254 CHAPTER 6 Normal Probability Distributions Finding z Scores from Known Areas Examples 3, 4, and 5 all involved the standard normal distribution, and they were all examples with this same format: Given z scores, find areas (or probabilities). In many cases, we need a method for reversing the format: Given a known area (or probability), find the corresponding z score. In such cases, it is really important to avoid confusion between z scores and areas. Remember, z scores are distances along the horizontal scale, but areas (or probabilities) are regions under the density curve. (Table A-2 lists z-scores in the left column and across the top row, but areas are found in the body of the table.) We should also remember that z scores positioned in the left half of the curve are always negative. If we already know a probability and want to find the corresponding z score, we use the following procedure. Procedure for Finding a z Score from a Known Area 1. Draw a bell-shaped curve and identify the region under the curve that corresponds to the given probability. If that region is not a cumulative region from the left, work instead with a known region that is a cumulative region from the left. 2. Use technology or Table A-2 to find the z score. With Table A-2, use the cumulative area from the left, locate the closest probability in the body of the table, and identify the corresponding z score. Special Cases In the solution to Example 6 that follows, Table A-2 leads to a z score of 1.645, which is midway between 1.64 and 1.65. When using Table A-2, we can usually avoid interpolation by simply selecting the closest value. The accompanying table lists special cases that are often used in a wide variety of applications. (For one of those special cases, the value of z = 2.576 gives an area slightly closer to the area of 0.9950, but z = 2.575 has the advantage of being the value exactly midway between z = 2.57 and z = 2.58.) Except in these special cases, we can usually select the closest value in the table. (If a desired value is midway between two table values, select the larger value.) For z scores above 3.49, we can use 0.9999 as an approximation of the cumulative area from the left; for z scores below -3.49, we can use 0.0001 as an approximation of the cumulative area from the left. Special Cases in Table A-2 z Score Cumulative Area from the Left 1.645 0.9500 -1.645 0.0500 2.575 0.9950 -2.575 0.0050 Above 3.49 0.9999 Below -3.49 0.0001 0 Area 5 0.95 z 5 ? FIGURE 6-9 Finding the 95th Percentile Bone Density Test: Finding a Test Score EXAMPLE 6 Use the same bone density test scores used in earlier examples. Those scores are normally distributed with a mean of 0 and a standard deviation of 1, so they meet the requirements of a standard normal distribution. Find the bone density score corresponding to P95, the 95th percentile. That is, find the bone density score that separates the bottom 95% from the top 5%. See Figure 6-9. SOLUTION Figure 6-9 shows the z score that is the 95th percentile, with 95% of the area (or 0.95) below it.
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