Calculating the Completion Time of a Project with PERT

Given:
- Project length: 50 weeks
- Variance: 9
- Probability factor Z: 1.65 for 95% probability

Calculating the Expected Completion Time:
To calculate the expected completion time of the project, we can use the formula:

Expected Completion Time = Project Length + (Probability Factor Z * Square Root of Variance)

Let's plug in the given values into the formula:

Expected Completion Time = 50 + (1.65 * √9)
Expected Completion Time = 50 + (1.65 * 3)
Expected Completion Time = 50 + 4.95
Expected Completion Time = 54.95 weeks

Therefore, the expected completion time of the project with a 95% probability is 54.95 weeks.

Explanation:
The PERT (Program Evaluation and Review Technique) is a project management tool that helps in estimating the time required to complete a project. It takes into account three time estimates for each activity: optimistic time (O), pessimistic time (P), and most likely time (M). From these estimates, the expected time (TE) and variance (V) can be calculated using the following formulas:

TE = (O + 4M + P) / 6
V = ((P - O) / 6)²

Once the expected time and variance for each activity are determined, the project's expected completion time can be calculated by summing up the expected times of all activities. The project's variance can be calculated by summing up the variances of all activities.

In this case, we are given the project length (50 weeks) and variance (9). The variance represents the measure of uncertainty or risk associated with the project. A higher variance indicates a greater degree of uncertainty.

To calculate the expected completion time with a specific probability, we can use the probability factor Z. The Z value represents the number of standard deviations from the mean in a standard normal distribution. For example, a Z value of 1.65 corresponds to a 95% probability.

By multiplying the probability factor Z with the square root of the variance and adding it to the project length, we can estimate the expected completion time with a given probability. In this case, the expected completion time with a 95% probability is calculated to be 54.95 weeks.

Conclusion:
The PERT calculations provide an estimate of the expected completion time of a project, taking into account the uncertainty or risk associated with the project. By using the probability factor Z, we can calculate the expected completion time with a specific probability. In this case, the project is expected to be completed within 54.95 weeks with a 95% probability.