336 Chapter 7 The Normal Distribution 4. The following boxplot represents a sample from a population. Is it reasonable to assume that this population is approximately normal? Yes 50 55 60 65 70 Answers are on page 341. Objective 5 Use normal quantile plots to assess normality Normal Quantile Plots Normal quantile plots are somewhat more complex than dotplots, histograms, and stemand leaf plots. We will present the idea behind normal quantile plots with an example. A simple random sample of size n = 5 is drawn, and we want to determine whether the population it came from is approximately normal. The five sample values, in increasing order, are 3.0 3.3 4.8 5.9 7.8 We proceed by using the following steps: Step 1: Let n be the number of values in the data set. Spread the n values evenly over the interval from 0 to 1. This is done by assigning the value 1∕(2n) to the first sample value, 3∕(2n) to the second, and so forth. The last sample value will be assigned the value (2n − 1)∕(2n). These values, denoted ai, represent areas under the normal curve. For n = 5, the values are 0.1, 0.3, 0.5, 0.7, and 0.9. i xi ai 1 3.0 0.1 2 3.3 0.3 3 4.8 0.5 4 5.9 0.7 5 7.8 0.9 Step 2: The values assigned in Step 1 represent left-tail areas under the normal curve. We now find the z-scores corresponding to each of these areas. The results are shown in the following table. i xi ai zi 1 3.0 0.1 −1.28 2 3.3 0.3 −0.52 3 4.8 0.5 0.00 4 5.9 0.7 0.52 5 7.8 0.9 1.28 Step 3: Plot the points (xi, zi). The plot is shown in Figure 7.44. A straight line has been added to the plot to help in interpreting the results. If the points approximately follow a straight line, then the population may be treated as being approximately normal. If the points deviate substantially from a straight line, the population should not be treated as normal. In this case, the points do approximately follow a straight line, so we may treat this population as approximately normal. Why do the points on a normal quantile plot tend to follow a straight line when the population is normal? If the population is normal, then, on the average, the values zi will be close to the actual z-scores of the xi. Now for any sample, the actual z-scores will follow
navidi_monk_elementary_statistics_2e_ch7-9
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