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 B$1.$

Path Expected Duration

$($weeks$)$ Variance

A $101\mathrm{.}21$

B $82\mathrm{.}00$

C $121\mathrm{.}00$

D $152\mathrm{.}89$

E $141\mathrm{.}44$

Click here for the Excel Data File

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.$)$

a$.$ Graduation time?

b$.$ The end of week $15?$

c$.$ The end of week $13?$