Question

A tasks completion time is normally distributed with a mean of 200 work days and a SD of 10 work days. What is the probability that:

(a) The task finishes within 185 work days.

(b) The task takes longer than 212 work days

(c) In order to be 99% sure that the task will not be completed late, the contractor must request a completion time of _______ days

Answer 1

a)

X ~ N ( µ = 200 , σ = 10 )
We convert this to standard normal as
P ( X < x ) = P ( Z < ( X - µ ) / σ )
P ( ( X < 185 ) = P ( Z < 185 - 200 ) / 10 )
= P ( Z < -1.5 )
P ( X < 185 ) = 0.0668

b)

X ~ N ( µ = 200 , σ = 10 )
We covert this to standard normal as
P ( X < x) = P ( (Z < X - µ ) / σ )
P ( X > 212 ) = P(Z > (212 - 200 ) / 10 )
= P ( Z > 1.2 )
= 1 - P ( Z < 1.2 )
= 1 - 0.8849

= 0.1151

c)

X ~ N ( µ = 200 , σ = 10 )
P ( X < x ) = 99% = 0.99
To find the value of x
Looking for the probability 0.99 in standard normal table to calculate critical value Z = 2.3263
Z = ( X - µ ) / σ
2.3263 = ( X - 200 ) / 10
X = 223.26 days