The New Director Of Special Events At A Large University Has Decided
The new director of special events at a large university has decided to completely revamp graduation ceremonies. Toward that end, a PERT chart of the major activities has been developed. The chart has five paths with expected completion times and variances as shown in the table. Graduation day is 16 weeks from now.
Use Table B and Table B1. Path Expected Duration (weeks) Variance
A 10 1.21
B 8 2.00
C 12 1.00
D 15 2.89
E 14 1.44
Assuming the project begins now, what is the probability that the project will be completed before: (Round your z-value to 2 decimal places and all intermediate probabilities to 4 decimal places. Round your final answers to 4 decimal places.)
Paper For Above instruction
The scenario involves assessing the probability of completing a project—revamping graduation ceremonies—within a specified timeline using PERT (Program Evaluation and Review Technique). To analyze this, we consider the expected durations and variances of different project paths, focusing on the critical path, which determines the overall project completion time.
The given data specify five different paths with their respective expected durations and variances. For probability calculations, it is essential to identify the critical path—the path with the longest expected duration—since the project's completion time hinges upon that pathway. Here, Path D has the maximum expected duration of 15 weeks, suggesting that it is the critical path with the highest influence on project completion time.
To determine the probability that the project completes before 16 weeks, we first calculate the cumulative mean and variance for the critical path. Given that, the total expected project duration (µ) and total variance (σ^2) are determined by summing the expected durations and variances of the activities along the
critical path. Although the actual activity-based structure of paths is not explicitly provided, based on typical PERT analysis, we assume Path D is the critical path due to its longest duration, and summing the relevant activity times provides the overall expected duration.
The expected duration for the critical path is 15 weeks, with a total variance of 2.89 (from the table). The standard deviation (σ) is the square root of the variance, which is approximately √2.89 ≈ 1.70 weeks. Using the normal distribution as an approximation for the sum of activities, we now compute the z-score, which measures how many standard deviations the completion deadline (16 weeks) is away from the mean.
The z-score is calculated as:
z = (X - µ) / σ = (16 - 15) / 1.70 ≈ 0.5882
Consulting a standard normal distribution table, the cumulative probability associated with z ≈ 0.59 is approximately 0.7224. This indicates there's about a 72.24% chance that the project will finish on or before 16 weeks.
For the second question, considering the penalty of $5,000 per week for delays, the focus shifts to calculating the probability that the project exceeds the 16-week deadline, which would trigger penalties. The probability of incurring a penalty of at least $5,000 corresponds to the probability that the project surpasses this threshold time—that is, the probability it completes beyond 16 weeks.
Given that, the probability of exceeding 16 weeks can be found as:
P(T > 16) = 1 - P(T ≤ 16) = 1 - 0.7224 = 0.2776
Thus, there is approximately a 27.76% chance that the project will incur at least a $5,000 penalty, which occurs if the project extends beyond the scheduled 16 weeks.
In summary, calculating these probabilities involves understanding the distribution of total project completion times based on activity estimates, variances, and the critical path. This approach underscores the importance of precise activity duration estimates in project management and risk assessment.
References
Shtub, A., Bard, J. F., & Globerson, S. (2014). Project Management: Engineering, Technology, and Implementation. Pearson."