Question

A random variable Y is normally distributed with µ = 20.0, σ = 4.0 Find:

P( Y < 22.0 )

P( 12.0 < Y < 28.0 )

P( 20.0 < Y )

P( 18.0 < Y < 22.0 )

P( 24.0 < Y )

Answer 1

If x is normally distributed with μ = 20.0 and σ =4.0,

a. P (Y< 22.0 )
P(Z<(Y-μ)/σ)=P(Z<(20-22)/4)=P(Z<-0.5) = 0.3085

b. P ( 12.0 <Y< 28.0 )
= P(Z<28)-P(Z<12. 0)
=P(Z<(28-20)/4) - P(Z<(12 -20)/4)
=P(Z<2 )-P(Z<-2 )
=0.9773- 0.0228 = 0.9545

C. P ( Y > 20.0 )= P( 20.0 < Y )
P(Z>(x-μ)/σ)=P(Z>(20.0 -20)/4)=P(Z> 0) = 1-0.5 =0.5

D. P ( 18.0 <Y< 22.0 )
= P(Z<22)-P(Z<18. 0)
=P(Z<(22-20)/4) - P(Z<(18 -20)/4)
=P(Z<0.5)-P(Z<-0.5)
=0.6915- 0.3085 =0.3857

E. P ( Y > 24.0 )= P( 24.0 < Y )
P(Z>(x-μ)/σ)=P(Z>(24.0 -20)/4)=P(Z> 1) = 1-0.8413 =0.1587