Chapter 3: Describing Data: Summary Measuresdscaas/testbank/TB Ch … · Web view ·...

CHAPTER 3

DESCRIBING DATA: SUMMARY MEASURES

MULTIPLE CHOICE QUESTIONS

In the following multiple-choice questions, please circle the correct answer.

1. Which of the following are the three most common measures of central location?

a. Mean, median and modeb. Average, variance and standard deviationc. Mode, sample mean, and sample varianced. Mean, median, and averageANSWER: a

2. Which of the following are considered measures of association?

a. Mean and varianceb. Variance and correlationc. Covariance and correlationd. Covariance and varianceANSWER: c

3. The difference between the first and third quartile is called the .

a. interquartile rangeb. interdependent rangec. unimodal ranged. common occurrence range e. none of the aboveANSWER: a

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4. If a value represents the 95th percentile, this means:

a. 95% of all values are below this pointb. 95% of all values are above this point c. 95% of the time you will observe this valued. there is a 5% chance that this value is incorrecte. none of the aboveANSWER: a

5. For a boxplot, the point inside the box indicates the location of the __________.

a. meanb. medianc. minimum valued. maximum valueANSWER: a

6. For a boxplot, the vertical line inside the box indicates the location of the______.

a. meanb. medianc. moded. minimum valuee. maximum valueANSWER: b

7. Suppose that a histogram of a data set is approximately symmetric and "bell shaped". Approximately percent of the observations are within three standard deviation of the mean.

a. 50b. 68c. 95d. 99.7e. 100ANSWER: d

8. The correlation coefficient is always:

a. between – 1 and 0b. between 0 and +1c. between 0.0 and 100d. between –1 and +1ANSWER: d

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Describing Data: Summary Measures

9. Which of the following are the two most commonly used measures of variability?

a. Variance and modeb. Variance and standard deviationc. Sample mean, and sample varianced. Mean and rangeANSWER: b

10. Boxplots are probably most useful for .

a. calculating the mean of the datab. calculating the median of the datac. comparing the mean to the mediand. comparing two populations graphicallye. comparing the correlation and the covarianceANSWER: d

11. Suppose that a histogram of a data set is approximately symmetric and "bell shaped". Approximately percent of the observations are within one standard deviation of the mean.

a. 50b. 68c. 95d. 99.7e. 100ANSWER: b

12. Suppose that a histogram of a data set is approximately symmetric and "bell shaped". Approximately percent of the observations are within two standard deviation of the mean.

a. 50b. 68c. 95d. 99.7e. 100ANSWER: c

13. The mode is best described as:

a. the middle observationb. the same as the averagec. the 50th percentiled. the most frequently occurring valueANSWER: d

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14. The median can also be described as:

a. the middle observation when the data is arranged in ascending orderb. the second quartilec. the 50th percentiled. all of the abovee. none of the aboveANSWER: d

15. For a boxplot, the box itself represents percent of the observations.

a. 25b. 50c. 75d. 90e. 100ANSWER: b

16. Which of the following best describes a data warehouse?

a. A physical facility full of databasesb. A storage facility for government informationc. A database specifically structured to enable data miningd. A database used to track warehousing informatione. None of the aboveANSWER: c

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TEST QUESTIONS

QUESTIONS 17 AND 18 ARE BASED ON THE FOLLOWING INFORMATION:

Statistics professor has just given a final examination in his statistical inference course. He is particularly interested in learning how his class of 40 students performed on this exam. The scores are shown below.

75 79 72 75 77 71 78 83 84 7181 82 79 71 73 89 74 75 93 7488 83 90 82 79 62 73 88 76 76

80 76 84 84 80 68 74 76 70 91

17. What are the mean and median scores on this exam?

ANSWER:Mean = 78.40, Median = 77.50

18. Explain why the mean and median are different.

ANSWER:There are few higher exam scores that tend to pull the mean away from the middle of the distribution. While there is a slight amount of positive skewness in the distribution (skewness = 0.182), the mean and the median are essentially equivalent in this case.


