6-2 Real Applications of Normal Distributions 261 Most statistics calculators and software do not require the use of Formula 6-2 to convert to z scores because probabilities can be found directly. However, if using Table A-2, we must first convert values to standard z scores. When finding areas with a nonstandard normal distribution, use the following procedure. Procedure for Finding Areas with a Nonstandard Normal Distribution 1. Sketch a normal curve, label the mean and any specific x values, and then shade the region representing the desired probability. 2. For each relevant value x that is a boundary for the shaded region, use Formula 6-2 to convert that value to the equivalent z score. (With many technologies, this step can be skipped.) 3. Use technology (software or a calculator) or Table A-2 to find the area of the shaded region. This area is the desired probability. The following example illustrates the above procedure. x m m s z 5 x 2 z P P 0 (b) Standard Normal Distribution (a) Nonstandard Normal Distribution FIGURE 6-11 Converting Distributions What Proportion of Men Are Taller Than the 72 in. Height Requirement for Showerheads (According to Most Building Codes)? EXAMPLE 1 Heights of men are normally distributed with a mean of 68.6 in. and a standard deviation of 2.8 in. (based on Data Set 1 “Body Data” in Appendix B). Find the percentage of men who are taller than a showerhead positioned at 72 in. above the floor. SOLUTION Step 1: See Figure 6-12, which incorporates this information: Men have heights that are normally distributed with a mean of 68.6 in. and a standard deviation of 2.8 in. The shaded region represents the men who are taller than the showerhead height of 72 in. 0.1123 72 in. m5 68.6 in. x (height) z 5 0 z 5 1.21 z scale FIGURE 6-12 Heights of Men continued Go Figure 293,000,000,000: Number of e-mails sent each day.
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