SECTION 5.3 Normal Distributions: Finding Values 253 z −1.645 0 Area = 0.05 0 z Area = 0.5 z 0 1.28 Area = 0.8997 TRY IT YOURSELF 1 1. Find the z@score that has 96.16% of the distribution’s area to its right. 2. Find the positive z@score for which 95% of the distribution’s area lies between -z and z. Answer: Page A39 In Example 1, the given areas correspond to entries in the Standard Normal Table. In most cases, the area will not be an entry in the table. In these cases, use the entry closest to it (or use technology, as shown at the left and in Example 2). When the area is halfway between two area entries, use the z@score halfway between the corresponding z@scores. In Section 2.5, you learned that percentiles divide a data set into 100 equal parts. To find a z@score that corresponds to a percentile, you can use the Standard Normal Table. Recall that if a value x represents the 83rd percentile P83, then 83% of the data values are below x and 17% of the data values are above x. Finding a z-Score Given a Percentile Find the z@score that corresponds to each percentile. 1. P5 2. P50 3. P90 SOLUTION 1. To find the z@score that corresponds to P5, find the z@score that corresponds to an area of 0.05 (see upper figure) by locating 0.05 in the Standard Normal Table. The areas closest to 0.05 in the table are 0.0495 1z = -1.652 and 0.0505 1z = -1.642. Because 0.05 is halfway between the two areas in the table, use the z@score that is halfway between -1.64 and -1.65. So, the z@score that corresponds to an area of 0.05 is -1.645. 2. To find the z@score that corresponds to P50, find the z@score that corresponds to an area of 0.5 (see middle figure) by locating 0.5 in the Standard Normal Table. The area closest to 0.5 in the table is 0.5000, so the z@score that corresponds to an area of 0.5 is 0. 3. To find the z@score that corresponds to P90, find the z@score that corresponds to an area of 0.9 (see lower figure) by locating 0.9 in the Standard Normal Table. The area closest to 0.9 in the table is 0.8997, so the z@score that corresponds to an area of 0.9 is about 1.28. You can use technology to find the z-score that corresponds to each percentile, as shown below. Remember that when you use technology, your answers may differ slightly from those found using the Standard Normal Table. EXCEL −1.644853627 B A 1. 2. 1 2 3 0 3. 1.281551566 =NORM.INV(0.05,0,1) =NORM.INV(0.5,0,1) =NORM.INV(0.9,0,1) TRY IT YOURSELF 2 Find the z@score that corresponds to each percentile. 1. P10 2. P20 3. P99 Answer: Page A39 EXAMPLE 2 Tech Tip You can use technology to find the z@scores that correspond to cumulative areas. For instance, you can use a TI-84 Plus to find the z@scores in Example 1, as shown below. invNorm(.3632,0,1) invNorm(.8925,0,1) -0.3499183227 1.239933478
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