1| 9
2| 0 1 2 2 6 9
3| 0 1 2 3 4 5 5 7 8 8 9
4| 2 6
5| 0 1 2 4 5 7 8 9
6| 5 9
Answer:
Minimum=19
1st quartile = 29.75
median = 37.5
3rd quartile = 52.5
Maximum = 69.
Interquartile range is 52.5-29.75 = 22.75. It means that 50% or 15 of the 30 observations are between 52.5 and 29.75
Fill-in-the-Blank
67. For a stem-and-leaf display, what is the leaf for the value 123? ____.
Answer: 3
68. If you are constructing a stem-and-leaf display, the "3" in 19.3 would be the ______.
Answer: leaf
69. If you are constructing a stem-and-leaf display, the "20" in 20.5 would be the ______.
Answer: stem
70. What is the best way to display the relationship between two variables measured on an interval or ratio level?
Answer: scatter diagram
71. What is the main advantage of a stem-and-leaf chart over a histogram? ______
Answer: The identity of each observation is not lost
72. The percentile range is the distance between any two ______.
Answer: percentiles
73. In a symmetric distribution, where is the 99th percentile located? ______
Answer: In the far right tail
74. In a positively skewed distribution, where is the 99th percentile located? ______
Answer: In the far right tail
75. In a negatively skewed distribution, where is the 1st percentile located? ______
Answer: In the far left tail
76. If the mean of a distribution is smaller than the median and mode, what is the sign of Pearson's coefficient of skewness? ______
Answer: negative
77. A frequency distribution may be divided into how many percentiles? ___
Answer: 99
78. For a set of data, how many quartiles are there? _____
Answer: three
79. If two sets of data are measured in different units, what statistic can be used to compare their dispersions? ______
Answer: coefficient of variation
80. What unit of measurement is used to express the coefficient of variation? ______
Answer: percent
81. The coefficient of variation is a measure of ______.
Answer: relative dispersion
82. The research director of a large oil company conducted a study of the buying habits of consumers with respect to the amount of gasoline purchased at full-service pumps. The arithmetic mean amount is 11.5 gallons and the median amount is 11.95 gallons. The standard deviation of the sample is 4.5 gallons. What is the Pearson's coefficient of skewness? ______
Answer: -0.30
83. Rainbow Trout, Inc. feeds fingerling trout in special ponds and markets them when they attain a certain weight. A group of 9 trout (considered the population) were isolated in a pond and fed a special food mixture called Grow Em Fast. At the end of the experimental period, the weights of the trout were (in grams): 124, 125, 123, 120, 124, 127, 125, 126 and 121. Another special mixture, Fatso 1B, was used in another pond. The mean of the population was computed to be 126.9 grams and the standard deviation was 1.20 grams. Which food results in a more uniform weight? ______
Answer: Fatso 1B
84. The annual incomes of the five vice presidents of Elly's Industries are: $41,000, $38,000, $32,000, $33,000 and $50,000. The annual incomes of Unique, another firm similar to Elly's Industries, were also studied and found to have a mean of $38,900 and a standard deviation of $6,612. What company has the greater coefficient of variation? ______
Answer: Elly, (19.0) > Unique (17.0)
85. The spread in the annual prices of stocks selling under $10 and those selling over $60 are to be compared. The mean price of the stocks selling under $10 is $5.25 and the standard deviation is $1.52. The mean price of those stocks selling over $60 is $92.50 and the standard deviation is $5.28. Why should the coefficient of variation be used to compare the dispersion in the prices? ______
Answer: means differ vastly
86. The lengths of stay on the cancer floor of Community Hospital were organized into a frequency distribution. The mean length was 28 days, the median 25 days and the modal length 23 days. The standard deviation was computed to be 4.2 days. What is the Pearson's coefficient of skewness? ______
Answer: 2.14
87. A sample of the homes currently offered for sale revealed that the mean asking price is $75,900, the median $70,100 and the modal price is $67,200. The standard deviation of the distribution is $5,900. What is the Pearson's coefficient of skewness? ______
Answer: 2.95
88. The Pearson's coefficient of skewness (Sk) measures the amount of skewness and may range from -3.0 to +3.0. It is computed by subtracting the median from the mean, multiplying the result by 3 and dividing by? ______
Answer: standard deviation
Essay
89. Given the sample information in the following table regarding public opinion on gun control, who is more likely to favor gun control?
Answer: Republicans are more likely to favor gun control with 58% favoring gun control. Only 38% of democrats favor gun control.
Fill-in-the-Blank
Use the following to answer questions 90-94:
A telemarketing firm is monitoring the performance of its employees based on the number of sales per hour. One employee had the following sales for the last 20 hours
90. What is the median for the distribution of number of sales per hour? ______
Answer: Median = 5 sales per hour
91. What is the first quartile for the distribution of number of sales per hour? ______
Answer: Q1 = 4 sales per hour
92. What is the third quartile for the distribution of number of sales per hour? ______
Answer: Q3 = 6.5 sales per hour
93. For the distribution of number of sales per hour, 50% are greater than ______
Answer: The median or 5 sales per hour
94. For the distribution of number of sales per hour, 50% of the observations are between ______and ______.
Answer: Q1 (4) and Q3 (6.5)
Use the following to answer questions 95-101:
The following stem and leaf display reports the number of boat shipments per week by Ottertail Boats, Inc.
11| 1 5 9
12| 0 1 2 2 6 9
13| 0 1 2 3 4 5 5 7 8 8 9
14| 2 6 8
15| 0 1 2 4 5 7 8 9
16| 1 5 7 9
95. How many weeks were included in the study?______
Answer: 35 weeks
96. How many observations are in the third class?______
Answer: 11 weeks
97. What are the smallest and largest values?______
Answer: 111 and 169 orders
98. List the actual values in the fourth class.______
Answer: 142, 146, and 148 orders
99. How often did the company complete 111 shipments? ______
Answer: 1 or once
100. How often did the company complete more than 140 shipments? ______
Answer: 15 times
101. What is the median value? ______
Answer: 138 shipments,
Essay
102. What is the common purpose of a scatter diagram and a contingency table?
Answer: Both are used to summarize two variables: ,7
103. What is the difference between a scatter diagram and a contingency table?
Answer: A scatter diagram requires interval or ratio scaled variables, a contingency table requires nominal or ordinal variables. ,7
104. Draw a negatively or positively skewed distribution and show the relative locations of the mean, median, and mode.
Answer: See Text: