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CHAPTER 4 NUMERICAL DESCRIPTIVE TECHNIQUES SECTIONS 1 MULTIPLE CHOICE QUESTIONS In the following multiple-choice questions, please circle the correct answer. 1. Which measure of central location is appropriate whenever we wish to estimate the expected mean return, or growth rate, for a single year in the future? a. The arithmetic mean b. The geometric mean c. The median d. The mode ANSWER: a 2. Which of the following statistics is not a measure of central tendency? a. Mean b. Median c. Mode d. Interquartile range ANSWER: d 69 70 Chapter Four 3. Which measure of central location is meaningful when the data are nominal? a. The arithmetic mean b. The geometric mean c. The median d. The mode ANSWER: d 4. Which measure of central location is appropriate whenever we wish to find the average growth rate, or rate of change, in a variable over time? a. The arithmetic mean b. The geometric mean c. The median d. The mode ANSWER: b 5. Which of the following statements about the arithmetic mean is not always correct? a. The sum of the deviations from the mean is zero b. Half of the observations are on either side of the mean c. The mean is a measure of the middle (center) of a distribution d. The value of the mean times the number of observations equals the sum of all observations ANSWER: b 6. Since the population is always larger than the sample, the population mean: a. is always larger than or equal to the sample mean b. is always smaller than or equal to the sample mean c. can be smaller than, or larger than, or equal to the sample mean d. None of the above ANSWER: c 7. Which of the following statements is true for the following data values: 7, 5, 6, 4, 7, 8, and 12? a. The mean, median and mode are all equal b. Onlythe mean and median are equal c. Onlythe mean and mode are equal d. Onlythe median and mode are equal ANSWER: a 8. In a histogram, the proportion of the total area which must be to the left of the median is: a. exactly 0.50 b. less than 0.50 if the distribution is skewed to the left c. more than 0.50 if the distribution is skewed to the right d. between 0.25 and 0.60 if the distribution is symmetric and unimodal ANSWER: a Numerical Descriptive Techniques 71 9. Which measure of central tendency can be used for both numerical and categorical variables? a. The arithmetic mean b. The median c. The mode d. The geometric mean ANSWER: c 10. Which of the mean, median, mode, and geometric mean are resistant measures of central tendency? a. The mean and median only b. The median and mode only c. The mode and geometric mean only d. The mean and mode only ANSWER: b 11. In a positively-skewed distribution, a. the median equals the mean b. the median is less than the mean c. the median is larger than the mean d. the mean, median, and mode are equal ANSWER: b 12. Which of the following statements about the median is not true? a. It is more affected by extreme values than the mean b. It is a measure of central tendency c. It is equal t2 d. It is equal to the mode in bell-shaped “normal” distributions ANSWER: a 13. Which of the following summary measures is the easiest to compute? a. The mean b. The median c. The mode d. Allof the above ANSWER: c 14. Which of the following is not a measure of central tendency? a. The arithmetic mean b. The geometric mean c. The mode d. The first quartile ANSWER: d 15. Which of the following summary measures is sensitive to extreme values? 72 Chapter Four a. The median b. The interquartile range c. The arithmetic mean d. The first quartile ANSWER: c 16. In a perfectly symmetrical bell-shaped “normal” distribution a. the mean equals the median b. the median equals the mode c. the mean equals the mode d. Allof the above ANSWER: d 17. Suppose you make a 2 – year investment of $2,500 and it grows by 100% to $5,000 during the first year. During the second year, however, the investment suffers a 50% loss from $5,000 back to $2,500. The arithmetic mean (and the median) is a. 75% b. 50% c. 25% d. 0% ANSWER: c 18. Suppose you make a 2 – year investment of $2,500 and it grows by 100% to $5,000 during the first year. During the