114 Chapter 3 Numerical Summaries of Data Solution a. We use the Empirical Rule. Approximately 68% of the data will be between ¯x − s and ¯x + s. We compute ¯x − s = 50 − 10 = 40 ¯x + s = 50 + 10 = 60 It is likely that approximately 68% of the data values are between 40 and 60. b. The value 30 is two standard deviations below the mean, since ¯x − 2s = 50 − 20 = 30 and the value 70 is two standard deviations above the mean, since ¯x + 2s = 50 + 20 = 70 Therefore, it is likely that approximately 95% of the data values are between 30 and 70. EXAMPLE 3.19 Determining whether the Empirical Rule is appropriate Following is a histogram for a data set. Should the Empirical Rule be used? 0 1 2 3 4 5 6 7 8 9 10 12 10 8 6 4 2 0 Frequency Solution No. The distribution is skewed, rather than bell-shaped. Therefore, the Empirical Rule should not be used. Check Your Understanding 4. A data set has a mean of 20 and a standard deviation of 3.Ahistogram is shown here. Is it appropriate to use the Empirical Rule to approximate the proportion 0.3 0.2 0.1 of the data between 14 and 26? If so, find the approximation. If not, explain why not. 0 10 12 14 16 18 20 22 24 26 28 30 32 Relative Frequency 5. A data set has a mean of 50 and a standard deviation of 8. A histogram is shown here. Is it appropriate to use the Empirical Rule to approximate the proportion of the data between 42 and 58? If so, find the approximation. If not, explain why not. 40 45 50 55 60 65 70 75 0.4 0.3 0.2 0.1 0 Relative Frequency Answers are on page 123. Objective 6 Use Chebyshev’s Inequality to describe a data set Chebyshev’s Inequality When a distribution is bell-shaped, the Empirical Rule gives us an approximation to the proportion of data that will be within one or two standard deviations of the mean. Chebyshev’s Inequality is a rule that holds for any data set.
navidi_monk_essential_statistics_1e_ch1_3
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