Problem $3(30$ points$)$

HealthTech Innovations has developed a new health monitoring device called "VitaBand". The

research and development department believes that this innovative device will expand the

company's footprint in the personal healthcare device market and significantly enhance its

profitability by promoting proactive health management.

The executive team has decided to conduct a comprehensive research study to explore the

potential success of VitaBand. This study will provide crucial insights into the production

processes and marketing strategies needed for successful launch and adoption. Below is a list of

the required activities $($in weeks$)$ for the research study and their durations.

a$)$ Complete the table above for all activities' Expected Times and Time Variance columns.

$(8$ points$)$

b$)$ Draw the project activity network diagram in the space below. $(10$ points$)$

c$)$ Determine all critical path$($s$),$ the expected project completion time of the new product

research study and its standard deviation. $(6$ points$)$

d$)$ Determine the probability that the new product research study will be completed in $13$

weeks or less. Show your work. $(6$ points$)$