Chapter 11.5, Problem 2E

To determine

To find: The probability that no more than $380$ of the surveyed upperclassmen have summer jobs.

Answer to Problem 2E

The probability that no more than $380$ of the surveyed upperclassmen have summer jobs is approximately $0.3239$ .

Explanation of Solution

Given:

According to an online poll, $80\%$ of high school upper-class have summer jobs.

Survey is conducted on random $480$ upper-class.

Calculation:

Consider an online poll which states that $80\%$ of high school upper-class have summer jobs. So,

  $\begin{array}{c}p=80\%\\ =0.8\end{array}$

Find the value of $q$ as follows.

  $\begin{array}{c}q=1-p\\ =1-0.8\\ =0.2\end{array}$

Tadeo conducts a survey of $480$ random upper-class. So,

  $n=480$

Find the value of $np$ .

  $\begin{array}{c}np=480\left(0.8\right)\\ =384\end{array}$

Find the value of $npq$ .

  $\begin{array}{c}npq=480\left(0.8\right)\left(0.2\right)\\ =76.8\end{array}$

As the value of $np$ and $npq$ are both greater than $5$ , so the normal distribution can be used to approximate the binomial distribution.

Find the mean as follows.

  $\begin{array}{c}\mu =np\\ =384\end{array}$

Find the standard deviation as follows.

  $\begin{array}{c}\sigma =\sqrt{npq}\\ =\sqrt{76.8}\\ =8.76\end{array}$

The probability that no more than $380$ of the surveyed upperclassmen have summer jobs will be represented by $P\left(X\le 380\right)$ .

Calculate the corresponding $z$ value for $X$ as follows.

  $\begin{array}{c}z=\frac{X-\mu }{\sigma }\\ =\frac{380-384}{8.76}\\ =-\frac{4}{8.76}\\ =-0.4566\end{array}$

Calculate the required probability by using ti-83 calculator as follows.

1. Press “ $2^{\text{nd}}$ ” key and then “ $\text{VARS}$” key.
2. Press “ $2$” as shown below.

  

3. Type $-4,-0.4566$
4. Press “ENTER” to obtain the result as shown below.

  

Therefore, the probability that no more than $380$ of the surveyed upperclassmen have summer jobs is approximately $0.3239$ .