A project has the three paths. The paths and three estimated times $($most likely, optimistic and pessimistic are as follows:

Expected time: to $+4$tm$+$ tp$/6$

Variance $($tp $-$ to$)2/36$

Path mean $=$ sum of expected times on path

Path standard deviation $=$ square root of path mean

Z $=$ Specified time $$ Path mean$/$ path standard deviation

Path XYZ

Optimistic $($to$)$ Most likely $($tm$)$ Pessimistic $($tp$)$ Expected time$($te$)$

Activity X $134$

Activity Y $246$

Activity Z $235$

Path QRS

Activity Q $345$

Activity R $357$

Activity S $579$

Path TUV

Activity T $236$

Activity U $468$

Activity V $346$

$1.$ Determine the expected time for each activity X $2\mathrm{.}83$ Y $4\mathrm{.}00$ Z $3\mathrm{.}17$ Q $4\mathrm{.}00$ R $5\mathrm{.}00$ S $7\mathrm{.}00$ T $3\mathrm{.}33$ U $6\mathrm{.}00$ V$4\mathrm{.}17$

$2.$ What is the critical path $($QRS$)$

$3.$ What is the variance of each activity on the critical path $(\mathrm{.}944,1\mathrm{.}00,1\mathrm{.}139)$

$4.$ What is the path standard deviation $(1\mathrm{.}00)$

$5.$ What is the probability of completing the project $($critical path$)$ in $15$ weeks, and $17$ weeks $(\mathrm{.}1587,\mathrm{.}8413)$