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Unformatted text preview: Chapter 6: The Normal Distribution • Normal Distributions • Applications of the Normal Distribution • The Central Limit Theorem • The Normal Approximation to the Binomial Distribution. The Normal Distribution : • A normal distribution is a continuous, symmetric, bellshaped distribution of a variable with curve equation g1877 g3404 1 g2026 2g2024 g1857 g2879 g2869 g2870g3097 g3118 g4666g3051g2879g3091g4667 g3118 where μ is the population mean and σ is the standard deviation. The Properties of the Theoretical Normal Distribution: 1. A normal distribution curve is bellshaped. 2. The mean, median, and mode are equal and are located at the center of the distribution. 3. A normal distribution curve is unimodal. 4. The curve is symmetric about the mean. 5. The curve is continuous ; that is, there are no gaps or holes. For each value of X , there is a corresponding value of Y. 6. The curve never touches the x axis . 7. The total area under a normal distribution curve is equal to 1.00 , or 100%. 8. The area under the part of a normal curve that lies within 1 standard deviation of the mean is approximately 0.68, or 68%; within 2 standard deviations, about 0.95, or 95%; and within 3 standard deviations, about 0.997, or 99.7%. The standard Normal distribution: • The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1, i.e. has eq. g1877 g3404 1 2g2024 g1857 g2879g3051 g3118 Note: any normally distributed variable x can be transformed to a standard normal variable z using the transformation g1878 g3404 g3051g2879g3091 g3097 . • Table E in Appendix C gives the area under the standard normal curve to the left of any z value from 3.49 to 3.49 . Finding Areas Under (the probability) the Standard Normal Distribution Curve: 1. To the left of any z value: Look up the z value in the table and use the area given . 2. To the right of any z value: Look up the z value and subtract the area from 1. 3. Between any two z values: Look up both z values and subtract the corresponding areas. Important: The area under the normal curve within two values (a,b) equals the probability that the normal variable x takes values within these two values (a,b)....
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 Spring '12
 johnanderson
 Statistics, Binomial, Central Limit Theorem, Normal Distribution

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