© 2010 Pearson Prentice Hall. All rights reserved Chapter The Normal Probability Distribution © 2010 Pearson Prentice Hall. All rights reserved 3 7

Section 7.2 The Standard Normal Distribution 7-2

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© 2010 Pearson Prentice Hall. All rights reserved The table gives the area under the standard normal curve for values to the left of a specified Z-score, z o, as shown in the figure. 7-7

© 2010 Pearson Prentice Hall. All rights reserved Find the area under the standard normal curve to the left of z = EXAMPLE Finding the Area Under the Standard Normal Curve Area left of z = is

© 2010 Pearson Prentice Hall. All rights reserved Area under the normal curve to the right of z o = 1 – Area to the left of z o 7-9

© 2010 Pearson Prentice Hall. All rights reserved EXAMPLE Finding the Area Under the Standard Normal Curve Find the area under the standard normal curve to the right of Z = Area right of 1.25 = 1 – area left of 1.25 = 1 – =

© 2010 Pearson Prentice Hall. All rights reserved Find the area under the standard normal curve between z = and z = EXAMPLE Finding the Area Under the Standard Normal Curve Area between and 2.94 = (Area left of z = 2.94) – (area left of z = -1.02) = – =

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© 2010 Pearson Prentice Hall. All rights reserved Find the z-score such that the area to the left of the z-score is EXAMPLE Finding a z-score from a Specified Area to the Left The z-score such that the area to the left of the z-score is is z =

© 2010 Pearson Prentice Hall. All rights reserved EXAMPLE Finding a z-score from a Specified Area to the Right Find the z-score such that the area to the right of the z-score is The area left of the z-score is 1 – = The approximate z-score that corresponds to an area of to the left ( to the right) is Therefore, z =

© 2010 Pearson Prentice Hall. All rights reserved Find the z-scores that separate the middle 80% of the area under the normal curve from the 20% in the tails. EXAMPLE Finding a z-score from a Specified Area Area = 0.8 Area = 0.1 z 1 is the z-score such that the area left is 0.1, so z 1 = z 2 is the z-score such that the area left is 0.9, so z 2 =

© 2010 Pearson Prentice Hall. All rights reserved The notation z α (prounounced “z sub alpha”) is the z-score such that the area under the standard normal curve to the right of z α is α. 7-17

© 2010 Pearson Prentice Hall. All rights reserved Find the value of z 0.25 EXAMPLE Finding the Value of z  We are looking for the z-value such that the area to the right of the z-value is This means that the area left of the z-value is z 0.25 =

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© 2010 Pearson Prentice Hall. All rights reserved Notation for the Probability of a Standard Normal Random Variable P(a < Z < b) represents the probability a standard normal random variable is between a and b P(Z > a)represents the probability a standard normal random variable is greater than a. P(Z < a) represents the probability a standard normal random variable is less than a. 7-20

© 2010 Pearson Prentice Hall. All rights reserved Find each of the following probabilities: (a) P(Z < -0.23) (b) P(Z > 1.93) (c) P(0.65 < Z < 2.10) EXAMPLE Finding Probabilities of Standard Normal Random Variables (a) P(Z < -0.23) = (b) P(Z > 1.93) = (c) P(0.65 < Z < 2.10) =

© 2010 Pearson Prentice Hall. All rights reserved For any continuous random variable, the probability of observing a specific value of the random variable is 0. For example, for a standard normal random variable, P(a) = 0 for any value of a. This is because there is no area under the standard normal curve associated with a single value, so the probability must be 0. Therefore, the following probabilities are equivalent: P(a < Z < b) = P(a < Z < b) = P(a < Z < b) = P(a < Z < b) 7-22