Question

The diameters of a batch of ball bearings are known to follow a normal distribution with a mean 4.0 in and a standard deviation of 0.15 in. If a ball bearing is chosen randomly, find the probability of realizing the following event: (a) a diameter between 3.8 in and 4.3 in, (b) a diameter smaller than 3 9 in, (c) a diameter larger than 4.2 in.

Answer 1

Solution :

Given that,

mean = = 4.0

standard deviation = =0.15

a ) P (3.8 < x < 4.3 )

P ( 3.8 - 4.0 /0.15) < ( x -  / ) < ( 4.3 - 4.0 / 0.15)

P ( - 0. 2 /0.15 < z < 0.3 / 0.15 )

P (-1.33 < z < 2 )

P ( z < 2 ) - P ( z < -1.33)

Using z table

= 0.9772 - 0.0918

= 0.8854

Probability =

b ) P ( x < 3.9 )

P ( x -  / ) < ( 3.9 - 4.0 / 0.15)

P ( z < - 0.1 / 0.15 )

P ( z < - 0.67 )

Using z table

= 0.2514

Probability =0.2514

c ) P ( x > 4.2 )

= 1 - P ( x < 4.2 )

= 1 - P( x -  / ) < ( 4.2 - 4.0 / 0.15)

= 1 - P( z < 0.2 / 0.15 )

= 1 - P ( z < 1.33 )

Using z table

= 1 - 0.9082

= 0.0918

Probability = 0.0918