264 CHAPTER 6 Normal Probability Distributions 3. Choose the correct (right>left) side of the graph. A value separating the top 10% from the others will be located on the right side of the graph, but a value separating the bottom 10% will be located on the left side of the graph. 4. A z score must be negative whenever it is located in the left half of the normal distribution. 5. Areas (or probabilities) are always between 0 and 1, and they are never negative. Procedure for Finding Values from Known Areas or Probabilities 1. Sketch a normal distribution curve, write the given probability or percentage in the appropriate region of the graph, and identify the x value(s) being sought. 2. If using technology, refer to the Tech Center instructions at the end of this section. If using Table A-2, refer to the body of Table A-2 to find the area to the left of x, then identify the z score corresponding to that area. 3. If you know z and must convert to the equivalent x value, use Formula 6-2 by entering the values for m, s, and the z score found in Step 2, then solve for x. Based on Formula 6-2, we can solve for x as follows: x = m + 1z # s2 1another form of Formula 6922 (If z is located to the left of the mean, be sure that it is a negative number.) 4. Refer to the sketch of the curve to verify that the solution makes sense in the context of the graph and in the context of the problem. The following example uses this procedure for finding a value from a known area. Designing a Front Door for a Home EXAMPLE 3 When designing equipment, one common criterion is to use a design that accommodates at least 95% of the population. What is the height of a door that would allow 95% of adults to walk through the doorway without bending or hitting their heads? Based on Data Set 1 “Body Data” in Appendix B, assume that heights of adults are normally distributed with a mean of 66.2 in. and a standard deviation of 3.8 in. How does the result compare to the door height of 80 in. required by the International Residential Code? SOLUTION Step 1: Figure 6-14 on the next page shows the normal distribution with the height x that we want to identify. The shaded area represents the shortest 95% of adults. Step 2: Technology: Technology will provide the value of x in Figure 6-14. For example, see the Excel display on the next page showing that x = 72.45044378 in., or 72.5 in. when rounded. Table A-2: If using Table A-2, search for an area of 0.9500 in the body of the table. (The area of 0.9500 shown in Figure 6-14 is a cumulative area from the left, and that is exactly the type of area listed in Table A-2.) The area of 0.9500 is between the Table A-2 areas of 0.9495 and 0.9505, but there is an asterisk and footnote indicating that an area of 0.9500 corresponds to z = 1.645.
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