Question

A project analyzed using PERT has an expected completion time of 65 days and a variance of completion time equal to 16 days. (a) What is the probability of completing the project within 60 days? (b) What is the probability of completing the project within 72 days? (c) What is the completion time that yields a 99.0% chance of completion?

Answer 1

a) 10.56% probability of completing the project within 60 days.

b) 95.99% probability of completing the project within 60 days.

c) A completion time of 74.3 days yields a 99.0% chance of completion.

Explanation:

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

`\mu = 65, \sigma = √(16) = 4`.

(a) What is the probability of completing the project within 60 days?

This probability is the pvalue of Z when
`X = 60`. So:

`Z = (X - \mu)/(\sigma)`

`Z = (60 - 65)/(4)`

`Z = -1.25`

`Z = -1.25` has a pvalue of 0.1056. This means that there is a 10.56% probability of completing the project within 60 days.

(b) What is the probability of completing the project within 72 days?

This probability is the pvalue of Z when
`X = 72`. So:

`Z = (X - \mu)/(\sigma)`

`Z = (72- 65)/(4)`

`Z = 1.75`

`Z = 1.75` has a pvalue of 0.9599. This means that there is a 95.99% probability of completing the project within 60 days.

(c) What is the completion time that yields a 99.0% chance of completion?

This is the value of X when Z has a pvalue of 0.99. So it is X when
`Z = 2.325`.

`Z = (X - \mu)/(\sigma)`

`2.325 = (X - 65)/(4)`

`X - 65 = 4*2.325`

`X = 74.3`

A completion time of 74.3 days yields a 99.0% chance of completion.