Answer
The Venn diagram,
Work Step by Step
(a)
In the Venn diagram the sum of all the regions represents the total number of students surveyed. So, the required number of students is:
\[\begin{align}
& n\left( \text{total students} \right)=n\left( \text{I} \right)+n\left( \text{II} \right)+n\left( \text{III} \right)+n\left( \text{IV} \right)+n\left( \text{V} \right)+n\left( \text{VI} \right)+n\left( \text{VII} \right)+n\left( \text{VIII} \right) \\
& =56+17+97+22+453+327+219+309 \\
& =1500
\end{align}\]
Hence, the number of students were surveyed is 1500.
(b)
In the Venn diagram the sum of the regionsII, III, IV, V, VI and VII represents the students who drank alcohol regularly or smoked cigarettes. So, the required number of students is:
\[\begin{align}
& n\left( \text{alcohol or cigarettes} \right)=n\left( \text{II} \right)+n\left( \text{III} \right)+n\left( \text{IV} \right)+n\left( \text{V} \right)+n\left( \text{VI} \right)+n\left( \text{VII} \right) \\
& =17+97+22+453+327+219 \\
& =1135
\end{align}\]
Hence, the number of students drank alcohol regularly or smoked cigarettes is 1135.
(c)
In the Venn diagram the region I represents the students used illegal drugs only. So, the required number of students is:
\[\begin{align}
& n\left( \text{illegal drugs only} \right)=n\left( \text{I} \right) \\
& =56
\end{align}\]
Hence, the number of students used illegal drugs only is 56.
(d)
In the Venn diagram the region VI represents the number of students who drank alcohol regularly and smoked cigarettes, but did not use illegal drugs. So, the required number of students is:
\[\begin{align}
& n\left( \text{alcohol and cigarettes not drugs} \right)=n\left( \text{VI} \right) \\
& =327
\end{align}\]
The number of students who drank alcohol regularly and smoked cigarettes, but did not use illegal drugs is 327.
(e)
In the Venn diagram the sum of the regions I, II andV represents the number of students who drank alcohol regularly or used illegal drugs, but did not smoke cigarettes. So, the required number of students is:
\[\begin{align}
& n\left( \text{drank alcohol or used illegal drugs but not cigarettes} \right)=n\left( \text{I} \right)+n\left( \text{II} \right)+n\left( \text{V} \right) \\
& =56+17+453 \\
& =526
\end{align}\]
The number of students who drank alcohol regularly or used illegal drugs, but did not smoke cigarettes is 526.
(f)
In the Venn diagram the sum of the regions II, IV and VI represents the number of students engaged in exactly two of these behaviors. So, the required number of students is:
\[\begin{align}
& n\left( \text{exactly two behaviors} \right)=n\left( \text{II} \right)+n\left( \text{IV} \right)+n\left( \text{VI} \right) \\
& =17+22+327 \\
& =366
\end{align}\]
The number of students engaged in exactly two of these behaviors is 366.
(g)
Explanation:
In the Venn diagram the sum of the regions I, II, III, IV, V, VI and VII represents the number of students engaged in at least one of these behaviors. So, the required number of students is:
\[\begin{align}
& n\left( \text{at least one behaviors} \right)=1500-n\left( \text{none} \right) \\
& =1500-n\left( \text{VIII} \right) \\
& =1500-309 \\
& =1191
\end{align}\]
The number of students engaged in at least one behaviors is 1191.
