Question

1. A marketing survey of consumers' soft-drink preferences collected the following data:75 liked cola65 liked ginger ale32 liked spring water43 liked cola and ginger ale13 liked cola and spring water15 liked ginger ale and spring water7 liked all three12 liked none of them(a) Construct a proper Venn diagram to display the above information.(b) What percentage of those surveyed liked exactly two of the choices?

Answer 1

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.

Step-by-step 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.

`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{ }`

Consider the formula.

`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)`
`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.}`
`n(C\cup G\cup S)=75+65+32-43-13-15+7`
`n(C\cup G\cup S)=108`

`\text{ The number of customers = n\lparen C}\cup G\cup S)+the\text{ number of custerms like non of them}`
`\text{ The number of customers = 108+12}`
`\text{ The number of customers = 120 customers}`
`\text{ The number of customers liked exactly two of the choices =}n(C\cap G)+n(C\cap S)+n(G\cap S)`
`\text{ The number of customers liked exactly two of the choices =43+13+15}`
`\text{ The number of customers liked exactly two of the choices =71}`

The percentage of those surveyed liked exactly two of the choices

`=(71)/(120)*100`
`=59.17\text{ \%}`

Final answer:

The percentage of those surveyed who liked exactly two of the choices is 59.17%.