Explanation Calculation: It is given that the random variables X , Y , and Z are independent normal random variable with the parameters 0 and σ 2 . That is, X , Y , Z ~ N ( 0 , σ 2 ) . The density function of the random variable X is, f ( x ) = 1 σ 2 π e − x 2 2 σ 2 , x ∈ ℝ The density function of the random variable Y is, f ( y ) = 1 σ 2 π e − y 2 2 σ 2 , y ∈ ℝ The density function of the random variable Z is, f ( z ) = 1 σ 2 π e − z 2 2 σ 2 , z ∈ ℝ Let Θ , Φ and R represents the spherical coordinates of ( X , Y , Z ) . That is, x = r sin ϕ cos θ , y = r sin ϕ sin θ and z = r cos ϕ . This indicates that r = x 2 + y 2 + z 2 , θ = arctan ( y x ) and φ = arccos ( z x ) . The joint density of Θ , Φ and R is, f R , Θ , Φ ( r , θ , φ ) = f X , Y , Z ( r sin ϕ cos θ , r sin ϕ sin θ , r cos ϕ ) | J ( r , θ , φ ) | If the random variables X , Y and Z are independent, then the joint density function of X , Y and Z is, f X , Y , Z = f X ( x ) f Y ( y ) f Z ( z ) ; x , y , z ∈ ℝ = ( 1 σ 2 π e − x 2 2 σ 2 ) ( 1 σ 2 π e − y 2 2 σ 2 ) ( 1 σ 2 π e − z 2 2 σ 2 ) = 1 2 π σ 3 2 π e − x 2 + y 2 + z 2 2 σ 2 The joint density for the spherical coordinates of ( X , Y , Z ) is f X , Y , Z ( r sin ϕ cos θ , r sin ϕ sin θ , r cos ϕ ) = 1 2 π σ 3 2 π e − x 2 + y 2 + z 2 2 σ 2 = 1 2 π σ 3 2 π e − r 2 2 σ 2 . The jacobian matrix is, [ ∂ x ∂ r ( r , θ , φ ) ∂ x ∂ θ ( r , θ , φ ) ∂ x ∂ φ ( r , θ , φ ) ∂ y ∂ r ( r , θ , φ ) ∂ y ∂ θ ( r , θ , φ ) ∂ y ∂ φ ( r , θ , φ ) ∂ z ∂ r ( r , θ , φ ) ∂ z ∂ θ ( r , θ , φ ) ∂ z ∂ φ ( r , θ , φ ) ] = [ cos θ sin φ − r sin θ sin φ r cos θ cos φ sin θ sin φ r cos θ sin φ r sin θ cos φ cos φ 0 − r sin φ ] The determinant of jacobian matrix is, det [ J ( r , θ , φ ) ] = det [ cos θ sin φ − r sin θ sin φ r cos θ cos φ sin θ sin φ r cos θ sin φ r sin θ cos φ cos φ 0 − r sin φ ] = { cos φ [ ( − r sin θ sin φ ) ( r sin θ cos φ ) − ( r cos θ sin φ ) ( r cos θ cos φ ) ] − 0 [ ( cos θ sin φ ) ( r sin θ cos φ ) − ( sin θ sin φ ) ( r cos θ cos φ ) ] + − r sin φ [ ( cos θ sin φ ) ( r cos θ sin φ ) − ( − r sin θ sin φ ) ( sin θ sin φ ) ] } = { cos φ [ ( − r 2 sin 2 θ sin φ cos φ ) − ( r 2 cos 2 θ sin φ cos φ ) ] − 0 − r sin φ [ r cos 2 θ sin 2 φ + r sin 2 θ sin 2 φ ] } = { − r 2 sin φ cos 2 φ ( sin 2 θ + cos 2 θ ) − r 2 sin 3 φ [ cos 2 θ + sin 2 θ ] } On simplification, det [ J ( r , θ , φ ) ] = − r 2 sin φ cos 2 φ − r 2 sin 3 φ = − r 2 sin φ ( cos 2 φ + sin 2 φ ) = − r 2 sin φ Therefore, | J ( r , θ , φ ) | = r 2 sin φ

