Chapter 18, Problem 12E

To determine

To use $68-95-99.7$ rule and describe the sampling distribution model for the proportion of smokers among a randomly selected group of $50$ adults and also discuss your assumptions and conditions.

Answer to Problem 12E

All the appropriate conditions necessary for your analysis are met.

Explanation of Solution

It is given in the question that public health statistics indicate that $26.4\%$ of American adults smoke cigarettes. Thus, using the rule $68-95-99.7$ , we get,

  $\begin{array}{l}\mu =p=0.264\\ \sigma =\sqrt{\frac{pq}{n}}\\ =\sqrt{\frac{0.264(0.736)}{50}}\\ =0.062\end{array}$

Thus, about $68\%$ of the sample proportions are expected to be between $\text{0}\text{.202 and 0}\text{.326}$ and about $95\%$ are expected to be between $\text{0}\text{.14 and 0}\text{.388}$ and about $99.7\%$ are expected to be between $\text{0}\text{.078 and 0}\text{.45}$ , by using the rule $68-95-99.7$ to describe the sampling distribution model. Thus, the diagram for this model is as:

  

Thus, the appropriate conditions necessary for your analysis are as:

Random condition: $\text{50}$ people were selected randomly.

  $10\%$ condition: $\text{50}$ people represents less than $10\%$ of all people.

Success/failure condition: as $np=13.2>10$ and $n(1-p)=36.8>10$ then this is also true.

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