Question

# Given that x is a normal variable with mean μ = 111 and standard deviation σ...

Given that x is a normal variable with mean μ = 111 and standard deviation σ = 14, find the following probabilities. (Round your answers to four decimal places.) (a) P(x ≤ 120) (b) P(x ≥ 80) (c) P(108 ≤ x ≤ 117)

Part A)

X ~ N ( µ = 111 , σ = 14 )
P ( X <= 120 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 120 - 111 ) / 14
Z = 0.6429
P ( ( X - µ ) / σ ) < ( 120 - 111 ) / 14 )
P ( X <= 120 ) = P ( Z < 0.6429 )
P ( X <= 120 ) = 0.7399

Part b)

X ~ N ( µ = 111 , σ = 14 )
P ( X >= 80 ) = 1 - P ( X < 80 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 80 - 111 ) / 14
Z = -2.2143
P ( ( X - µ ) / σ ) > ( 80 - 111 ) / 14 )
P ( Z > -2.2143 )
P ( X >= 80 ) = 1 - P ( Z < -2.2143 )
P ( X >= 80 ) = 1 - 0.0134
P ( X >= 80 ) = 0.9866

Part c)

X ~ N ( µ = 111 , σ = 14 )
P ( 108 <= X <= 117 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 108 - 111 ) / 14
Z = -0.2143
Z = ( 117 - 111 ) / 14
Z = 0.4286
P ( -0.21 < Z < 0.43 )
P ( 108 <= X <= 117 ) = P ( Z < 0.43 ) - P ( Z < -0.21 )
P ( 108 <= X <= 117 ) = 0.6659 - 0.4152
P ( 108 <= X <= 117 ) = 0.2507

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