Question

A production process has a cycle time, time to produce a part, that is normally distributed with a mean of 12.6 minutes and a standard deviation of 2.4 minutes. Let X represent the cycle time for a part being produced.

P( X > 17.4) = ______ Answer accurately to 4 decimal places.

P( 11.4 < X < 16.2)= _______ Answer accurately to 4 decimal places.

97.5 percent of all parts are produced in less than _______ minutes. Enter your answer accurate to 1 decimal place.

Answer 1

Given,

= 12.6 , = 2.4

We convert this to standard normal as

P(X < x) = P(Z < ( x - ) / )

a)

P(X > 17.4 ) = P(Z > ( 17.4 - 12.6) / 2.4)

= P(Z > 2)

= 0.0228

b)

P(11.4 < X < 16.2 ) = P(X < 16.2) - P(X < 11.4)

= P(Z < ( 16.2 - 12.6) / 2.4 ) - P(Z < ( 11.4 - 12.6) / 2.4 )

= P(Z < 1.5) - P(Z < -0.5)

= 0.9332 - 0.3085

= 0.6247

c)

We have to calculate x such that P(X < x) = 0.975

That is

P(Z < ( x - ) / ) = 0.975

From Z table, z-score for the probability of 0.975 is 1.96

( x - ) / = 1.96

( x - 12.6 ) / 2.4 = 1.96

Solve for x

x = 17.3