The average time in minutes, for a sample of 40 people living in Detroit metropolitan area to travel to work and back home each day is shown below.

42.8 47.1 46.3 42.2 55.9 35.0 41.4 37.8 33.9 36.651.8 43.1 51.5 45.5 40.7 57.6 36.5 32.3 41.9 37.041.3 32.6 34.1 31.9 42.9 48.5 46.7 40.9 46.6 29.561.5 36.5 48.9 40.6 48.1 39.8 35.6 44.4 38.6 45.6

19. Find the most representative average commute time.

ANSWER:The most representative average commute time for Detroit metropolitan area is slightly greater than 41 minutes. The median is the best measure of central location due to the distribution positive skewness (skewness = 0.573).

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20. Does it appear that the distribution of average commute time is approximately symmetric? Explain why or why not.

ANSWER:The distribution is slightly skewed to the right; it is essentially not symmetric. Notice that the mean is slightly greater than the median here; the mean is influenced by the existence of some relatively large commute times in the sample.


A production manager for Bell Computers is interested in determining the variability of the proportion defective items in a shipment of one of the computer components used in their product. The results below were calculated using information on the percent defective from 50 randomly selected shipments collected over a 6-week period.

Count 50.000Mean 0.210Median 0.206Standard deviation 0.057Minimum 0.104Maximum 0.299Variance 0.003First quartile 0.153Third quartile 0.248

21. Use the above information to determine the rules of thumb in this case.

ANSWER:Approximately 68% of the values will fall between 0.210 0.057, that is between 0.153 and 0.267.Approximately 95% of the values will fall between 0.210 2 (0.057), that is between 0.096 and 0.324.Approximately 99.7% of the values will fall between 0.210 3 (0.057), that is between 0.039 and 0.381.

22. Explain what would cause the mean to be higher than the median in this case.

ANSWER:The data seems to be skewed to the right. This is causing the mean to be slightly higher than the median.

QUESTIONS 23 THROUGH 25 ARE BASED ON THE FOLLOWING INFORMATION:

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A manager for Marko Manufacturing, Inc. has recently been hearing some complaints that women are being paid less than men for the same type of work in one of their manufacturing plants. The boxplots shown below represent the annual salaries for all salaried workers in that facility (40 men and 34 women).

23. Would you conclude that there is a difference between the salaries of women and men in this plant? Justify your answer.

ANSWER:Yes. The men seem to have higher salaries than the women do in many cases. We can see from the boxplots that the mean and median values for the men are both higher than for the women. You can also see from the boxplots that the middle 50% of salaries for men is above the median for women. This means that if you were in the 25th percentile for men, you would be above the 50th percentile for women. You can also see that the mean and median salaries for the men are about $10,000 above those for the women.

24. How large must a person’s salary should be to qualify as an outlier on the high side? How many outliers are there in these data?

ANSWER:A person’s salary should be somewhere above $70,000. There is one male salary that would be considered an outlier (at approximately $80,000)

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25. What can you say about the shape of the distributions given the boxplots above?

ANSWER:They both appear to be slightly skewed to the right (both have a mean > median). The total variation seems to be close for both distributions (with one outlier for the male salaries), but there seems to be more variation in the middle 50% for the women than for the men. There seem to be more men’s salaries clustered more closely around the mean than for the women.


Below you will find current annual salary data and related information for 30 employees at Gamma Technologies, Inc. These data include each selected employees gender (1 for female; 0 for male), age, number of years of relevant work experience prior to employment at Gamma, number of years of employment at Gamma, the number of years of post-secondary education, and annual salary. The tables of correlations and covariances are presented below.

