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Lesson The Standard Normal Distribution

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Quiz Homework Problem: Chapter 6 review The Stanley Cup is a best of seven series to determine the NHL champions. The following data represents the number of games played, X, in the Stanley Cup to determine a champion from 1939 to 2004 a) Determine the mean b) Determine the Std Dev Reading questions: –Area to the right in a standard normal curve = ____________ –What methods are used to evaluate standard normal probabilities? X Frequency

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Objectives Find the area under the standard normal curve Find Z-scores for a given area Interpret the area under the standard normal curve as a probability

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Vocabulary Z α – the Z-score that corresponds to the area under the standard normal curve to the right of Z α is α

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Properties of the Standard Normal Curve It is symmetric about its mean, μ = 0, and has a standard deviation of σ = 1 Because mean = median = mode, the highest point occurs at μ = 0 It has inflection points at μ – σ = -1 and μ + σ = 1 Area under the curve = 1 Area under the curve to the right of μ = 0 equals the area under the curve to the left of μ, which equals ½ As Z increases the graph approaches, but never reaches 0 (like approaching an asymptote). As Z decreases the graph approaches, but never reaches, 0. The Empirical Rule applies

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Calculate the Area Under the Standard Normal Curve There are several ways to calculate the area under the standard normal curve –What does not work – some kind of a simple formula –We can use a table (such as Table IV on the inside back cover) –We can use technology (a calculator or software) Using technology is preferred Three different area calculations –Find the area to the left of –Find the area to the right of –Find the area between

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ApproachGraphicallySolution Find the area to the left of z a P(Z < a) Shade the area to the left of z a Use Table IV to find the row and column that correspond to z a. The area is the value where the row and column intersect. Normcdf(-E99,a,0,1) Find the area to the right of z a P(Z > a) or 1 – P(Z < a) Shade the area to the right of z a Use Table IV to find the area to the left of z a. The area to the right of z a is 1 – area to the left of z a. Normcdf(a,E99,0,1) or 1 – Normcdf(-E99,a,0,1) Find the area between z a and z b P(a < Z < b) Shade the area between z a and z b Use Table IV to find the area to the left of z a and to the left of z a. The area between is area zb – area za. Normcdf(a,b,0,1) Obtaining Area under Standard Normal Curve aa a b

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Example 1 Determine the area under the standard normal curve that lies to the left of A.Z = B.Z = C.Z = 0.92 D.Z = 2.90 a Normalcdf(-E99,-3.49) = Normalcdf(-E99,-1.99) = Normalcdf(-E99,0.92) = Normalcdf(-E99,2.90) =

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Example 2 Determine the area under the standard normal curve that lies to the right of a)Z = b)Z = c)Z = 2.23 d)Z = 3.45 Normalcdf(-3.49,E99) = Normalcdf(-0.55,E99) = Normalcdf(2.23,E99) = Normalcdf(3.45,E99) = a

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Example 3 Find the indicated probability of the standard normal random variable Z a)P(-2.55 < Z < 2.55) b)P(-0.55 < Z < 0) c)P(-1.04 < Z < 2.76) Normalcdf(-2.55,2.55) = Normalcdf(-0.55,0) = Normalcdf(-1.04,2.76) = a b

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Example 4 Find the Z-score such that the area under the standard normal curve to the left is 0.1. Find the Z-score such that the area under the standard normal curve to the right is aa invNorm(0.1) = = a invNorm(1-0.35) = 0.385

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Summary and Homework Summary –Calculations for the standard normal curve can be done using tables or using technology –One can calculate the area under the standard normal curve, to the left of or to the right of each Z-score –One can calculate the Z-score so that the area to the left of it or to the right of it is a certain value –Areas and probabilities are two different representations of the same concept Homework –pg 381 – 383; 5 - 6, 9, 14, 21 – 22, 34, 37, 40

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