6-1 The Standard Normal Distribution 251 z 5 1.27 0 Area 5 0.8980 (from Table A-2) FIGURE 6-5 Finding Area to the Left of z = 1.27 TABLE A-2 (continued) Cumulative Area from the LEFT z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 0.0 .5000 .5040 .5080 .5120 .5160 .5199 .5239 .5279 .5319 .5359 0.1 .5398 .5438 .5478 .5517 .5557 .5596 .5636 .5675 .5714 .5753 0.2 .5793 .5832 .5871 .5910 .5948 .5987 .6026 .6064 .6103 .6141 1.0 .8413 .8438 .8461 .8485 .8508 .8531 .8554 .8577 .8599 .8621 1.1 .8643 .8665 .8686 .8708 .8729 .8749 .8770 .8790 .8810 .8830 1.2 .8849 .8869 .8888 .8907 .8925 .8944 .8962 .8980 .8997 .9015 1.3 .9032 .9049 .9066 .9082 .9099 .9115 .9131 .9147 .9162 .9177 1.4 .9192 .9207 .9222 .9236 .9251 .9265 .9279 .9292 .9306 .9319 in the adjoining row of probabilities that is directly below 0.07, as shown in the accompanying excerpt. Table A-2 shows that there is an area of 0.8980 corresponding to z = 1.27. We want the area below 1.27, and Table A-2 gives the cumulative area from the left, so the desired area is 0.8980. Because of the correspondence between area and probability, we know that the probability of a z score below 1.27 is 0.8980. INTERPRETATION The probability that a randomly selected person has a bone density test result below 1.27 is 0.8980, shown as the shaded region in Figure 6-5. Another way to interpret this result is to conclude that 89.80% of people have bone density levels below 1.27. YOUR TURN. Do Exercise 9 “Standard Normal Distribution.” Bone Density Test: Finding the Area to the Right of a Value EXAMPLE 4 Using the same bone density test from Example 3, find the probability that a randomly selected person has a result above -1.00. A value above -1.00 is considered to be in the “normal” range of bone density readings. continued SOLUTION We again find the desired probability by finding a corresponding area. We are looking for the area of the region to the right of z = -1.00 that is shaded in Figure 6-6 on the next page. The Statdisk display on the next page shows that the area to the right of z = -1.00 is 0.841345. If we use Table A-2, we should know that it is designed to apply only to cumulative areas from the left. Referring to the page with negative z scores, we find that the cumulative area from the left up to z = -1.00 is 0.1587, as shown in Figure 6-6. Because the total area under the curve is 1, we can find the shaded area by
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