Section 3.2 Measures of Spread 123 50. Price of electricity: The Energy Information Administration records the price of electricity in the United States each month. In January 2010, the average price of electricity was 10.54 cents per kilowatt-hour. Suppose that the standard deviation is 2.1 cents per kilowatt-hour. What can you determine about these data by using Chebyshev’s Inequality with K = 3? 51. Possible or impossible? A data set has a mean of 20 and a standard deviation of 5. Which of the following might possibly be true, and which are impossible? a. Less than 50% of the data values are between 10 and 30. b. Only 1% of the data values are greater than 35. c. More than 15% of the data values are less than 5. d. More than 90% of the data values are between 5 and 35. 52. Possible or impossible? A data set has a mean of 50 and a standard deviation of 10. Which of the following might possibly be true, and which are impossible? a. More than 10% of the data values are negative. b. Only 5% of the data values are greater than 70. c. More than 20% of the data values are less than 30. d. Less than 75% of the data values are between 30 and 70. 53. Standard deviation and mean: For a list of positive numbers, is it possible for the sample standard deviation to be greater than the mean? If so, give an example. If not, explain why not. 54. Standard deviation equal to 0? Is it possible for the sample standard deviation of a list of numbers to equal 0? If so, give an example. If not, explain why not. 55. Height and weight: A National Center for Health Statistics study states that the mean height for adult men in the United States is 69.4 inches with a standard deviation of 3.1 inches, and the mean weight is 194.7 pounds with a standard deviation of 68.3 pounds. a. Compute the coefficient of variation for height. b. Compute the coefficient of variation for weight. c. Which has greater spread relative to its mean, height or weight? 56. Test scores: Scores on a statistics exam had a mean of 75 with a standard deviation of 10. Scores on a calculus exam had a mean of 60 with a standard deviation of 9. a. Compute the coefficient of variation for statistics exam scores. b. Compute the coefficient of variation for calculus exam scores. c. Which has greater spread relative to their mean, statistics scores or calculus scores? Extending the Concepts 57. Mean absolute deviation: A measure of spread that is an alternative to the standard deviation (SD) is the mean absolute deviation (MAD). For a data set containing values x1, ..., xn, the mean absolute deviation is given by Mean absolute deviation = |x − ¯x| n a. Compute the mean ¯x for the data set 1, 3, 4, 7, 9. b. Construct a table like Table 3.5 that contains an additional column for the values |x − ¯x|. c. Use the table to compute the SD and the MAD. d. Now consider the data set 1, 3, 4, 7, 9, 30. Compute the SD and the MAD for this data set. e. Which measure of spread is more resistant, the SD or the MAD? Explain. Answers to Check Your Understanding Exercises for Section 3.2 1. The variance of the St. Louis temperatures is 291.9. This is greater than the variance of the San Francisco temperatures, which indicates that there is greater spread in the St. Louis temperatures. 2. a. Variance is 3.7143; standard deviation is 1.9272. b. Variance is 281.8667; standard deviation is 16.7889. 3. Variance is 4.8594; standard deviation is 2.2044. 4. Approximately 95% of the data values are between 14 and 26. 5. The Empirical Rule should not be used because the data are skewed. 6. No. The interval between 50 and 90 is the interval within two standard deviations of the mean. At least 75% of the data must be between 50 and 90. 7. Yes. The interval between 122 and 128 is the interval within three standard deviations of the mean. At least 8/9 (88.9%) of the data must be between 122 and 128. 8. 0.0945
navidi_monk_essential_statistics_1e_ch1_3
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