304 Chapter 7 The Normal Distribution EXAMPLE 7.17 Finding a normal value corresponding to an area Mensa is an organization whose membership is limited to people whose IQ is in the top 2% of the population. Assume that scores on an IQ test are normally distributed with mean �� = 100 and standard deviation �� = 15. What is the minimum score needed to qualify for membership in Mensa? Area = 0.98 Area = 0.02 x 100 Figure 7.17 Solution Step 1: Figure 7.17 presents a sketch of the normal curve, showing the value x separating the upper 2% from the lower 98%. Step 2: The area 0.02 is on the right, so we subtract from 1 and work with the area 0.98 on the left. Step 3: The area closest to 0.98 in Table A.2 is 0.9798, which corresponds to a z-score of 2.05. Step 4: The IQ score that separates the upper 2% from the lower 98% is x = �� + z�� = 100 + (2.05)(15) = 130.75 Since IQ scores are generally whole numbers, we will round this to x = 131. Finding the normal value corresponding to a given area by using technology To find the percentile of a normal distribution with technology, follow Steps 1 and 2 of the method for using the table. What is done after that depends on the technology being used. Example 7.18 illustrates the use of the TI-84 Plus calculator. EXAMPLE 7.18 Finding the normal value corresponding to an area Recall: The pth percentile of a population is the value that separates the lowest p% of the population from the highest (100 − p)%. by using technology IQ scores have a mean of 100 and a standard deviation of 15. Use technology to find the 90th percentile of IQ scores; in other words, find the IQ score that separates the upper 10% from the lower 90%. Solution Step 1: Figure 7.18 presents a sketch of the normal curve, showing the value x separating the upper 10% from the lower 90%. Step 2: We work with the area 0.90 on the left. Step 3: For the TI-84 Plus calculator, use the invNorm command with area 0.90, mean 100, and standard deviation 15. Step-by-step instructions are given in the Using Technology section on page 305. Figure 7.19 presents the results from the TI-84 Plus calculator. The IQ score corresponding to the top 10% is 119. x Area = 0.10 100 Area = 0.90 Figure 7.18 Figure 7.19
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