Answer
The Venn diagram
Work Step by Step
(a)
In the Venn diagram, the sum of all the regions represents the total readers suffering from depression were included in the report. So, the required number of students is:
\[\begin{align}
& n\left( \text{readers surveyed} \right)=n\left( \text{I} \right)+n\left( \text{II} \right)+n\left( \text{III} \right)+n\left( \text{IV} \right)+n\left( \text{V} \right)+n\left( \text{VI} \right)+n\left( \text{VII} \right)+n\left( \text{VIII} \right) \\
& =8+2+15+20+453+80+105+150 \\
& =833
\end{align}\]
(b)
In the Venn diagram, the sum of the regions II, III, IV, V, VI and VII represent the readers who felt better using prescription drugs or meditation. So, the required number of readers is:
\[\begin{align}
& n\left( \text{drugs or meditation} \right)=n\left( \text{II} \right)+n\left( \text{III} \right)+n\left( \text{IV} \right)+n\left( \text{V} \right)+n\left( \text{VI} \right)+n\left( \text{VII} \right) \\
& =2+15+20+453+80+105 \\
& =675
\end{align}\]
(c)
In the Venn diagram, the region I represents the readers who felt better using St. John’s wort only. So, the required number of readers is:
\[\begin{align}
& n\left( \text{St}\text{. John }\!\!'\!\!\text{ s wort only} \right)=n\left( \text{I} \right) \\
& =8
\end{align}\]
(d)
In the Venn diagram, the region VI represents the number of readers who improved using prescription drugs and meditation, but not St. John’s wort. So, the required number of readers is:
\[\begin{align}
& n\left( \text{prescription drugs and meditation, but not St}\text{. Johns wort} \right)=n\left( \text{VI} \right) \\
& =80
\end{align}\]
(e)
In the Venn diagram, the sum of the regions I, II and V represents the number of readers improved using prescription drugs or St. John’s wort, but not meditation. So, the required number of students is:
\[\begin{align}
& n\left( \text{drank alcohol or used illegal drugs but not cigarettes} \right)=n\left( \text{I} \right)+n\left( \text{II} \right)+n\left( \text{V} \right) \\
& =8+2+453 \\
& =463
\end{align}\]
(f)
In the Venn diagram, the sum of the regions II, IV and VI represents the number of readers improved using exactly two of these treatments. So, the required number of students is:
\[\begin{align}
& n\left( \text{exactly two behaviors} \right)=n\left( \text{II} \right)+n\left( \text{IV} \right)+n\left( \text{VI} \right) \\
& =2+20+80 \\
& =102
\end{align}\]
(g)
In the Venn diagram, the sum of the regions I, II, III, IV, V, VI and VII representing the number of readers improved using at least one of these treatments. So, the required number of students is:
\[\begin{align}
& n\left( \text{at least one treatments} \right)=833-n\left( \text{none} \right) \\
& =833-n\left( \text{VIII} \right) \\
& =833-150 \\
& =683
\end{align}\]
