Chapter PIV, Problem 22RE

(a)

To determine

To find: the number of smokers expected.

Answer to Problem 22RE

36

Explanation of Solution

Given:

  $\begin{array}{l}n=120\\ p=0.3\end{array}$

Formula used:

  $\mu =np$

Calculation:

The number of students chosen for high school is 120.

The probability of high school students smoking is 0.3

The number of smokers expected is

  $\begin{array}{c}np=\left(120\right)\left(0.3\right)\\ =36\end{array}$

(b)

To determine

To find: the standard number deviation for smokers.

Answer to Problem 22RE

5.02

Explanation of Solution

Given:

  $\begin{array}{l}n=120\\ p=0.3\end{array}$

Formula used:

  $\sigma =\sqrt{npq}$

Calculation:

Standard deviation is

  $\begin{array}{c}\sqrt{npq}=\sqrt{\left(120\right)\left(0.3\right)\left(0.7\right)}\\ =5.02\end{array}$

(c)

To determine

To Explain: the reason that number could be described with the normal model.

Explanation of Solution

Given:

  $\begin{array}{l}n=120\\ p=0.3\end{array}$

Calculation:

The trails are independent

There is

  $\begin{array}{c}np=\left(120\right)\left(0.3\right)\\ =36\end{array}$

  $\begin{array}{c}nq=\left(120\right)\left(0.7\right)\\ =84\end{array}$

Both $np\ge 10\text{ and }nq\ge \text{10}$

Therefore, X could be explained with normal model.

(d)

To determine

To Construct: and interpret a model for the number of smokers in 120 student groups on the basis of 68-95-99.7 rule

Explanation of Solution

By this rule

The probability of the number of smokers being between with one standard deviation

  $\begin{array}{c}\left(\mu -\sigma ,\mu +\sigma \right)=\left[36-5.02,36+5.02\right]\\ =30.98,41.02\end{array}$

The probability of the number of smokers being between with two standard deviation

  $\begin{array}{c}\left(\mu -2\sigma ,\mu +2\sigma \right)=\left[36-2\left(5.02\right),36+2\left(5.02\right)\right]\\ =25.98,46.04\end{array}$

The probability of the number of smokers being between with three standard deviation

  $\begin{array}{c}\left(\mu -3\sigma ,\mu +3\sigma \right)=\left[36-3\left(5.02\right),36+3\left(5.02\right)\right]\\ =20.94,51.06\end{array}$