Question

Suppose X is a normal random variable with μ = 300 and σ = 40. Find the values of the following probabilities. (Round your answers to four decimal places.)

(a)    P(X < 414)


(b)    P(340 < X < 408)


(c)    P(X > 340)

You may need to use the appropriate table in the Appendix of Tables to answer this question.

Answer 1

a)

X ~ N ( µ = 300 , σ = 40 )
P ( X < 414 ) = ?
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 414 - 300 ) / 40
Z = 2.85
P ( ( X - µ ) / σ ) < ( 414 - 300 ) / 40 )
P ( X < 414 ) = P ( Z < 2.85 )
P ( X < 414 ) = 0.9978

b)

X ~ N ( µ = 300 , σ = 40 )
P ( 340 < X < 408 ) = ?
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 340 - 300 ) / 40
Z = 1
Z = ( 408 - 300 ) / 40
Z = 2.7
P ( 1 < Z < 2.7 )
P ( 340 < X < 408 ) = P ( Z < 2.7 ) - P ( Z < 1 )
P ( 340 < X < 408 ) = 0.9965 - 0.8413
P ( 340 < X < 408 ) = 0.1552

c)

X ~ N ( µ = 300 , σ = 40 )
P ( X > 340 ) = 1 - P ( X < 340 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 340 - 300 ) / 40
Z = 1
P ( ( X - µ ) / σ ) > ( 340 - 300 ) / 40 )
P ( Z > 1 )
P ( X > 340 ) = 1 - P ( Z < 1 )
P ( X > 340 ) = 1 - 0.8413
P ( X > 340 ) = 0.1587