Question

A random variable is normally distributed with a mean of 80 and a standard deviation of 6.

a) Find P(X < 75.5)

b) Find P(X > 82)

c) Find P(77 < X < 84.8)

Answer 1

Solution :

Given that ,

mean = = 80

standard deviation = = 6

P(x < 75.5 ) = P[(x - ) / < ( 75.5 - 80) / 6 ]

= P(z < -0.75 )

Using z table,

= 0.2266

Probability = 0.2266

( b )

P(x > 82 ) = 1 - P( x < 82 )

= 1- P[(x - ) / < ( 82 - 80) / 6 ]

= 1- P(z < 0.33 )

Using z table,

= 1 - 0.6293

= 0.3707

Probability = 0.3707

( c )

P( 77 < x < 84.8 )

= P[( 77 - 80) / 6 ) < (x - ) /  < ( 84.8 - 80) / 6 ) ]

= P( -0.5 < z < 0.8 )

= P(z < 0.8 ) - P(z < -0.5 )

Using z table,

= 0.7881 - 0.3085

= 0.4796

Probability = 0.4796