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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