Chapter 7, Problem R7.5RE

(a)

To determine

To find: the probability that the proportion of travellers who get a red light in a random sample of 100 travellers is less than or equal to the result in this sample.

Answer to Problem R7.5RE

0.0146

Explanation of Solution

Given:

  $\begin{array}{c}p=0.30\\ x=20\\ n=100\end{array}$

Formula used:

  ${\sigma }_{\widehat{p}}=\sqrt{\frac{p\left(1-p\right)}{n}}$

Calculation:

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

  ${\mu }_{\widehat{p}}=p=0.30$

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

  $\begin{array}{c}{\sigma }_{\widehat{p}}=\sqrt{\frac{p\left(1-p\right)}{n}}\\ =\sqrt{\frac{0.30\left(1-0.30\right)}{100}}\\ =0.0458\end{array}$

The sampling distribution of $\widehat{p}$ is about normal if $np$ and $n\left(1-p\right)$ are both at least 10.

  $\begin{array}{c}np=100\times 0.30=30\\ n\left(1-p\right)=100\times \left(1-0.30\right)=70\end{array}$

  $\widehat{p}=\frac{x}{n}=\frac{20}{100}=0.2$

The z-score is

  $\begin{array}{c}z=\frac{x-\mu }{\sigma }\\ =\frac{0.2-0.30}{0.0458}\\ =-2.18\end{array}$

Determine the corresponding probability using table A:

  $P\left(\widehat{p}\le 0.20\right)=P\left(z<-2.18\right)=0.0146$

(b)

To determine

To Explain: that there is convincing evidence that less than 30% of all travellers will be stopped.

Answer to Problem R7.5RE

Yes

Explanation of Solution

The probability in part (a) was 0.0146.

A probability in part (a) was then small, which means that the it is unusual that only 20 of the 100 travellers get a red light and thus there is convincing evidence that less than 30% of all travellers will be stopped.