Question

Who owns a Harley? 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?

Short answer

Required probability is$P(\hat{p}\ge 0.20)=0.00008$

Step-by-step solution

Given Information

It is given that $p=14\%=0.14$

$n=500$

Calculation

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

We know that ${\sigma }_{\hat{p}}=\sqrt{\frac{p(1-p)}{n}}$

$=\sqrt{\frac{0.14(1-0.14)}{500}}=0.0155$

Also, $z=\frac{x-\mu }{\sigma }$

$=\frac{0.20-0.14}{0.0155}=3.87$

Required probability is $P(\hat{p}\ge 0.20)$

$=P(z\ge 3.78)=1-P(z\le 3.78)$

$=1-0.99992=0.00008$

Required probability isrole="math" localid="1654901989353" $0.00008$