Section 3.2 Measures of Spread 117 SECTION 3.2 Exercises Exercises 1–8 are the Check Your Understanding exercises located within the section. Understanding the Concepts In Exercises 9–12, fill in each blank with the appropriate word or phrase. 9. If all values in a data set are the same, then the sample variance is equal to . 10. The standard deviation is the square root of the . 11. For a bell-shaped data set, approximately of the data will be in the interval μ − σ to μ + σ. 12. Chebyshev’s Inequality states that for any data set, the proportion of data within K standard deviations of the mean is at least . In Exercises 13–16, determine whether the statement is true or false. If the statement is false, rewrite it as a true statement. 13. The variance and standard deviation are measures of center. 14. The range of a data set is the difference between the largest value and the smallest value. 15. In a bell-shaped data set with μ = 15 and σ = 5, approximately 95% of the data will be between 10 and 20. 16. For some data sets, Chebyshev’s Inequality may be used but the Empirical Rule should not be. Practicing the Skills 17. Find the sample variance and standard deviation for the following sample: 17 40 24 18 16 18. Find the sample variance and standard deviation for the following sample: 59 25 12 29 16 8 26 30 17 19. Find the sample variance and standard deviation for the following sample: 15 9 5 12 9 21 4 24 18 20. Find the population variance and standard deviation for the following population: 16 6 18 3 25 22 21. Find the population variance and standard deviation for the following population: 20 8 11 23 27 29 62 4 22. Find the population variance and standard deviation for the following population: 26 25 29 23 14 20 12 18 24 31 22 32 23. Approximate the sample variance and standard deviation given the following frequency distribution: Class Frequency 0–9 13 10–19 7 20–29 10 30–39 9 40–49 11 24. Approximate the sample variance and standard deviation given the following frequency distribution: Class Frequency 0–15 2 16–31 14 32–47 6 48–63 13 64–79 15 25. Approximate the population variance and standard deviation given the following frequency distribution: Class Frequency 0–49 17 50–99 26 100–149 14 150–199 34 200–249 26 250–299 8 26. Approximate the population variance and standard deviation given the following frequency distribution: Class Frequency 0–19 18 20–39 11 40–59 6 60–79 6 80–99 10 100–119 5 27. The following TI-84 Plus display presents some population parameters. a. Assume the population is bell-shaped. Approximately what percentage of the population values are between 26 and 38? b. Assume the population is bell-shaped. Between what two values will approximately 95% of the population be? c. If we do not assume that the population is bell-shaped, at least what percentage of the population will be between 20 and 44? 28. The following TI-84 Plus display presents some population parameters. a. Assume the population is bell-shaped. Approximately what percentage of the population values are between 124 and 144?
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