second year, however, the investment suffers a 50% loss from $5,000 back to $2,500. The geometric mean is a. 75% b. 50% c. 25% d. 0% ANSWER: d 19. Which of the following statement is false? a. A parameter is a descriptive measurement about a population b. A statistic is a descriptive measurement about a sample c. Students may be more familiar with the average as the other name for the arithmetic mean d. None of the above ANSWER: d 20. Which of the following statements is true? a. When the distribution is skewed to the left, mean > median > mode b. When the distribution is skewed to the right, mean < median < mode c. When the distribution is symmetric and unimodal, mean = median = mode d. When the distribution is symmetric and bimodal, mean = median = mode ANSWER: c 21. In a histogram, the proportion of the total area which must be to the right of the mean is a. less than 0.50 if the distribution is skewed to the left Numerical Descriptive Techniques 73 b. exactly 0.50 c. more than 0.50 if the distribution is skewed to the right d. exactly 0.50 if the distribution is symmetric and unimodal ANSWER: d 22. The average score for a class of 30 students was 75. The 20 male students in the class averaged 70. The 10 female students in the class averaged: a. 75 b. 85 c. 65 d. 70 ANSWER: b 74 Chapter Four TRUE / FALSE QUESTIONS 23. Two classifications of statistical descriptions are measures of central location and measures of variability. ANSWER: T 24. The mean is one of the most frequently used measures of variability. ANSWER: F 25. In a histogram, the proportion of the total area which must be to the left of the median is more than 0.50 if the distribution is skewed to the right. ANSWER: F 26. A data sample has a mean of 107, a median of 122, and a mode of 134. The distribution of the data is positivelyskewed. ANSWER: F 27. A student scores 87, 73, 92, and 86 on four exams during the semester and 95 on the final exam. If the final is weighted double and the four others weighted equally, the student's final average score would be 90. ANSWER: F 28. In a bell-shaped distribution, there is no difference in the values of the mean, median, and mode. ANSWER: T 29. Lily has been keeping track of what she spends to eat out. The last week's expenditures for meals eaten out were $5.69, $5.95, $6.19, $10.91, $7.49, $14.53, and $7.66. The mean amount Lilyspends on meals is $8.35. ANSWER: T 30. In a negatively skewed distribution, the mean is smaller than the median and the median is smaller than the mode. ANSWER: T 31. The median of a set of data would be more representative than the mean of that data when the average of the data values is larger than most of the values. ANSWER: T 32. Since the sample is always smaller than the population, the sample mean is always smaller than the population mean. ANSWER: F 33. The geometric mean is useful in measuring the rate of change of a variable over time. ANSWER: T 34 The value of the mean times the number of observations equals the median of the observations. Numerical Descriptive Techniques 75 ANSWER: F 35. In a histogram, the proportion of the total area which must be to the left of the median is less than 0.50 if the distribution is skewed to the left. ANSWER: F 36. The mean is a measure of the deviation in a data set. ANSWER: F 37. In a histogram, the proportion of the total area which must be to the right of the mean is exactly 0.50 if the distribution is symmetric and unimodal. ANSWER: T 38. The geometric mean is a measure of variation or dispersion in a set of data. ANSWER: F 39. In a sample of size 50, the sample mean is 20. In this case, the sum of all observations in the sample is 1,000. ANSWER: T 40. A statistics professor bases his final grade on homework, two midterm examinations, and a final examination. The homework counts 10% toward the final grade, while each midterm examination counts 30%. The remaining portion consists of the final examination. If a student scored 95% in homework, 75% on the first midterm examination, 95% on the second midterm examination, and 80% on the final, his final average is 84.5%. ANSWER: T 41. The median of a data set with 30 