Chapter 18, Problem 11E

(a)

To determine

To draw and label the distribution of the proportion of these cars the police will observe speeding using $68-95-99.7$ rule.

Explanation of Solution

It is given in the question that state police believes that $70\%$ of the drivers travelling on a major interstate highway exceed the speed limit. They plan to set up a radar trap and check the speeds of $80$ cars. Thus, using the rule $68-95-99.7$ , we get,

  $\begin{array}{l}\mu =p=0.7\\ \sigma =\sqrt{\frac{pq}{n}}\\ =\sqrt{\frac{0.7(0.3)}{80}}\\ =0.051\end{array}$

Thus, about $68\%$ of the sample proportions are expected to be between $0.649\text{ and 0}\text{.751}$ and about $95\%$ are expected to be between $0.598\text{ and 0}\text{.802}$ and about $99.7\%$ are expected to be between $0.547\text{ and 0}\text{.853}$ , by using the rule $68-95-99.7$ to describe the sampling distribution model. Thus, the diagram for this model is as:

  

(b)

To determine

To explain do you think the appropriate conditions necessary for your analysis are met.

Answer to Problem 11E

The appropriate conditions necessary for your analysis are met.

Explanation of Solution

It is given in the question that state police believes that $70\%$ of the drivers travelling on a major interstate highway exceed the speed limit. They plan to set up a radar trap and check the speeds of $80$ cars. Thus, the appropriate conditions necessary for your analysis are as:

Random condition: the sample may not be very representative. If the flow of the traffic is fast the speed of the other cars around may have some effect on the speed of other drivers. Similarly, if the traffic is slow police officers may find a smaller proportion of speeders than expected.

  $10\%$ condition: $80$ cars represents less than $10\%$ of all cars.

Success/failure condition: as $np=56>10$ and $n(1-p)=24>10$ then this is also true.

Thus, all the appropriate conditions necessary for your analysis are met.