Use the following project information for Problems $4$ to $8.$

$\backslash$begin$\{$tabular$\}\{|$c$|$c$|$c$|$c$|$c$|\}$

$\backslash$hline Activity & $\backslash$begin$\{$tabular$\}\{$l$\}$

Optimistic Time $\backslash \backslash$

Estimate $($weeks$)$

$\backslash$end$\{$tabular$\}$ & Most Likely Time Estimates $($weeks$)$ & Pessimistic Time Estimates $($weeks$)$ & Immediate Predecessor $($s$)\backslash \backslash$

$\backslash$hline A & $2$ & $3$ & $5$ & none $\backslash \backslash$

$\backslash$hline B & $2\mathrm{.}5$ & $4$ & $5$ & A $\backslash \backslash$

$\backslash$hline C & $5$ & $7$ & $8$ & A $\backslash \backslash$

$\backslash$hline D & $3$ & $5$ & $7$ & B$,$ C $\backslash \backslash$

$\backslash$hline E & $4$ & $7$ & $14$ & B$,$ C $\backslash \backslash$

$\backslash$hline F & $2$ & $5$ & $8$ & D $\backslash \backslash$

$\backslash$hline G & $4$ & $5$ & $6$ & E $\backslash \backslash$

$\backslash$hline H & $6$ & $6$ & $8$ & F $\backslash \backslash$

$\backslash$hline I & $4$ & $8$ & $11$ & G $\backslash \backslash$

$\backslash$hline J & $2$ & $3$ & $3$ & H$,$ I $\backslash \backslash$

$\backslash$hline

$\backslash$end$\{$tabular$\}$

$4.$ Using the information given, construct a network diagram using AON notation.

$5.$ Using the information given, calculate the expected time for each of the project activities.

$6.$ Using the information given, calculate the variance for each of the project activities.

$7.$ Using your results from Problems $4$ and $5,$

a$.$ Calculate the completion time for this project.

b$.$ Identify the activities included on the critical path of this project.

$8.$ Using your results from Problem $6,$

a$.$ Calculate the probability that the project will be completed in $32$ weeks.

b$.$ Calculate the probability that the project will be completed in $35$ weeks.