Table of Correlations

Gender Age Prior Exp Gamma Exp Education SalaryGender 1.000Age -0.111 1.000Prior_Exp 0.054 0.800 1.000Gamma_Exp -0.203 0.916 0.587 1.000Education -0.039 0.518 0.434 0.342 1.000Salary -0.154 0.923 0.723 0.870 0.617 1.000

Table of Covariances (variances on the diagonal)

Gender Age Prior Exp Gamma Exp Education SalaryGender 0.259Age -0.633 134.051Prior Exp 0.117 39.060 19.045Gamma Exp -0.700 72.047 17.413 49.421Education -0.033 9.951 3.140 3.987 2.947Salary -1825.97 249702.35 73699.75 143033.29 24747.68 584640062

26. Which two variables have the strongest linear relationship with annual salary?

ANSWER:Age at 0.923 and Gamma experience at 0.870 have the strongest linear relationship with annual salary.

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27. For which of the two variables, number of years of prior work experience or number of years of post-secondary education, is the relationship with salary stronger? Justify your answer.

ANSWER:Prior work experience is stronger at 0.723 versus 0.617 for number of years of post-secondary education.

28. How would you characterize the relationship between gender and annual salary?

ANSWER:It is a somewhat weak relationship at –0.154. Also, the negative value tells us that the salaries are decreasing as the gender value increases. This indicates that the salaries are lower for females (1) than for males (0).


The data shown below contains family incomes (in thousands of dollars) for a set of 50 families; sampled in 1980 and 1990. Assume that these families are good representatives of the entire United States.

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1980 1990 1980 1990 1980 1990 58 54 33 29 73 69 6 2 14 10 26 22 59 55 48 44 64 70 71 57 20 16 59 55 30 26 24 20 11 7 38 34 82 78 70 66 36 32 95 97 31 27 33 29 12 8 92 88 72 68 93 89 115 111100 96 100 102 62 58 1 0 51 47 23 19 27 23 22 18 34 30 22 47 50 75 36 61141 166 124 149 125 150 72 97 113 138 121 146165 190 118 143 88 113 79 104 96 121

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29. Find the mean, median, standard deviation, first and third quartiles, and the 95 th

percentile for family incomes in both years.

ANSWER: Income 1980 Income 1990

MeanMedianStandard deviationFirst quartileThird quartile95th percentile

62.82059.00039.78630.25092.750124.550

67.12057.50048.08727.50097.000149.55

30. The Republicans claim that the country was better off in 1990 than in 1980, because the average income increased. Do you agree?

ANSWER:It is true that the mean increased slightly, but the median decreased and the standard deviation increased. The 95th percentile shows that the mean increase might be because the rich got riches.

31. Generate a boxplot to summarize the data. What does the boxplot indicate?

ANSWER:The boxplot shows that there is not much difference between the two populations.

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Income_1980

Income_1990

0 40 80 120 160 200



In an effort to provide more consistent customer service, the manager of a local fast-food restaurant would like to know the dispersion of customer service times about their average value for the facility’s drive-up window. The data below represents the customer service times (in minutes) for a sample of 47 customers collected over the past week.


32. Interpret the variance and standard deviation of these sample data.

ANSWER:The variance = 0.261 and this represents the average of the squared deviations from the mean. The standard deviation = 0.511 and is the square root of the variance. The standard deviation is in minutes (the original units).

33. Determine the rules of thumb in this case.

ANSWER:Approximately 68% of the values will fall between 0.914 0.511, that is between 0.403 and 1.425.Approximately 95% of the values will fall between 0.914 2(0.511), that is between (-0.108 and 1.936.Approximately 99.7% of the values will fall between 0.914 3(0.511), that is between -0.632 and 2.447.

34. Does it appear as though most of the data falls within 3 standard deviations of the mean? Explain.

ANSWER:Since the minimum value is 0.095 and the maximum value is 2.372, then all of the data falls within three standard deviations of the mean.

