Question $7[20$ Points$]$: A health information systems company plans to design, market, and implement an information system that caters specifically to nephrology practices and interfaces with dialysis clinics, so that the physician's role in patient care can be proactive across the care continuum. Activities and their optimistic, most likely, and pessimistic activity durations in hours are shown below.

$\backslash$table$[[$Activity$,$Activity Description,Predecessor Activity,Optimistic,Most Likely,Pessimistic$],[$A$,$Exploratory Phase,$-,65,157\mathrm{.}5,250],[$B$,$Maestro Modifications,A$,590,940,1,290],[$C$,$Dialysis Clinic Interface,A$,690,990,1,290],[$D$,$HL$7$ Lab Order Interface,A$,390,590,790],[$E$,$Scanned$-$In Documents,A$,190,390,590],[$F$,$Standard Phrase Code Set,A$,8,29,50],[$G$,$Family History Module,A$,590,740,890],[$H$,$Outlook Integration,A$,1,090,1,690,2,290],[$I$,$Formulary Import and Check,A$,390,590,790],[$J$,$Scheduler Module,A$,190,390,590],[$K$,$Physician Encounter Sheet,B$,2,090,3,690,5,290],[$L$,$PDA Module,K$,1,090,3,190,5,290],[$M$,$Drug$-$Drug and Drug$-$Allergy Module,I,$390,590,790],[$N$,$Coordination of Benefits,J$,190,340,490],[0,$Referral Tracking Module,N$,390,590,790],[$P$,$User Training Plan,C$,$D$,$E$,$F$,$G$,$H$,$L$,$M$,0,40,50,60],[$Q$,$Marketing Plan,L$,$P$,60,150,240]]$

a$)$ Calculate the mean duration for each activity.

b$)$ Calculate the variance for each activity time.

c$)$ Identify the mean and the standard deviation for each path.

d$)$ Calculate project completion probability for $8,200,9,200,$ and $9,500$ hours.