items would be the average of the 15 and the 16 items in the ordered array. ANSWER: T 42. The value of the median times the number of observations equals the sum of all of the observations. ANSWER: F 43. In a positively skewed distribution, the mean is larger than the median and the median is larger than the mode. ANSWER: T 44. The median is an appropriate measure of central location for nominal data, whereas the mode is more appropriate for ordinal data. ANSWER: F 76 Chapter Four STATISTICAL CONCEPTS & APPLIED QUESTIONS FOR QUESTIONS 45 THROUGH 47, USE THE FOLLOWING NARRATIVE: Narrative: Monthly Rent Monthly rent data in dollars for a sample of 10 one-bedroom apartments in a small town in Iowa are as follows: 220, 216, 220, 205, 210, 240, 195, 235, 204, and 250. 45. {Monthly Rent Narrative} Compute the sample monthlyaverage rent. ANSWER: x = $219.50 46. {Monthly Rent Narrative} Compute the sample median. ANSWER: $218 47. {Monthly Rent Narrative} What is the mode? ANSWER: $220 FOR QUESTIONS 48 AND 49, USE THE FOLLOWING NARRATIVE: Narrative: Number of Pets A sample of 25 families were asked how many pets they owned. Their responses are summarized in the following table. Number of Pets 0 1 2 3 4 5 Number of Families 3 10 5 4 2 1 48. {Number of Pets Narrative} Determine the mean, the median, and the mode of the number of pets owned per family. ANSWER: x = 1.80 pet, median = 1 pet, mode = 1 pet 49. {Number of Pets Narrative} Describe brieflywhat each statistic tells you about the data. ANSWER: The “average” number of pets owned was 1.80 pets. Half the families own at most one pet, and the other half own at least one pet. The most frequent number of pets owned was one pet. Numerical Descriptive Techniques 77 50. What are the relative magnitudes of the mean, median, and mode for a unimodal distribution that is a. symmetrical? b. skewed to the left? c. skewed to the right? ANSWER: a. mean = median = mode b. mean < median < mode c. mode < median < mean 51. A basketball player has the following points for seven games: 20, 25, 32, 18, 19, 22, and 30. Compute the following measures of central location: a. mean b. median c. mode ANSWER: a. x = 23.714 b. median = 22.0 c. There is no mode FOR QUESTIONS 52 THROUGH 54, USE THE FOLLOWING NARRATIVE: Narrative: Number of Children The following data represent the number of children in a sample of 10 families from Chicago: 4, 2, 1, 1, 5, 3, 0, 1, 0, and 2. 52. {Number of Children Narrative} Compute the mean number of children. ANSWER: x = 1.90 53. {Number of Children Narrative} Compute the median number of children. ANSWER: Median = 1.5 54. {Number of Children Narrative} Is the distribution of the number of children symmetrical or skewed? Why? ANSWER: The distribution is positively skewed because the mean is larger than the median. FOR QUESTIONS 55 THROUGH 58, USE THE FOLLOWING NARRATIVE: 78 Chapter Four Narrative: Weights of Workers The following data represent the weights in pounds of a sample of 25 workers: 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. 55. {Weights of Workers Narrative} Construct a stem and leaf display for the weights. ANSWER: Stem Leaf 13 47 14 0568 15 124667 16 2345889 17 123447 56. {Weights of Workers Narrative} Find the median weight. ANSWER: Median = 162 pounds 57. {Weights of Workers Narrative} Find the mean weight. ANSWER: x =159.04 58. {Weights of Workers Narrative} Is the distribution of the weights of workers symmetrical or skewed? Why? ANSWER: The distribution is negatively skewed because the mean is smaller than the median. 59. The number of hours a college student spent studying during the final exam week was recorded as follows: 76, 4, 9, 8, 5, and 10. Compute for the data and the value in an appropriate unit. ANSWER: x =7 hours FOR QUESTIONS 60 THROUGH 63, USE THE FOLLOWING NARRATIVE: Narrative: Ages of Employees Numerical Descriptive Techniques 79 The following data represent the ages in years of a sample of 25 employees from a government department: 31, 43, 56, 23, 49, 42, 33, 61, 44, 28, 48, 38, 44, 35, 40, 64, 52, 42, 47, 39, 53, 27, 36, 35, and 20. 