12.3 Measures of Central Tendency and Position 795 56. Physical Education Class Samantha took a flexibility test in her physical education class and scored at the 75th percentile. In a different class, Jose took the same flexibility test and scored at the 60th percentile. a) Is it possible to determine which of the two had the higher flexibility score on their respective exams? No b) Is it possible to determine which of the two was in a better relative position with regard to their respective classes? Explain. Yes. Samantha was in better relative position. 57. Employee Salaries The following statistics represent weekly salaries at the Midtown Construction Company: Mean $620 First quartile $580 Median $610 Third quartile $645 Mode $600 83rd percentile $685 a) What is the most common salary? $600 b) What salary did half the employees’ salaries surpass? $610 c) About what percent of employees’ salaries surpassed $645? 25% d) About what percent of employees’ salaries were less than $580? 25% e) About what percent of employees’ salaries surpassed $685? 17% f) If the company has 100 employees, what is the total weekly salary of all employees? $62,000 58. Employee Salaries The following statistics represent annual salaries of physical therapists at Lattimore Physical Therapy: Mean $88,000 First quartile $72,000 Median $86,520 Third quartile $93,000 Mode $80,000 90th percentile $99,000 a) What is the most common salary? $80,000 b) What salary did half of the physical therapists’ salaries surpass? $86,520 c) About what percent of physical therapists’ salaries surpassed $93,000? 25% d) About what percent of physical therapists’ salaries were less than $72,000? 25% e) About what percent of physical therapists’ salaries surpassed $99,000? 10% f) If Lattimore Physical Therapy employs 25 physical therapists, what is the total annual salary of all employees? $2,200,000 Concept/Writing Exercises 59. Life Expectancy In 2021, the National Center for Health Statistics indicated an “average life expectancy” of 76 years for the total U.S. population. The average life expectancy for men was 73.2 years, and for women it was 79.1 years. Which “average” do you think the National Center for Health Statistics is using? * 60. A Grade of B To get a grade of B, a student must have a mean average of 80 or greater. Jim has a mean average of 79 for 10 quizzes. He approaches his teacher and asks for a B, reasoning that he missed a B by only one point. What is wrong with Jim’s reasoning? * 61. Creating a Data Set Construct a set of five pieces of data in which the mode has a lower value than the median and the median has a lower value than the mean. One example is 1, 1, 2, 5, 6. 62. Creating a Data Set Construct a set of six pieces of data with a mean, median, and midrange of 75 and where no two pieces of data are the same. One example is 72, 73, 74, 76, 77, 78. 63. Creating a Data Set Construct a set of six pieces of data with a mean of 84 and where no two pieces of data are the same. One example is 81, 82, 83, 85, 86, 87. 64. Creating a Data Set Construct a set of six pieces of data such that if only one piece of data is changed, the mean, median, and mode will all change. One example: 1, 2, 3, 3, 4, 5 changed to 1, 2, 3, 4, 4, 5 65. Central Tendencies Which of the measures of central tendency must be an actual piece of data in the distribution? Mode 66. Changing One Piece of Data Is it possible to construct a set of six different pieces of data such that by changing only one piece of data you cause the mean, median, mode, and midrange to change? Explain. No. By changing only one piece of the 6 pieces of data, you cannot alter both the median and the midrange. 67. Percentiles For any set of data, what must be done to the data before percentiles can be determined? The data must be ranked. 68. Percentiles Josie scored in the 73rd percentile on the verbal part of her College Board test. What does that mean? She scored above approximately 73 percent of all the students who took the test. 69. Percentiles When a national sample of heights of kindergarten children was taken, Kevin was told that he was in the 35th percentile. Explain what that means. He is taller than approximately 35 percent of all kindergarten children. 70. Quartiles A union leader is told that, when all workers’ salaries are considered, the first quartile is $27,750. Explain what that means. About 25% of the workers earn $27,750 or less. 71. The 50th Percentile Give the names of two other statistics that have the same value as the 50th percentile. Second quartile, median Challenge Problems/Group Activities 72. The Mean of the Means Consider the following five sets of data. i) 5 6 7 7 8 9 14 ii) 3 6 8 9 iii) 1 1 1 2 5 iv) 6 8 9 12 15 v) 50 51 55 60 80 100 $ $ $
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