Question

# A normal random variable x has mean μ = 1.6 and standard deviation σ = 0.19....

A normal random variable x has mean μ = 1.6 and standard deviation σ = 0.19. Find the probabilities of these X-values. (Round your answers to four decimal places.)

1.00 < X < 1.20

X > 1.37

1.35 < X < 1.50

Solution :

Given that,

mean = = 1.6

standard deviation = = 15

a) P (1.00 < x < 1.20 )

P ( 1.00 - 1.6 / 0.19) < ( x -  / ) < ( 1.20 - 1.6 / 0.19)

P ( -0.60 / 0.19 < z < 0.40 / 0.19 )

P (-3.16 < z < 2.10 )

P ( z < - 0.259 ) - P ( z < -3.16 )

Using z table

= 0.9821 - 0.0008

= 0.9813

Probability = 0.9813

b ) P (x > 1.37 )

= 1 - P (x < 1.37 )

= 1 - P ( x -  / ) < ( 1.37 - 1.6 / 0.19)

= 1 - P ( z < -0.23 / 0.19 )

= 1 - P ( z < -1.21 )

Using z table

= 1 - 0.1131

= 0.8869

Probability =0.8869

c ) P (1.35 < x < 1.50)

P ( 1.35 - 1.6 / 0.19) < ( x -  / ) < ( 1.50 - 1.6 / 0.19)

P ( -0. 25 / 0.19 < z < - 0.10 / 0.19 )

P (-1.31 < z < - 0.53)

P ( z < - 0.53 ) - P ( z < -1.31)

Using z table

= 0.2981 -0.0951

= 0.2030

Probability = 0.2030

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