Section 3.1 Measures of Center 103 49. How many numbers? A data set has a median of 17, and six of the numbers in the data set are less than 17. The data set contains a total of n numbers. a. If n is odd, and exactly one number in the data set is equal to 17, what is the value of n? b. If n is even, and none of the numbers in the data set are equal to 17, what is the value of n? 50. How many numbers? A data set has a median of 10, and eight of the numbers in the data set are less than 10. The data set contains a total of n numbers. a. If n is odd, and exactly one of the numbers in the data set is equal to 10, what is the value of n? b. If n is even, and two of the numbers in the data set are equal to 10, what is the value of n? 51. What’s the score? Jermaine has entered a bowling tournament. To prepare, he bowls five games each day and writes down the score of each game, along with the mean of the five scores. He is looking at the scores from one day last week and finds that one of the numbers has accidentally been erased. The four remaining scores are 201, 193, 221, and 187. The mean score is 202. What is the missing score? 52. What’s your grade? Addison has been told that her average on six homework assignments in her history class is 85. She can find only five of the six assignments, which have scores of 91, 72, 96, 88, and 75. What is the score on the lost homework assignment? 53. Mean or median? The Smith family in Example 3.5 had the good fortune to win a million-dollar prize in a lottery. Their annual income for each of the five years leading up to their lottery win are as follows: 15,000 18,000 20,000 25,000 1,025,000 a. Compute the mean annual income. b. Compute the median annual income. c. Which provides a more appropriate description of the Smiths’ financial position, the mean or the median? Explain. 54. Mean or median? The incomes in a certain town of 1000 households are strongly skewed to the right. The mean income is $60,000, and the median income is only $40,000. The town is going to impose a 1% income tax, and the town council wants to estimate how much revenue will be generated. Which is the more relevant measure of center for the town council, the mean income or the median income? Explain. 55. Properties of the mean: Make up a data set in which the mean is equal to one of the numbers in the data set. 56. Properties of the median: Make up a data set in which the median is equal to one of the numbers in the data set. 57. Properties of the mean: Make up a data set in which the mean is not equal to one of the numbers in the data set. 58. Properties of the median: Make up a data set in which the median is not equal to one of the numbers in the data set. 59. The midrange: The midrange is a measure of center that is computed by averaging the largest and smallest values in a data set. In other words, Midrange = Largest value + Smallest value 2 Is the midrange resistant? Explain. 60. Mean, median, and midrange: A data set contains only two values. Are the mean, median, and midrange all equal? Explain. Extending the Concepts 61. Changing units: A sample of five college students have heights, in inches, of 65, 72, 68, 67, and 70. a. Compute the sample mean. b. Compute the sample median. c. Convert each of the heights to units of feet, by dividing by 12. d. Compute the sample mean of the heights in feet. Is this equal to the sample mean in inches divided by 12? e. Compute the sample median of the heights in feet. Is this equal to the sample median in inches divided by 12? 62. Effect on the mean and median: Four employees in an office have annual salaries of $30,000, $35,000, $45,000, and $70,000. a. Compute the mean salary. b. Compute the median salary. c. Each employee gets a $1000 raise. Compute the new mean. Does the mean increase by $1000? d. Each employee gets a 5% raise. Compute the new mean. Does the mean increase by 5%? 63. Nonresistant median: Consider the following data set: 0 0 1 1 1 1 1 1 2 2 8 8 9 9 9 9 9 9 10 10 a. Show that the mean and median are both equal to 5. b. Suppose that a value of 26 is added to this data set. Which is affected more, the mean or the median? c. Suppose that a value of 100 is added to this data set. Which is affected more, the mean or the median? d. It is possible for an extreme value to affect the median more than the mean, but if the value is extreme enough, the mean will be affected more than the median. Explain.
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