6-2 Real Applications of Normal Distributions 263 Step 1: See Figure 6-13, which incorporates this information: Women have heights that are normally distributed with a mean of 63.7 in. and a standard deviation of 2.9 in. The shaded region represents the women with heights between 64 in. and 77 in. Step 2: With some technologies, the shaded area in Figure 6-13 can be found directly and it is not necessary to convert the x scores of 64 in. and 77 in. to z scores. (See Step 3.) If using Table A-2, we cannot find the shaded area directly, but we can find it indirectly by using the same procedures from Section 6-1, as follows: (1) Find the cumulative area from the left up to 77 in. (or z = 4.59); (2) find the cumulative area from the left up to 64 in. (or z = 0.10); (3) find the difference between those two areas. The heights of 77 in. and 64 in. are converted to z scores by using Formula 6-2 as follows: For x = 77 in.: z = x - m s = 77 - 63.7 2.9 = 4.59 1z = 4.59 yields an area of 0.9999.2 For x = 64 in.: z = x - m s = 64 - 63.7 2.9 = 0.10 1z = 0.10 yields an area of 0.5398.2 Step 3: Technology: To use technology, refer to the Tech Center instructions on page 267. Technology will show that the shaded area in Figure 6-13 is 0.4588 Table A-2: Refer to Table A-2 with z = 4.59 and find that the cumulative area to the left of z = 4.59 is 0.9999. (Remember, Table A-2 is designed so that all areas are cumulative areas from the left.) Table A-2 also shows that z = 0.10 corresponds to an area of 0.5398. Because the areas of 0.9999 and 0.5398 are cumulative areas from the left, we find the shaded area in Figure 6-13 as follows: Shaded area in Figure 6913 = 0.9999 - 0.5398 = 0.4601 There is a relatively small discrepancy between the area of 0.4588 found from technology and the area of 0.4601 found from Table A-2. The area obtained from technology is more accurate because it is based on unrounded z scores, whereas Table A-2 requires z scores rounded to two decimal places. YOUR TURN. Do Exercise 15 “Pulse Rates.” INTERPRETATION Expressing the result as a percentage, we conclude that about 46% of women satisfy the requirement of having a height between 64 in. and 77 in. About 54% of women did not meet that requirement and they were not eligible to be pilots in the U.S. Air Force. Finding Values from Known Areas Here are helpful hints for those cases in which the area (or probability or percentage) is known and we must find the relevant value(s): 1. Graphs are extremely helpful in visualizing, understanding, and successfully working with normal probability distributions, so they should always be used. 2. Don’t confuse z scores and areas. Remember, z scores are distances along the horizontal scale, but areas are regions under the normal curve. Table A-2 lists z scores in the left columns and across the top row, but areas are found in the body of the table. continued
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