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1. Normal Curve 2. Normally Distributed Outcomes 3. Properties of Normal Curve 4. Standard Normal Curve 5. The Normal Distribution 6. Percentile 7. Probability for General Normal Distribution 1

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The bell-shaped curve, as shown below, is call a normal curve. 2

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Examples of experiments that have normally distributed outcomes: 1. Choose an individual at random and observe his/her IQ. 2. Choose a 1-day-old infant and observe his/her weight. 3. Choose a leaf at random from a particular tree and observe its length. 3

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A certain experiment has normally distributed outcomes with mean equal to 1. Shade the region corresponding to the probability that the outcome ( a ) lies between 1 and 3; ( b ) lies between 0 and 2; ( c ) is less than.5; ( d ) is greater than 2. 5

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The equation of the normal curve is 7 The standard normal curve has

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8 A(z) is the area under the standard normal curve to the left of a normally distributed random variable z.

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Use the normal distribution table to determine the area corresponding to ( a ) z < -.5; ( b ) 1< z < 2; ( c ) z > 1.5. 9

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( a ) A (-.5) =.3085 ( b ) A (2) - A (1) =.9772 - 8413 =.1359 ( c ) 1 - A (1.5) = 1 -.9332 =.0668 10

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If a score S is the p th percentile of a normal distribution, then p % of all scores fall below S, and (100 - p )% of all scores fall above S. The p th percentile is written as z p. 11

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What is the 95 th percentile of the standard normal distribution? In the normal distribution, find the value of z such that A ( z ) =.95. A (1.65) =.9506 Therefore, z 95 = 1.65. 12

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If X is a random variable having a normal distribution with mean and standard deviation then where Z has the standard normal distribution and A ( z ) is the area under that distribution to the left of z. 13

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Find the 95 th percentile of infant birth weights if infant birth weights are normally distributed with = 7.75 and = 1.25 pounds. The value for the standard normal random variable is z 95 = 1.65. Then x 95 = 7.75 + (1.65)(1.25) = 9.81 pounds. 14

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A normal curve is identified by its mean ( ) and its standard deviation ( ). The standard normal curve has = 0 and = 1. Areas of the region under the standard normal curve can be obtained with the aid of a table or graphing calculator. 15

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A random variable is said to be normally distributed if the probability that an outcome lies between a and b is the area of the region under a normal curve from x = a to x = b. After the numbers a and b are converted to standard deviations from the mean, the sought-after probability can be obtained as an area under the standard normal curve. 16

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