Given:
75 liked cola
65 liked ginger ale
32 liked spring water
43 liked cola and ginger ale
13 liked cola and spring water
15 liked ginger ale and spring water
7 liked all three
12 liked none of them
Required:
a) We need to draw a Venn diagram for the given data.
b) We need to find the percentage of those surveyed who liked exactly two of the choices.
Explanation:
a)
Let C be the cola, G be the ginger ale and S be the spring water.
The Venn diagram
b)
We get the following from the Venn diagram.
[tex]n(C)=75,n(G)=65,n(S)=32,n(C\cap G)=43,n(C\cap S)=12,n(G\cap S)=15,n(C\cap G\cap S)=15.\text{ }[/tex]
Consider the formula.
[tex]n(C\cup G\cup S)=n(C)+n(G)+n(S)+n(C\cap G)-n(C\cap S)-n(G\cap S)+(C\cap G\cap S)[/tex][tex]Substitute\text{ }n(C)=75,n(G)=65,n(S)=32,n(C\cap G)=43,n(C\cap S)=12,n(G\cap S)=15,n(C\cap G\cap S)=15\text{ in the equation.}[/tex][tex]n(C\cup G\cup S)=75+65+32-43-13-15+7[/tex][tex]n(C\cup G\cup S)=108[/tex][tex]\text{ The number of customers = n\lparen C}\cup G\cup S)+the\text{ number of custerms like non of them}[/tex][tex]\text{ The number of customers = 108+12}[/tex][tex]\text{ The number of customers = 120 customers}[/tex][tex]\text{ The number of customers liked exactly two of the choices =}n(C\cap G)+n(C\cap S)+n(G\cap S)[/tex][tex]\text{ The number of customers liked exactly two of the choices =43+13+15}[/tex][tex]\text{ The number of customers liked exactly two of the choices =71}[/tex]The percentage of those surveyed liked exactly two of the choices
[tex]=\frac{71}{120}\times100[/tex][tex]=59.17\text{ \%}[/tex]Final answer:
The percentage of those surveyed who liked exactly two of the choices is 59.17%.