35. Explain what would cause the mean to be higher than the median in this case.

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ANSWER:The data is skewed to the right. This is causing the mean to be higher than the median. It is important to understand that service times are bounded on the lower end by zero (or it is impossible for the service time to be negative). However, there is no bound on the maximum service time. Therefore, the larger service times are causing the mean to be somewhat higher than the median.


The percentage of the US population without health insurance coverage is shown below with samples from the 50 states and District of Columbia for both 1997 and 1998.

State % in 97 % in 98 State % in 97 % in 98Alabama 19.9 14.2 Montana 14.3 13.4Alaska 14.0 13.2 Nebraska 11.4 9.7Arizona 20.9 21.1 Nevada 16.4 19.4Arkansas 18.1 18.6 New Hampshire 12.6 10.7California 21.8 21.3 New Jersey 13.7 14.9Colorado 13.1 15.5 New Mexico 23.8 26.3Connecticut 11.1 9.5 New York 16.7 15.9Delaware 14.2 16.4 North Carolina 14.0 15.0District of Columbia 17.1 18.0 North Dakota 9.1 9.0Florida 17.9 19.0 Ohio 11.7 12.6Georgia 16.9 18.6 Oklahoma 18.5 19.9Hawaii 9.9 9.6 Oregon 13.8 13.2Idaho 14.7 14.7 Pennsylvania 11.3 10.6Illinois 12.1 11.7 Rhode Island 12.2 13.6Indiana 11.2 13.3 South Carolina 14.9 15.3Iowa 10.4 12.0 South Dakota 10.7 10.1Kansas 13.6 13.1 Tennessee 10.9 15.5Kentucky 15.9 15.3 Texas 24.9 25.2Louisiana 19.9 21.2 Utah 12.2 12.4Maine 13.8 14.2 Vermont 9.3 13.9Maryland 13.3 16.0 Virginia 12.7 14.2Massachusetts 13.2 11.8 Washington 13.4 13.1Michigan 11.5 10.4 West Virginia 16.9 16.6Minnesota 10.2 8.7 Wisconsin 9.6 8.0Mississippi 18.5 20.4 Wyoming 16.1 16.6Missouri 12.9 15.3

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36. Describe the distribution of state percentages of Americans without health insurance coverage in 1998. Be sure to employ both measures of central location and dispersion in developing your characterization of this sample.

ANSWER:

Count 51.000Mean 14.855Median 14.200Standard deviation 4.098Minimum 8.000Maximum 26.300First quartile 12.200Third quartile 16.500Skewness 0.699

This distribution is slightly skewed to the right, as evidenced by the fact that its mean (14.855%) exceeds its median (14.2%). The standard deviation of the 1998 distribution is approximately 4.10%.

37. Compare the 1998 distribution of percentages with the corresponding set of percentages taken in 1997. How are these two sets similar? In what ways are they different?

ANSWER:

Percentage in 97 Percentage in 98Count 51.000 51.000Mean 14.455 14.855Median 13.700 14.200Standard deviation 3.724 4.098Minimum 9.100 8.000Maximum 24.900 26.300First quartile 11.600 12.200Third quartile 16.800 16.500Skewness 0.910 0.699

The two distributions are similar in that both are somewhat positively skewed (even though the 1997 distribution is more skewed than the 1998 distribution). They are different inasmuch as the 1998 distribution has higher measures of central location and dispersion than does the 1997 distribution.

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38. Compute a correlation measure for the two given sets of percentages. What does the correlation tell you in this case?

ANSWER:Table of correlations

Percent 97 Percent 98

There is a very large positive correlation between these two sets of percentages. This indicates that the percentages tend to move together, although not perfectly.

39. Based on your answers to Questions 37 and 38, what would you expect to find upon analyzing similar data for 1999?

ANSWER:Based on the answers to Questions 37 and 38 above, it is reasonable to expect that the percent of the U.S. population without health insurance coverage in 1999 will be slightly greater than in the previous year. Also, we expect that there will be greater variability about the mean in 1999 than was the case in 1998.


