6-1 The Standard Normal Distribution 255 Table A-2: If using Table A-2, search for the area of 0.95 in the body of the table and then find the corresponding z score. In Table A-2 we find the areas of 0.9495 and 0.9505, but there’s an asterisk with a special note indicating that 0.9500 corresponds to a z score of 1.645. We can now conclude that the z score in Figure 6-9 is 1.645, so the 95th percentile is z = 1.645. YOUR TURN. Do Exercise 37 “Finding Bone Density Scores.” Excel INTERPRETATION For bone density test scores, 95% of the scores are less than or equal to 1.645, and 5% of them are greater than or equal to 1.645. Bone Density Test EXAMPLE 7 Using the same bone density test described in Example 3, we have a standard normal distribution with a mean of 0 and a standard deviation of 1. Find the bone density test score that separates the bottom 2.5% and find the score that separates the top 2.5%. continued SOLUTION The required z scores are shown in Figure 6-10. Those z scores can be found using technology. If using Table A-2 to find the z score located to the left, we search the body of the table for an area of 0.025. The result is z = -1.96. To find the z score located to the right, we search the body of Table A-2 for an area of 0.975. (Remember that Table A-2 always gives cumulative areas from the left.) The result is z = 1.96. The values of z = -1.96 and z = 1.96 separate the bottom 2.5% and the top 2.5%, as shown in Figure 6-10. 0 z 5 1.96 z 521.96 Area 5 0.025 Area 5 0.025 To find this z score, locate the cumulative area to the left in Table A–2. Locate 0.975 in the body of Table A–2. FIGURE 6-10 Finding z Scores Lefties Die Sooner? A study by psychologists Diane Halpern and Stanley Coren received considerable media attention and generated considerable interest when it concluded that left-handed people don’t live as long as right-handed people. Based on their study, it appeared that left-handed people live an average of nine years less than righties. The Halpern>Coren study has been criticized for using flawed data. They used second-hand data by surveying relatives about people who had recently died. The myth of lefties dying younger became folklore that has survived many years. However, more recent studies show that left-handed people do not have shorter lives than those who are right-handed. Technology: We could find the z score using technology. The accompanying Excel display shows that the z score with an area of 0.95 to its left is z = 1.644853627, or 1.645 when rounded.
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