60. {Ages of Employees Narrative} Construct a stem and leaf displayfor the ages. ANSWER: Stem Leaf 2 0378 3 1355689 4 022344789 5 236 6 14 61. {Ages of Employees Narrative} Find the median age. ANSWER: Median = 42 years 62. {Ages of Employees Narrative} Compute the sample mean age. ANSWER: x =41.2 years 63. {Ages of Employees Narrative} Find the modal age. ANSWER: Modes are 35, 42, and 44 FOR QUESTIONS 64 THROUGH 66, USE THE FOLLOWING NARRATIVE: Narrative: Salaries of Employees The following data represent the salaries (in thousands of dollars) of a sample of 13 employees of a firm: 26.5, 23.5, 29.7, 24.8, 21.1, 24.3, 20.4, 22.7, 27.2, 23.7, 24.1, 24.8, and 28.2. 64. {Salaries of Employees Narrative} Compute the mean salary. ANSWER: x = 24.692 65. {Salaries of Employees Narrative} Compute the median salary. ANSWER: 80 Chapter Four Median = 24.3 66. {Salaries of Employees Narrative} Consider the following population of measurements: 162, 152, 177, 157, 184, 176, 165, 181, 170, and 163. a. Compute the mean. b. Compute the median. ANSWER: a. μ = 168.7 b. Median = 167.5 67. A sample of 12 construction workers has a mean age of 25 years. Suppose that the sample is enlarged to 14 construction workers, by including two additional workers having common age of 25 each. Find the mean of the sample of 14 workers. ANSWER: x = 25 68. The mean of a sample of 15 measurements is 35.6. Suppose that the sample is enlarged to 16 measurements, by including one additional measurement having a value of 42. Find the mean of the sample of the16 measurements. ANSWER: x = 36 69. An investment you made in the years 2000-2003 has the following rates of return shown below: Year 2000 2001 2002 2003 Rate of Return .50 .30 -.10 -.15 Compute the geometric mean. ANSWER: 4 R g (+ R 1)(+ R 2)(+ R 3)(+ R 4) 4 = (+.50 )(.30 1−)(0 1−.)( ) -1 = 0.1052 FOR QUESTIONS 70 THROUGH 72, USE THE FOLLOWING NARRATIVE: Narrative: Investment Numerical Descriptive Techniques 81 Suppose you make a 2-year investment of $5,000 and it grows by 100% to $10,000 during the first year. During the second year, however, the investment suffers a 50% loss, from $10,000 back to $5,000. 70. {Investment Narrative} Calculate the arithmetic mean. ANSWER: R =(R + R )/ 2 = [100 + (-50)] / 2 = 25% 1 2 71. {Investment Narrative} Calculate the geometric mean. ANSWER: Rg= 2(1+ R1)(1+ R2)1= 2(1+1 )(.5 −)= 0. 72. {Investment Narrative} Interpret the value of the arithmetic mean. ANSWER: The figure of the arithmetic mean is misleading. Because there was no change in the value of the investment from the beginning to the end of the 2-year period, the “average” compounded rate of return is 0%, and this is the value of the geometric mean. FOR QUESTIONS 73 THROUGH 75, USE THE FOLLOWING NARRATIVE: Narrative: Ages of Senior Citizens A sociologist recently conducted a survey of citizens over 65 years of age whose net worth is too high to qualify for Medicaid and have no private health insurance. The ages of 20 uninsured senior citizens were as follows: 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 78, 79, 80, 81, 86, 87, 91, 92, 94, and 97. 73. {Ages of Senior Citizens Narrative} Calculate the mean age of the uninsured senior citizens to the nearest hundredth of a year. ANSWER: x= 78.15 74. {Ages of Senior Citizens Narrative} Calculate the median age of the uninsured senior citizens. ANSWER: 76.5 75. {Ages of Senior Citizens Narrative} Calculate the model age of the uninsured senior citizens. 82 Chapter Four ANSWER: There is no mode. 76. Suppose that a firm’s sales were $2,500,000 four years ago, and sales have grown annually by 25%, 15%, -5%, and 10% since that time. What was the geometric mean growth rate in sales over the past four years? ANSWER: R If gis the geometric mean, then (1+ R )4= (1+0.25)(1+0.15)(1-0.05)(1+0.10)=1.5022⇒ Rg= 0.1071 or 10.71% g 77. Suppose that a firm’s sales were $3,750,000 five years ago and are $5,250,000 today. What was the geometric mean growth rate in sales over the past five years? ANSWER: IfR gis the geometric mean, then 5 3,750,000 (1+ Rg) = 5,250,000 ⇒ R g= 0.0696 or 6.96% Numerical Descriptive Techniques 83 SECTION 2 MULTIPLE CHOICE QUESTIONS In the following multiple-choice questions, please circle the correct answer. 