Question

3. Given that x is a Normal random variable with a mean of 10 and standard

deviation of 4, find the following probabilities: (6 points)

1. P(x<6.7)
2. P(x>12.5)
3. P(8.8<x<12.5)

Answer 1

Solution :

Given that,

mean = = 10

standard deviation = = 4

a ) P( x < 6.7 )

P ( x - / ) < ( 6.7 - 10 / 4 )

P ( z < -3.3 / 4 )

P ( z < -0.82)

= 0.2061

Probability = 0.2061

b ) P (x > 12.5 )

= 1 - P (x < 12.5 )

= 1 - P ( x -  / ) < ( 12.5 - 10 / 4 )

= 1 - P ( z < 2.5 / 4 )

= 1 - P ( z < 0.62)

Using z table

= 1 - 0.7324

= 0.2676

Probability = 0.2676

c ) P (8.8 < x < 12.5 )

P ( 8.8 - 10 / 4) < ( x -  / ) < ( 12.5 - 10 / 4)

P ( -1.2 / 4 < z < 2.5 / 4 )

P (-0.3 < z < 0.62 )

P ( z < 0.62 ) - P ( z < -0.3 )

Using z table

= 0.7324 - 0.3821

= 0.3503

Probability = 0.3503