Question

The Pew Research Center asked a random sample of $2024$adult cell-phone owners from the United States their age and which type of cell phone they own: iPhone, Android, or other (including non-smartphones). The two-way table summarizes the data.

Suppose we select one of the survey respondents at random.

a. Find $P\left(iPhone\right|18-34)$.

b. Use your answer from part (a) to help determine if the events “iPhone” and $"18-34"$are independent.

Short answer

Part a. $0.327$

Part b. iPhone & $18-34$are not independent.

Step-by-step solution

Part a. Step 1. Given information

The Table is-

Part a. Step 2. Calculation

We have,

$$\begin{gathered} P\left(iPhone\right|18-34)=\frac{169}{517} \\ =0.327 \end{gathered}$$

Therefore, the solution is$0.327$.

Part b. Step 1. Calculation

We have,

$P\left(iPhone\right|18-34)=\frac{P(iPhone\cap 18-34)}{P(18-34)}$

If iPhone & $18-34$are independent

$$\begin{gathered} =\frac{P\left(iPhone\right|·P(18-34)}{P(18-34)} \\ =P\left(iPhone\right) \end{gathered}$$

Now,

$$\begin{gathered} P\left(iPhone\right)=\frac{467}{2024} \\ =0.2307 \\ \ne 0.327 \end{gathered}$$

Therefore, iPhone & $18-34$are not independent.