The expected completion time of the project is 40 weeks while the variance of the critical path is 9. It is worth mentioning that the standard time of the critical path is equivalent to the square root of variance (Mankiw 251). These values will be used for calculating the subsequent probabilistic questions (Baltagi 104). All the calculations in the questions will be answered using the standard normal table.
The probability of finishing the project in 40 weeks or less will be calculated using the formula below.
Z = (Specified time – Path expected time) / path standard time
= (40 – 40) √9
= 0/3
= 0
From the Z table, the area in the lower tail of a Z-value of 0 is 0.50. That is equivalent to 50%. Thus, the probability of completing the project in 40 weeks or less is 50%. This indicates that there is a 50% (100% – 50%) chance that the project will take longer than 40 weeks.
The probability that completion of the project will take longer than 40 weeks is calculated using the formula below.
Z = (Specified time – Path expected time) / path standard time
= (40 – 40) √9
= 0/3
= 0
From the Z table, the area in the upper tail of a Z-value of 0 is 0.50. This value is equivalent to 50%. Thus, the probability of that the project will take longer than 40 weeks is 50%. This can also be interpreted that the probability that the project will take less than 40 weeks is 50% (100% – 50%).
The probability that the project will be completed in 46 weeks or less is estimated using the formula below.
Z = (Specified time – Path expected time) / path standard time
= (46 – 40) √9
= 6/3
= 2
From the Z table, the area in the lower tail of a Z-value of 2.0 is 0.9772. The value is equivalent to 97.72% (0.9772 * 100). Thus, the probability of that the project will be completed in 46 weeks or less is 97.72% (Greene 191). Alternatively, it can be stated that the probability that it will take longer than 46 weeks to complete the project is 2.28% (100% – 97.72%).
The probability that the project will take longer than 46 weeks will be computed using the formula below.
Z = (Specified time – Path expected time) / path standard time
= (46 – 40) √9
= 6/3
= 2
From the Z table, the area in the upper tail of a Z-value of 2.0 is 0.0228. The value is equivalent to 2.28% (0.0228 * 100). Thus, the probability of that the project will take longer than 46 weeks is 2.28%.
In this case, time (in weeks) that will ensure that there is a 90% chance that the project is completed on schedule is calculated below. From the standard normal table the corresponding Z value for 90% (0.9) is 1.29 (Zikmund et al. 181; Verbeek 201).
Z = (Specified time – Path expected time) / path standard time
1.29 = (X – 40) √9
1.29 = (X – 40) / 3
1.29 * 3 = X – 40
3.87 = X – 40
X = 3.87 + 40
X = 43.87 weeks.
When rounded off, the value becomes 44 weeks. Therefore, the project manager should set the due date at 44 weeks. This will ensure that there is only a 10% chance that the project will take longer than this duration (Bade and Parkin 98).
Works Cited
Bade, Robin and Michael Parkin. Essential Foundations of Economics, USA: Pearson Education, 2013. Print.
Baltagi, Badi. Econometrics, New York: Springer Publishers, 2011. Print.
Greene, William. Econometric Analysis, Harlow: Prentice–Hall, 2003. Print.
Mankiw, Gregory. Principles of economics, USA: South-Western Cengage Learning, 2011. Print.
Verbeek, Marno. A Guide to Modern Econometrics, England: John Wiley & Sons, 2008. Print.
Zikmund, William, Barry Babin, Jon Carr, and Mitch Griffin. Business Research Methods. USA: Cengage Learning, 2012. Print.