Section 3.2 Measures of Spread 113 8 9 10 11 12 13 14 15 16 17 18 16 14 12 10 8 6 4 2 0 Frequency Percentage of people aged 65 and over Figure 3.7 Histogram for the data in Table 3.8 Solution Step 1: Figure 3.7 shows that the histogram is approximately bell-shaped, so we may use the Empirical Rule. Step 2: We use the TI-84 Plus to compute the mean and standard deviation. The display is shown here. Note that the 51 entries (corresponding to 50 states plus the District of Columbia) are an entire population. Therefore we will interpret the quantity ¯x = 13.24901961 produced by the TI-84 Plus as the population mean μ, and we will use σ = 1.682711694 for the standard deviation. Step 3: We compute the quantities μ−σ, μ+σ, μ−2 σ, μ+2 σ, μ−3 σ, and μ+3 σ. μ − σ = 13.24901961 − 1.682711694 = 11.57 μ + σ = 13.24901961 + 1.682711694 = 14.93 μ − 2σ = 13.24901961 − 2(1.682711694) = 9.88 μ + 2σ = 13.24901961 + 2(1.682711694) = 16.61 μ − 3σ = 13.24901961 − 3(1.682711694) = 8.20 μ + 3σ = 13.24901961 + 3(1.682711694) = 18.30 We conclude that the percentage of the population aged 65 and over is between 11.57 and 14.93 in approximately 68% of the states, between 9.88 and 16.61 in approximately 95% of the states, and between 8.20 and 18.30 in almost all the states. The Empirical Rule can be used for samples as well as populations. When we work with a sample, we use ¯x in place of μ and s in place of σ. EXAMPLE 3.18 Using the Empirical Rule to describe a data set A sample of size 200 has sample mean ¯x = 50 and sample standard deviation s = 10. The histogram is approximately bell-shaped. a. Find an interval that is likely to contain approximately 68% of the data values. b. Approximately what percentage of the data values will be between 30 and 70?
navidi_monk_essential_statistics_1e_ch1_3
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