78. Which of the following statements is true? a. The sum of the deviations from the arithmetic mean is always zero b. The sum of the squared deviations from the arithmetic mean is always zero c. The range mean is always smaller than the variance d. The standard deviation is always smaller than the variance ANSWER: a 79. A sample of 20 observations has a standard deviation of 3. The sum of the squared deviations from the sample mean is: a. 20 b. 23 c. 29 d. 171 ANSWER: d 80. If two data sets have the same range, a. the distances from the smallest to largest observations in both sets will be the same b. the smallest and largest observations are the same in both sets c. both sets willhave the same standard deviation d. both sets willhave the same interquartile range ANSWER: a 81. The Empirical Rule states that the approximate percentage of measurements in a data set (providing that the data set has a bell-shaped distribution) that fall within two standard deviations of their mean is approximately: a. 68% b. 75% c. 95% d. 99% ANSWER: c 82. Which of the following summary measures is affected most by outliers? a. The median b. The geometric mean c. The range d. The interquartile range ANSWER: c 84 Chapter Four 83. Chebysheff’s Theorem states that the percentage of measurements in a data set that fall within three standard deviations of their mean is: a. 75% b. at least 75% c. 89% d. at least 89% ANSWER: d 84. Which of the following is not a measure of variability? a. The range b. The variance c. The median d. The interquartile range ANSWER: c 85. The smaller the spread of scores around the mean, a. the smaller variance b. the smaller the standard deviation c. the smaller the coefficient of variation d. Allof the above ANSWER: d 86. Is a standard deviation of 10 a large number indicating great variability, or is it small number indicating little variability? To answer this question correctly, one should look carefullyat the value of the a. mean b. standard deviation c. standard deviation divided by the mean d. mean dividing by the standard deviation ANSWER: c 87. Which of the following statements is false? a. There is one measure of variabilityfor nominal data b. There is a measure that can be used to describe the variabilityof ordinal data c. Measures of variabilitycan be used only for interval data d. None of the above ANSWER: a 88. Which of the following statements is true regarding the data set 8, 8, 8, 8, and 8? a. The range equals 0 b. The standard deviation equals 0 c. The coefficient of variation equals 0 d. Allof the above ANSWER: d Numerical Descriptive Techniques 85 TRUE / FALSE QUESTIONS 89. The value of the standard deviation may be either positive or negative, while the value of the variance willalways be positive. ANSWER: F 90. The difference between the largest and smallest values in an ordered array is called the range. ANSWER: T 91. The standard deviation is expressed in terms of the original units of measurement but the variance is not. ANSWER: T 92. Measures of variability describe typical values in the data. ANSWER: F 93. While Chebysheff’s Theorem applies to any distribution, regardless of shape, the Empirical Rule applies only to distributions that are bell-shaped and symmetrical. ANSWER: T 94. The mean of fifty sales receipts is $65.75 and the standard deviation is $10.55. Using Chebysheff's Theorem, 75% of the sales receipts were between $44.65 and $86.85. ANSWER: T 95. A sample of 15 observations has a standard deviation of 4. The sum of the squared deviations from the sample mean is 60. ANSWER: F 96. According to Chebysheff’s Theorem, at least 93.75% of observations should fall within 4 standard deviations of the mean. ANSWER: T 97. Chebysheff’s Theorem states that the percentage of observations in a data set that should fallwithin five standard deviations of their mean is at least 96%. ANSWER: T 98. The Empirical Rule states that the percentage of observations in a data set (providing that the data set has a bell-shaped and symmetric distribution) that fall within one standard deviation of their mean is approximately 75%. ANSWER: F 99. A