Chapter 7, Problem 5CRE

a.

To determine

To compute: The probability that the proportion of travelers who get a red light less than 20.

Answer to Problem 5CRE

The probability is 0.0145.

Explanation of Solution

Given:

Probability of success ( p ) = 0.30

Number of events ( x ) = 20

Number of trials ( n ) = 100

Calculation:

If X is a random variable defined as the number of people get red light.

Since, np (mean) and np (1- p ) (variance) are greater than 5. Therefore, X will follow Normal approximation.

  $\begin{array}{l}X\sim N\left(np,np\left(1-p\right)\right)\\ X\sim N\left(100\left(0.30\right),100\left(0.30\right)\left(1-0.30\right)\right)\\ X\sim N\left(30,21\right)\end{array}$

The probability that the proportion of travelers who get a red light less than 20, can be calculated as:

  $\begin{array}{c}P\left(X<20\right)=P\left(\frac{X-np}{\sqrt{np\left(1-p\right)}}<\frac{20-30}{\sqrt{21}}\right)4.5826\\ =P\left(Z<-2.1822\right)\left(\text{From standard normal table}\right)\\ =0.0145\end{array}$

Thus, the required probability is 0.0145.

b.

To determine

To explain: If the researcher should believe on the agent’s claim or not.

Explanation of Solution

From the above part, the probability is 0.0146. Since, the probability is low thus it appears to be that the claim is false, Thus, the researcher does not believe that the customs agent claim.