$[30$ points$]$ An app design project consists of following activities. Times are given in weeks.

$\backslash$table$[[$Activity$,$Description,$\backslash$table$[[$Immediate$],[$Predecessor$]],\backslash$table$[[$Optimistic$],[$Time$]],\backslash$table$[[$Most Likely$],[$Time$]],\backslash$table$[[$Pessimistic$],[$Time$]]],[$A$,$Prob. Def.,$-,2,3,4\mathrm{.}5],[$B$,$Sys$.$ Analysis,A$,4,5,6],[$C$,$Design I$/$O$,$B$,4,4\mathrm{.}5,7\mathrm{.}5],[$D$,$Design DB$,$B$,13,14,21],[$E$,$Develop Input,C$,6,8,12],[$F$,$Develop Output,C$,8,9,15],[$G$,$Develop DB$,$D$,1\mathrm{.}5,1\mathrm{.}75,3\mathrm{.}5],[$H$,$Test System,E$,$ F$,$ G$,4\mathrm{.}5,5\mathrm{.}5,7\mathrm{.}5],[$I$,$Implement,H$,3,5,8]]$

a$.$ Calculate the expected activity duration $(t_e),$ variance for each activity, ES$,$ EF$,$ LS$,$ and LF times and the slack for each activity. Show the values in a table $($activity schedule$).$ What is the critical path for the project? Show all work to earn credit.

b$.$ Using MS Project $($or any other software$),$ create a Gantt chart for the project based on LS and LF times.

c$.$ What is the probability of completing the project in $33$ weeks? Show all work.

d$.$ Which activities would you focus on in order to get the project back on schedule? Why? Explain in $2-3$ sentences.

e$.$ Assume that "Systems Analysis" $($B$)$ actually finished at $8$ weeks, "Design I$/$O$"($C$)$ actually finished at $15$ weeks, and "Design Database" $($G$)$ actually finished at $29$ weeks. $($Note that this is not the duration of activity$).$ Recalculate the expected project completion time. Show all work.