Below you will find summary measures on salaries for classroom teachers across the United States. You will also find a list of selected states and their average teacher salary. All values are in thousands of dollars.

State SalaryAlabama 31.3Colorado 35.4Connecticut 50.3Delaware 40.5Nebraska 31.5Nevada 36.2New Hampshire 35.8New Jersey 47.9New Mexico 29.6South Carolina 31.6South Dakota 26.3Tennessee 33.1

Texas 32.0Utah 30.6Vermont 36.3Virginia 35.0Wyoming 31.6

40. Which of the states listed paid their teachers average salaries that exceed at least 75% of all average salaries?

ANSWER:

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Percent 97 1.000Percent 98 0.903 1.000

SalaryCount 51.000Mean 35.890Median 35.000Standard deviation 6.226Minimum 26.300Maximum 50.300Variance 38.763First quartile 31.550Third quartile 40.050


Connecticut at 50.3; Delaware at 40.5; and New Jersey at 47.9 (all those > 40.05).

41. Which of the states listed paid their teachers average salaries that are below 75% of all average salaries?

ANSWER:Alabama at 31.3; Nebraska at 31.5; New Mexico at 29.6; South Dakota at 26.3; and Utah at 30.6 (all those < 31.55).

42. What salary amount represents the second quartile?

ANSWER:$35,000 (median)

43. How would you describe the salary of Virginia’s teachers compared to those across the entire United States? Justify your answer.

ANSWER:Virginia = $35,000 which is also the median. Virginia is at the 50 th percentile or 50% of the teachers’ salaries across the U.S. are below Virginia and 50% of the salaries are above theirs.


A manager at Abby Technologies, Inc. (ATI) is interested in analyzing the compensation of employees at her company. ATI currently has 25 employees and below you will find summary measures for the 25 employees.


44. Interpret the mean and median values.

ANSWER:

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The mean value of $43,957.20 represents the average compensation for the 25 employees. The median value of $42,220.00 represents the 50th percentile or the second quartile. This means that 50% of the values are below $42,220.

45. Why is there a difference in the mean and median values? Which of these two values do you believe is more appropriate in describing this distribution?

ANSWER:The mean could be somewhat inflated due to one or two large values (such as the maximum compensation of $71,250). The median may be more appropriate.

46. Explain what the first quartile and third quartile values represent.

ANSWER:The first quartile means that 25% of the employees have a compensation package below $37,250 (also means that 75% of the employees have a compensation package above this amount). The third quartile means that 75% of the employees have a compensation package below $51,125 (also means that 25% of the employees have a compensation package above this amount).

47. What range of values would contain approximately 95% of the observations?

ANSWER:Approximately $43,957.20 2(12,478.24) or from $19,000.72 to $68,913.68.

48. Do any of the values fall outside of the range calculated in Question 47?

ANSWER:Yes, the minimum value ($18,440) and the maximum value ($71,250) fall outside of this range.


The following are two sets of data, each with a sample of size n = 9

Set 1 5 2 4 2 3 2 10 2 12Set 2 15 12 14 12 13 12 20 12 22

49. For each data, compute the mean, median, and mode

ANSWER:

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Set 1 Set 2Mean 4.667 14.667Median 3 13Mode 2 12

50. Compare your results in Question 49, and summarize your findings

ANSWER:The measures of central tendency (mean, median, and mode) for Set 2 are all 10 more than the comparable statistics for Set 1.

51. Compare the first sampled item in each set, compare the second sampled item in each set, and so on. Briefly describe your findings in light of your summary in Question 50.

ANSWER:The data values in Set 2 are each 10 more than the corresponding values in Set 1. The is the reason behind the results in Question 50.

52. For each set, compute the range, interquartile range, variance and standard deviation.

ANSWER:

Set 1 Set 2Range 10 10Interquartile range 5 5Variance 14.25 14.25Standard deviation 3.7749 3.7749

53. Describe the shape of each set.

ANSWER:Since the mean is greater than the median for each data set, the distributions are both right-skewed.

