the Solution
[tex]Pr(\text{STEM)}=\frac{158}{401}=0.394[/tex][tex]Pr(\text{Sophomore)}=\frac{87}{401}=0.217[/tex][tex]Pr(sophomore|Non-stem)=\frac{68}{401}=0.170[/tex][tex]Pr(\text{sophomore and Non-stem)=}\frac{87}{401}\times\frac{243}{401}=\frac{21141}{160801}=0.1315[/tex]To ascertain whether Sophomore and Non-stem are dependent, we have to test the following:
[tex]\begin{gathered} If\text{ Pr(sophomore or Non-stem) = Pr(sophomore and Non-stem),} \\ \text{then we conclude that both events are Independent,} \\ \text{otherwise, they are dependent.} \end{gathered}[/tex][tex]\begin{gathered} Pr(\text{sophomore or Non-stem)=Pr(sophomore)+Pr(Non-stem)} \\ -Pr(\text{sophomore and Non-stem)} \end{gathered}[/tex][tex]Pr(\text{Sophomore or Non-stem)=}\frac{87}{401}+\frac{243}{401}-(\frac{87}{401}\times\frac{234}{401})[/tex][tex]=\frac{330}{401}-\frac{21141}{160801}=0.82294-0.13147=0.69147[/tex]Cleary, we have that sophomore and non-stem events are dependent events since 0.69147 is not the same as 0.13147.