Question

Precision manufacturing: A process manufactures ball bearings with diameters that are normally distributed with mean 24.4 millimeters and standard deviation 0.07 millimeter. Round your answers to at least four decimal places. (a) What proportion of the diameters are less than 24.3 millimeters? (b) What proportion of the diameters are greater than 24.6 millimeters? (c) To meet a certain specification, a ball bearing must have a diameter between 24.2 and 24.5 millimeters. What proportion of the ball bearings meet the specification? Part: 0 / 30 of 3 Parts Complete Part 1 of 3 What proportion of the diameters are less than 24.3 millimeters? The proportion of the diameters that are less than 24.3 millimeters is .

Answer 1

Z = x - µ / ∂

= 24.2 – 24.6 / 0.07

=-0.4/ 0.07

= -5.71

Explanation:

Solution:

Mean = µ = 24.4mm

Standard deviation = ∂= 0.07 mm

(a) Diameter less than 24.3 mm

P(x < 24.3)

The formula for normalized z value:

Z = x - µ / ∂

= 24.3 - 24.4 / 0.07

= - 0.1 / 0.07

= -1.4286

P(z < -1.428) = 1 – p ( z > -1.428)

= 1 – p( z < 1.42)

=1 – 0.4233

= 0.5767

(b) Diameter greater than 24.6 mm

P( x > 24.6)

The formula for normalized z value:

Z = x - µ / ∂

= 24.3 – 24.6 / 0.07

=-0.3 / 0.07

= - 4.285

P( z > -4.285) = 1 – p(z < -4.285)

= 1 – p ( z > 4.285)

=

(c) Diameter between 24.2 and 24.5 mm.

P(x > 24.5)

Z = x - µ / ∂

= 24.5 – 24.6 / 0.07

=-0.1/ 0.07

= -1.4286

P(z < -1.428) = 1 – p ( z > -1.428)

= 1 – p( z < 1.42)

=1 – 0.4233

= 0.5767

P(x < 24.2)

Z = x - µ / ∂

= 24.2 – 24.6 / 0.07

=-0.4/ 0.07

= -5.71