Given : The critical path for the project has an expected duration of 112 days, and a standard deviation of 5 days.
i.e. [tex]\mu=112\ ;\ \sigma=5[/tex]
We assume that this a normal distribution.
Let x be the random variable that represents the time duration to complete the project.
z-score : [tex]z=\dfrac{x-\mu}{\sigma}[/tex]
For x= 115
[tex]z=\dfrac{115-112}{5}=0.6[/tex]
P-value : [tex]P(z<115)=P(z<0.6)=0.7257469\approx0.726[/tex]
Thus, the probability of finishing on time if the due date is 115 days is 0.726.
Also, for x= 120
[tex]z=\dfrac{120-112}{5}=1.6[/tex]
P-value : [tex]P(z>120)=P(z>1.6)=1-P(z<1.6)[/tex]
[tex]1-0.9452007=0.0547993\approx0.055[/tex]
Hence, the probability that they will have to pay the penalty is 0.055 .
