Q. 40 - The Practice of Statistics for AP

Question

Harley-Davidson motorcycles make up $14\%$ of all the motorcycles registered in the United States. You plan to interview an SRS of $500$ motorcycle owners. How likely is your sample to contain $20\%$ or more who own Harleys? Follow the four-step process. 

Step-by-Step Solution

Answer

$P(\hat{p}\ge 20\%)=0.0001$

Step 1: Given Information

Given

$p=14\%=0.14$

$\hat{p}=20\%=0.20$

$n=500$

Step 2: Explanation

The mean of the sampling distribution of $\hat{p}$ is equal to the population proportion $p$ :

${\mu }_{\hat{p}}=p=0.14$

The standard deviation of the sampling distribution of $\hat{p}$ is:

$$\begin{gathered} {\sigma }_{\hat{p}}=\sqrt{\frac{p(1-p)}{n}} \\ =\sqrt{\frac{0.14(1-0.14)}{500}} \\ \approx 0.0155177 \end{gathered}$$

The $z$-score is the value decreased by the mean, divided by the standard deviation:

$$\begin{gathered} z=\frac{x-\mu }{\sigma } \\ =\frac{0.20-0.14}{0.0155177} \\ \approx 3.87 \end{gathered}$$

Determine the corresponding probability using table $A$:

$$\begin{gathered} P(\hat{p}\ge 20\%)=P(z>3.87) \\ =P(Z<-3.87) \\ =0.0001 \end{gathered}$$