Normal Distributions: Finding Values 5.3 252 CHAPTER 5 Normal Probability Distributions Finding z@Scores Transforming a z@Score to an x@Value Finding a Specific Data Value for a Given Probability What You Should Learn How to find a z-score given the area under the normal curve How to transform a z-score to an x-value How to find a specific data value of a normal distribution given the probability Finding z-Scores In Section 5.2, you were given a normally distributed random variable x and you found the probability that x would lie in an interval by calculating the area under the normal curve for the interval. But what if you are given a probability and want to find a value? For instance, a university might want to know the lowest test score a student can have on an entrance exam and still be in the top 10%, or a medical researcher might want to know the cutoff values for selecting the middle 90% of patients by age. In this section, you will learn how to find a value given an area under a normal curve (or a probability), as shown in the next example. Finding a z-Score Given an Area 1. Find the z@score that corresponds to a cumulative area of 0.3632. 2. Find the z@score that has 10.75% of the distribution’s area to its right. SOLUTION 1. Find the z@score that corresponds to an area of 0.3632 by locating 0.3632 in the Standard Normal Table. The values at the beginning of the corresponding row and at the top of the corresponding column give the z@score. For this area, the row value is -0.3 and the column value is 0.05. So, the z@score is -0.35, as shown in the figure at the left. 20.5 20.4 20.3 20.2 z .09 .08 .07 .06 .05 .03 23.4 .0002 .0003 .0003 .0003 .0003 .0003 .0003 .2776 .2810 .2843 .2877 .2912 .2946 .2981 .3121 .3156 .3192 .3228 .3264 .3300 .3336 .3483 .3520 .3557 .3594 .3632 .3669 .3707 .3859 .3897 .3936 .3974 .4013 .4052 .4090 .04 2. Because the area to the right is 0.1075, the cumulative area is 1 - 0.1075 = 0.8925. Find the z@score that corresponds to an area of 0.8925 by locating 0.8925 in the Standard Normal Table. For this area, the row value is 1.2 and the column value is 0.04. So, the z@score is 1.24, as shown in the figure at the left. 1.0 .8413 .8438 .8461 .8485 .8508 .8531 .8554 1.1 .8643 .8665 .8686 .8708 .8729 .8749 .8770 1.2 .8849 .8869 .8888 .8907 .8925 .8944 .8962 z .00 .01 .02 .03 .04 .05 .06 0.0 .5000 .5040 .5080 .5120 .5160 .5199 .5239 1.3 .9032 .9049 .9066 .9082 .9099 .9115 .9131 EXAMPLE 1 z Area = 0.3632 − 0.35 0 1.24 z 0 Area = 0.1075
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