54. Compare your results in Questions 52 and 53, and discuss your findings.

ANSWER:Because the data values in Set 2 are each 10 more than the corresponding values in Set 1, the measures of spread among the data values remain the same across the two sets. Set 2 is a reflection of Set 1 simply shifted up the scale 10 units, so the distributions are also reflections of each other.

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55. On the basis of your answers to Questions 49 through 54, what can you generalize about the properties of central tendency, variation, and shape?

ANSWER:Generally stated, when a second data set is an additive shift from an original set, the measures of central tendency for the second set are equal to the comparable measures for the original set plus the value, or distance, of the shift; the measures of spread for the second set are equal to the corresponding measures for the original set, and the shape of the second distribution will be a reflection of the shape of the original distribution.

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TRUE / FALSE QUESTIONS

56. If all values in a data set are negative, the value of the standard deviation may be either positive or negative.ANSWER: F

57. Assume that the histogram of a data set is symmetric and bell shaped, with mean of 72 and standard deviation of 10. Then, using the “rules of thumb”, we can say that 95% of the data values were between 52 and 92.ANSWER: T

58. If a histogram has a single peak and looks approximately the same to the left and right of the peak, we should expect no difference in the values of the mean, median, and mode.ANSWER: T

59. The mean is a measure of central location.ANSWER: T

60. In a negatively skewed distribution, the mean is larger than the median and the median is larger than the mode.ANSWER: F

61. Since the population is always larger than the sample, the population mean is always larger than the sample mean.ANSWER: F

62. The length of the box in the boxplot portrays the interquartile range. ANSWER: T

63. In a positively skewed distribution, the mean is smaller than the median and the median is smaller than the mode.ANSWER: F

64. The interquartile range is considered the weakest measure of central location.ANSWER: F

65. The value of the standard deviation always exceeds that of the variance.ANSWER: F

66. The difference between the first and third quartiles is called the interquartile range.

ANSWER: T

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67. The standard deviation is measured in original units, such as dollars and pounds.ANSWER: T

68. The median is one of the most frequently used measures of variability.ANSWER: F

69. Each of the covariance and correlation measures the strength and direction of a linear relationship between two numerical variables.ANSWER: T

70. Abby has been keeping track of what she spends to rent movies. The last seven week's expenditures, in dollars, were 6, 4, 8, 9, 6, 12, and 4. The mean amount Abby spends on renting movies is $7. ANSWER: T

71. If two data sets have the same range, the smallest and largest observations in both sets will be the sameANSWER: F

72. The variance is a measure of the linear relationship between two variablesANSWER: F

73. Generally speaking, if two variables are unrelated, the covariance will be a positive or negative number close to zeroANSWER: T

74. The correlation between two variables is a unitless quantity that is always between –1 and +1.ANSWER: T

75. The variance is the positive square root of the standard deviation.ANSWER: F

76. It is possible that the data points are close to a curve and have a correlation close to 0, because correlation is relevant only for measuring linear relationships.ANSWER: T

77. Expressed in percentiles, the interquartile range is the difference between the 25th

and 75th percentiles.ANSWER: T

78. A data sample has a mode 0f 140, a median of 130, and a mean of 120. The distribution of the data is positively skewed.

ANSWER: F

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79. A student scores 85, 75, and 80 on three exams during the semester and 90 on the final exam. If the final is weighted double and the three others weighted equally, the student's final average would be 80.ANSWER: F

80. Suppose that a sample of 10 observations has a standard deviation of 3, then the sum of the squared deviations from the sample mean is 30.ANSWER: F

81. The value of the mean times the number of observations equals the sum of all of the data values.ANSWER: T

82. The difference between the largest and smallest values in a data set is called the range.ANSWER: T

83. There are four quartiles that divide the values in a data set into four equal parts.ANSWER: F

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