12.5 The Normal Curve 811 Determine the Percent of Data Between Any Two Data Values in a Normal Distribution If we are given any normal distribution with a known mean and standard deviation, it is possible through the use of Table 12.8 (the z-table) to determine the percent of data between any two given values. The total area under any normal curve is 1.00. Table 12.8 will be used to determine the cumulative area under the normal curve that lies to the left of a specified z-score. We will use Table 12.8(a) when we wish to determine area to the left of a negative z-score, and we will use Table 12.8(b) when we wish to determine area to the left of a positive z-score. Example 2 Determining the Area Under the Normal Curve Determine the area under the normal curve a) to the left of = − z 1.00. b) to the left of = z 1.19. c) to the right of = z 1.19. d) between = − z 1.62 and = z 2.57. Solution a) To determine the area under the normal curve to the left of = − z 1.00, as illustrated in Fig. 12.26, we use Table 12.8(a), since we are looking for an area to the left of a negative z-score. In the upper-left corner of the table, we see the letter z. The column under z gives the units and the tenths value for z. To locate the hundredths value of z, we use the column headings to the right of z. In this case, the hundredths value of = − z 1.00 is 0, so we use the first column labeled .00. To determine the area to the left of = − z 1.00, we use the row labeled −1.0 and move to the column labeled .00. The table entry, which is .1587, is circled in blue. Therefore, the total area to the left of = − z 1.00 is 0.1587. b) To determine the area under the normal curve to the left of = z 1.19 (Fig. 12.27), we use Table 12.8(b), since we are looking for an area to the left of a positive z-score. We first look for 1.1 in the column under z. Since the hundredths value of = z 1.19 is 9, we move to the column labeled .09. Using the row labeled 1.1 and the column labeled 0.09, the table entry is .8830, circled in green. Therefore, the total area to the left of = z 1.19 is 0.8830. c) To determine the area to the right of = z 1.19, we use the fact that the total area under the normal curve is 1. In part (b), we determined that the area to the left of = z 1.19 was 0.8830. To determine the area to the right of = z 1.19, we can subtract the area to the left of = z 1.19 from 1 (Fig. 12.28(a)). Therefore, the area to the right of = z 1.19 is −1 0.8830, or 0.1170. 0 z-scores 1.19 0.1170 0.8830 1– 0.8830 Figure 12.28 Example 2 illustrates the procedure to follow when using Table 12.8 to determine the area under the normal curve. When you are determining the area under the normal curve, it is often helpful to draw a picture and shade the area to be determined. 0 z-scores –1.00 0.1587 Figure 12.26 0 1.19 z-scores 0.8830 Figure 12.27 0 z-scores –1.19 1.19 0.1170 0.1170 Another way to determine the area to the right of = z 1.19 is to use the fact that the normal curve is symmetric about the mean. Therefore, the area to the left of = − z 1.19 is equal to the area to the right of = z 1.19 (Fig. 12.28(b)). Using Table 12.8(a), we see that the area to the left of = − z 1.19 is .1170. This value is circled in red in the table. Therefore, the area to the right of = z 1.19 is also .1170. (a) (b) (continued)
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