Chapter 11.5, Problem 3E

To determine

To find: The probability that more than 80 but fewer than 100 of those surveyed will say that the work week should be increased.

Answer to Problem 3E

The probability that more than 80 but fewer than 100 of those surveyed will say that the work week should be increased is approximately $0.67653$ .

Explanation of Solution

Given:

According to an online poll, $28\%$ of adults feel that the standard 40-hour work week should be increased.

Interview is taken for $300$ adults.

Calculation:

Consider an online poll which states that $28\%$ of adults feel that the standard 40-hour work week should be increased. So,

  $\begin{array}{c}p=28\%\\ =0.28\end{array}$

Find the value of $q$ as follows.

  $\begin{array}{c}q=1-p\\ =1-0.28\\ =0.72\end{array}$

Interview is taken for $300$ adults. So,

  $n=300$

Find the value of $np$ .

  $\begin{array}{c}np=300\left(0.28\right)\\ =84\end{array}$

Find the value of $npq$ .

  $\begin{array}{c}npq=300\left(0.28\right)\left(0.72\right)\\ =60.48\end{array}$

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.

  $\begin{array}{c}\mu =np\\ =84\end{array}$

Find the standard deviation as follows.

  $\begin{array}{c}\sigma =\sqrt{npq}\\ =\sqrt{60.48}\\ =7.78\end{array}$

The probability that more than 80 but fewer than 100 of those surveyed will say that the work week should be increased will be represented by $P\left(80<X<100\right)$ .

Calculate the corresponding $z$ value for $X=80$ as follows.

  $\begin{array}{c}z=\frac{X-\mu }{\sigma }\\ =\frac{80-84}{7.78}\\ =-\frac{4}{7.78}\\ =-0.514\end{array}$

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 $-0.514,2.057$
4. Press “ENTER” to obtain the result as shown below.

  

Therefore, the probability that more than 80 but fewer than 100 of those surveyed will say that the work week should be increased is approximately $0.67653$ .