population with 200 elements has a mean of 20. From this information, it can be shown that the population standard deviation is 10. ANSWER: F 86 Chapter Four 100. If two data sets have the same range, the distances from the smallest to largest observations in both sets willbe the same ANSWER: T 101. The sum of the deviations from the arithmetic mean is always zero. ANSWER: T 102. The sample mean is a measure of variability or spread or dispersion. ANSWER: F 103. The standard deviation is the positive square root of the variance. ANSWER: T 104. If two data sets have the same standard deviation, they must have the same coefficient of variation. ANSWER: T 105. The unit attached to the variance is the same unit attached to the original observations of a set of data. ANSWER: F 106. The unit attached to the standard deviation is the same unit of the original data set. ANSWER: T 107. The variance, as a measure of variability, is more meaningful and easier to interpret compared to the standard deviation. ANSWER: F 108. The range is considered the weakest measure of variability. ANSWER: T 109. Chebysheff's Theorem applies only to data sets that have a mound-shaped distribution. ANSWER: F 110. The coefficient of variation allows us to compare two sets of data based on different measurement units. ANSWER: T Numerical Descriptive Techniques 87 STATISTICAL CONCEPTS & APPLIED QUESTIONS 111. A basketball player has the following points for seven games: 20, 25, 32, 18, 19, 22, and 30. Compute the following measures of variability. a. Standard deviation b. Coefficient of variation ANSWER: a. s =5.499 b. cv = 0.232 112. The following data represent the number of children in a sample of 10 families from a certain community: 4, 2, 1, 1, 5, 3, 0, 1, 0, and 2. a. Compute the range b. Compute the variance c. Compute the standard deviation d. Compute the coefficient of variation. ANSWER: a. 5 b. 2= 2.77 s c. s = 1.66 d. 0.87 FOR QUESTIONS 113 AND 114, USE THE FOLLOWING NARRATIVE: Narrative: Weights of Workers The following data represent the weights in pounds of a sample of 25 workers: 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. 113. {Weights of Workers Narrative} Compute the sample variance, and sample standard deviation. A2SWER: s = 156.12, and s =12.49 114. {Weights of Workers Narrative} Compute the range and coefficient of variation. ANSWER: Range = 43, cv = 12.49 / 159.04 = 0.079 115. Is it possible for the standard deviation of a data set to be larger than its variance? Explain. 88 Chapter Four ANSWER: Yes. A standard deviation is larger than its corresponding variance when the variance is between 0 and 1 (exclusive). FOR QUESTIONS 116 THROUGH 120, USE THE FOLLOWING NARRATIVE: Narrative: Ages of Teachers Three samples, regarding the ages of teachers, are selected randomly as shown below: Sample A: 17 22 20 18 23 Sample B: 30 28 35 40 25 Sample C: 44 39 54 21 52 116. {Ages of Teachers Narrative} Examine the three samples. Without performing any calculations, indicate which sample has the largest amount of variability and which sample has the least amount of variability. ANSWER: Sample C has the largest variability, with values ranging from 21 to 54. Sample A has the least variability, with all values close to 20. (x − x) 117. {Ages of Teachers Narrative} Calcula∑e i for the three samples. What can you infer about this calculation in general? ANSWER: ∑ (xi− x) equals zero for each of the three sa∑plei.− x= 0 is always true. 118. {Ages of Teachers Narrative}Calculate the variance for the three samples. ANSWER: 2 s = 6.50, 35.3, and 174.5 for samples A, B, and C, respectively 119. {Ages of Teachers Narrative} Compute the range for the three samples. ANSWER: Range = 6, 12, and 33 for samples A, B, and C, respectively 120. {Ages of Teachers Narrative}Compute the coefficient of variation for the three samples. ANSWER: cv = 0.127, 0.188, and 0.315 for samples A, B, and C, respectively 121. The number of hours a college student spent studying during the final exam week was recorded as follows: 7, 6, 4, 9, 8, 5, and 10. Compute the range for the data, express the number in the appropriate unit. ANSWER: