Question

Analyzing Survey Data: In a consumer survey of 500 people,

200 indicated that they would be buying a major appliance

within the next month, 150 indicated that they would buy

a car and 25 said that they would purchase both a major

appliance and a car. How many will purchase neither? How

many will purchase only a car?

Short answer

125 people, will purchase neither

150 people, will purchase only a car

Step-by-step solution

Step 1. Given information

$$\begin{gathered} \mathrm{Survey}\mathrm{of}500\mathrm{people}: \\ 200\mathrm{indicated}\mathrm{that}\mathrm{they}\mathrm{would}\mathrm{be}\mathrm{buying}\mathrm{a}\mathrm{major}\mathrm{appliance}. \\ 150\mathrm{indicated}\mathrm{that}\mathrm{they}\mathrm{would}\mathrm{buy}\mathrm{a}\mathrm{car}. \\ 25\mathrm{said}\mathrm{that}\mathrm{they}\mathrm{would}\mathrm{purchase}\mathrm{both}\mathrm{a}\mathrm{major}\mathrm{appliance}\mathrm{and}\mathrm{a}\mathrm{car}. \end{gathered}$$

Step 2. Calculation

$$\begin{gathered} \mathrm{Let}: \\ \mathrm{Total}\mathrm{people}\mathrm{N}=500 \\ \mathrm{Number}\mathrm{of}\mathrm{people}\mathrm{buying}\mathrm{only}\mathrm{a}\mathrm{major}\mathrm{appliance}\mathrm{be}\mathrm{n}\left(\mathrm{A}\right)=200 \\ \mathrm{Number}\mathrm{of}\mathrm{people}\mathrm{buying}\mathrm{only}\mathrm{a}\mathrm{car}\mathrm{be}\mathbf{n}\mathbf{\left(}\mathbf{C}\mathbf{\right)}\mathbf{=}\mathbf{150} \\ \mathrm{Number}\mathrm{of}\mathrm{people}\mathrm{buying}\mathrm{both}\mathrm{a}\mathrm{major}\mathrm{appliance}\mathrm{and}\mathrm{a}\mathrm{car}\mathrm{be}\mathrm{n}(\mathrm{A}\cap \mathrm{C})=25 \\ \mathrm{So}, \\ \mathrm{Number}\mathrm{of}\mathrm{people}\mathrm{purchasing}\mathrm{neither}\mathrm{be}\mathrm{n}(\mathrm{A}\text{'}\cap \mathrm{B}\text{'})=\mathrm{N}-\left[\mathrm{n}\left(\mathrm{A}\right)+\mathrm{n}\left(\mathrm{C}\right)+\mathrm{n}(\mathrm{A}\cap \mathrm{C})\right] \\ \mathrm{n}(\mathrm{A}\text{'}\cap \mathrm{B}\text{'})=500-\left[200+150+25\right] \\ \mathrm{n}(\mathrm{A}\text{'}\cap \mathrm{B}\text{'})=500-375 \\ \mathbf{n}\mathbf{(}\mathbf{A}\mathbf{\text{'}}\mathbf{\cap }\mathbf{B}\mathbf{\text{'}}\mathbf{)}\mathbf{=}\mathbf{125} \end{gathered}$$