Question

A random sample of 401 student were recent survey regarding their class standing freshmen software junior senior and their major type stem versus nonstem the following contingency table gives the 2 way classification of the response suppose one student is randomly selected from a group calculate the following probability

Answer 1

the Solution

$Pr(\text{STEM)}=\frac{158}{401}=0.394$$Pr(\text{Sophomore)}=\frac{87}{401}=0.217$$Pr(sophomore|Non-stem)=\frac{68}{401}=0.170$$Pr(\text{sophomore and Non-stem)=}\frac{87}{401}\times\frac{243}{401}=\frac{21141}{160801}=0.1315$

To ascertain whether Sophomore and Non-stem are dependent, we have to test the following:

$\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}$$\begin{gathered} Pr(\text{sophomore or Non-stem)=Pr(sophomore)+Pr(Non-stem)} \\ -Pr(\text{sophomore and Non-stem)} \end{gathered}$$Pr(\text{Sophomore or Non-stem)=}\frac{87}{401}+\frac{243}{401}-(\frac{87}{401}\times\frac{234}{401})$$=\frac{330}{401}-\frac{21141}{160801}=0.82294-0.13147=0.69147$

Cleary, we have that sophomore and non-stem events are dependent events since 0.69147 is not the same as 0.13147.