Mathematical Statistics and Data Analysis (with CD Data Sets)

3rd Edition

ISBN: 9780534399429

Author: John A. Rice

Publisher: Cengage Learning

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Chapter 3.8, Problem 54P

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The population mean and standard deviation are given below. Find the required probability and determine whether the given sample mean would be considered unusual. For a sample of n = 65, find the probability of a sample mean being greater than 225 if μ = 224 and σ = 3.5. For a sample of n = 65, the probability of a sample mean being greater than 225 if μ=224 and σ = 3.5 is 0.0102 (Round to four decimal places as needed.)

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Chapter 3 Solutions

Mathematical Statistics and Data Analysis (with CD Data Sets)

Ch. 3.8 - Prob. 1P Ch. 3.8 - Prob. 2P Ch. 3.8 - Prob. 3P Ch. 3.8 - Prob. 4P Ch. 3.8 - Prob. 5P Ch. 3.8 - Prob. 6P Ch. 3.8 - Prob. 7P Ch. 3.8 - Prob. 8P Ch. 3.8 - Prob. 9P Ch. 3.8 - Prob. 10P

Ch. 3.8 - Prob. 11P Ch. 3.8 - Prob. 12P Ch. 3.8 - Prob. 13P Ch. 3.8 - Prob. 14P Ch. 3.8 - Prob. 15P Ch. 3.8 - Prob. 16P Ch. 3.8 - Prob. 17P Ch. 3.8 - Prob. 18P Ch. 3.8 - Prob. 19P Ch. 3.8 - Prob. 20P Ch. 3.8 - Prob. 22P Ch. 3.8 - Prob. 23P Ch. 3.8 - Prob. 24P Ch. 3.8 - Prob. 25P Ch. 3.8 - Prob. 27P Ch. 3.8 - Prob. 28P Ch. 3.8 - Prob. 29P Ch. 3.8 - Prob. 30P Ch. 3.8 - Prob. 31P Ch. 3.8 - Prob. 32P Ch. 3.8 - Prob. 33P Ch. 3.8 - Prob. 34P Ch. 3.8 - Prob. 35P Ch. 3.8 - Prob. 38P Ch. 3.8 - Prob. 39P Ch. 3.8 - Prob. 44P Ch. 3.8 - Prob. 45P Ch. 3.8 - Prob. 46P Ch. 3.8 - Prob. 47P Ch. 3.8 - Prob. 48P Ch. 3.8 - Prob. 50P Ch. 3.8 - Prob. 51P Ch. 3.8 - Prob. 52P Ch. 3.8 - Prob. 53P Ch. 3.8 - Prob. 54P Ch. 3.8 - Prob. 55P Ch. 3.8 - Prob. 56P Ch. 3.8 - Prob. 57P Ch. 3.8 - Prob. 58P Ch. 3.8 - Prob. 60P Ch. 3.8 - Prob. 61P Ch. 3.8 - Prob. 62P Ch. 3.8 - Prob. 63P Ch. 3.8 - Prob. 64P Ch. 3.8 - Prob. 65P Ch. 3.8 - Prob. 66P Ch. 3.8 - Prob. 67P Ch. 3.8 - Prob. 68P Ch. 3.8 - Prob. 69P Ch. 3.8 - Prob. 70P Ch. 3.8 - Prob. 71P Ch. 3.8 - Prob. 72P Ch. 3.8 - Prob. 73P Ch. 3.8 - Prob. 74P Ch. 3.8 - Prob. 75P Ch. 3.8 - Prob. 76P Ch. 3.8 - Prob. 77P Ch. 3.8 - Prob. 78P Ch. 3.8 - Prob. 79P Ch. 3.8 - Prob. 80P Ch. 3.8 - Prob. 81P

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