Chapter 7.2, Problem 40E

To determine

Tocalculate:The probability of the sample that 20% or more own Harleys.

Answer to Problem 40E

The required probability is 0.0001.

Explanation of Solution

Given information:

Proportion of registered Harley-Davidson motorcycles in the United States $=14\%$

Number of SRS motorcycle owners = 500

Formula used:

The mean of the sampling distribution of $\widehat{p}$is, ${\mu }_{\widehat{p}}=p$ .

The standard deviation of the sampling distribution of $\widehat{p}$is, ${\sigma }_{\widehat{p}}=\sqrt{\frac{p\left(1-p\right)}{n}}$ .

Consider the proportion of registered Harley-Davidson motorcycles in the United States be p and sample proportion of who own Harleys be $\widehat{p}$ .

So, we can write:

  $p=14\%=0.14$

  $\widehat{p}=20\%=0.20$

For the mean of the sampling distribution of $\widehat{p}$ , we can use the formula ${\mu }_{\widehat{p}}=p$ .

Substitute 0.14 for p in the formula ${\mu }_{\widehat{p}}=p$ .

  ${\mu }_{\widehat{p}}=0.14$

For the standard deviation of the sampling distribution, use the formula ${\sigma }_{\widehat{p}}=\sqrt{\frac{p\left(1-p\right)}{n}}$ .

Substitute 0.14 for p and 500 for n in the above formula and simplify.

  $\begin{array}{c}{\sigma }_{\widehat{p}}=\sqrt{\frac{0.14\left(1-0.14\right)}{500}}\\ =\sqrt{\frac{0.14\times 0.86}{500}}\\ =\sqrt{\frac{0.1204}{500}}\\ \approx 0.0155177\end{array}$

Now, find the z-score by using the formula $z=\frac{x-\mu }{\sigma }$ .

Substitute 0.20 for $x$ , 0.14 for $\mu$and 0.0155177 for $\sigma$in the formula $z=\frac{x-\mu }{\sigma }$and simplify.

  $\begin{array}{c}z=\frac{0.20-0.14}{0.0155177}\\ \approx 3.87\end{array}$

Find the probability by using table A.

  $\begin{array}{c}P\left(\widehat{p}\ge 20\%\right)=P\left(z>3.87\right)\\ =P\left(Z<-3.87\right)\\ =0.0001\end{array}$

Hence, the required probability is 0.0001.