Numerical Descriptive Techniques 89 Range = 6 hours 122. The number of hours a college student spent studying during the final exam week was recorded as follows: 7, 6, 4, 9, 8, 5, and 10. Computes2 and s for the data and express the numbers in the appropriate unit. ANSWER: s = 4.667 (hours)2 s = 2.160 hours 123. The annual percentage rates of return over the past 10 years for two mutual funds are as shown below. Which fund would you classifyas having the higher level of risk? Fund A: 7.1 -7.4 19.7 -3.9 32.4 41.7 23.2 4.0 1.9 29.3 Fund B: 10.8 -4.1 5.1 10.9 26.5 24.0 16.9 9.4 -2.6 10.1 ANSWER: Variance of returns will be used as the measure of risk of an investment. Since, 2 2 2 2 sA= 280.34(%) and sB= 99.37(%) , fund A has the higher level of risk. FOR QUESTIONS 124 THROUGH 126, USE THE FOLLOWING NARRATIVE: Narrative: Ages of Employees The following data represent the ages in years of a sample of 25 employees from a government department: 31, 43, 56, 23, 49, 42, 33, 61, 44, 28, 48, 38, 44, 35, 40, 64, 52, 42, 47, 39, 53, 27, 36, 35, and 20. 124. {Ages of Employees Narrative} Compute the range of the data, and express the number in the appropriate unit. ANSWER: Range = 44 years 125. {Ages of Employees Narrative} Compute the sample variance, and sample standard deviation, and express the numbers in the appropriate units. ANSWER: s = 124.83 (year)2, and s =11.17 years 126. {Ages of Employees Narrative} Compute the coefficient of variation, and express the number in the appropriate unite. ANSWER: cv = 11.17 years / 41.2 years = 0.271. CV has no units attached to it. 90 Chapter Four FOR QUESTIONS 127 THROUGH 130, USE THE FOLLOWING NARRATIVE: Narrative: Salaries of Employees The following data represent the salaries (in thousands of dollars) of a sample of 13 employees of a firm: 26.5, 23.5, 29.7, 24.8, 21.1, 24.3, 20.4, 22.7, 27.2, 23.7, 24.1, 24.8, and 28.2. 127. {Salaries of Employees Narrative} Compute the variance, and standard deviation of the salaries, and express the numbers in the appropriate units. ANSWER: 2 2 s = 7.097 (dolla)s, ands =2.664 dollars 128. {Salaries of Employees Narrative} Compute the coefficient of variation, and express the number in the appropriate unit. ANSWER: cv = 2.664 dollars / 24.692 dollars = 0.108. No units are attached to cv. 129. {Salaries of Employees Narrative} Compute the range. ANSWER: Range = 9.3 130. {Salaries of Employees Narrative} Consider the following population of measurements: 162, 152, 177, 157, 184, 176, 165, 181, 170, and 163. Compute the variance and the standard deviation. ANSWER: σ = 101.61 and σ = 10.08 FOR QUESTIONS 131 THROUGH 133, USE THE FOLLOWING NARRATIVE: Narrative: Egg Demand A supermarket has determined that daily demand for egg cartons has an approximate mound- shaped distribution, with a mean of 55 cartons and a standard deviation of six cartons. 131. {Egg Demand Narrative} For what percentage of days can we expect the number of cartons of eggs sold to be between 49 and 61? Numerical Descriptive Techniques 91 ANSWER: Approximately 68% 132. {Egg Demand Narrative} For what percentage of days can we expect the number of cartons of eggs sold to be more than 2 standard deviations from the mean? ANSWER: Approximately 5% 133. {Egg Demand Narrative} If the supermarket begins each morning with a stock of 77 cartons of eggs, for what percentage of days will there be an insufficient number of cartons to meet the demand? ANSWER: Approximately 2.5% 134. A sample of 13 high school teachers has a mean age of 30 years and a standard deviation of 5 years. Suppose that the sample is enlarged to 15 high school teachers, by including two additional teachers having common age of 30 each. Find the standard deviation of the new sample of 15 teachers. ANSWER: s =4.629 135. The price-earnings ratios of a sample of stocks have a mean value of 13.5 and a standard deviation of 2. If the ratios have a mound-shaped distribution, what can we say about the proportion of ratios that fallbetween a. 11.5 and 15.5? b. 9.5 and 17.5? c. 7.5 and 19.5? ANSWER: a. The interval contains approximately68% of the ratios.
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