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

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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

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

 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

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

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

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

 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

 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

 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

 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

 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

 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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