Chapter 12, Problem 42SGR

To determine

The probability that at least 60 of the polled students did not want to go to a theme park.

Answer to Problem 42SGR

The probability is1.7%.

Explanation of Solution

Given:

The poll of the Ryan’s school determined that 76% of the students wanted to go to theme park.

Ryan surveyed 180 random students from the school.

Considering x as the random variable shows the number of the students did not want to go to a theme park.

Here x has binomial distribution with the parameters:

  $n=180\text{and }s=1-0.76=0.24$

Using the normal distribution for approximation of the probability as follows:

Now, evaluating the mean of the normal approximation:

  $\begin{array}{l}\overline{x}=ns\\ =180\left(0.24\right)\\ =43.2\end{array}$

Evaluating the standard deviation of the normal approximation as shown below:

  $\begin{array}{l}\sigma =\sqrt{nsf}\\ =\sqrt{180\left(0.76\right)\left(0.24\right)}\\ =5.73\end{array}$

The z- score for x=60:

  $\begin{array}{l}z=\frac{x-\mu }{\sigma }\\ =\frac{60-43.2}{5.73}\end{array}$

Therefore, the probability that at least 60 of the students did want to go to theme park:

  $\begin{array}{l}P\left(x\ge 60\right)=1-P\left(x\le 60\right)\\ =1-P\left(z\le 2.93\right)\\ =1-0.9983\\ =0.0017\\ =1.7\%\end{array}$

Hence, the probability that at least 60 of student did not want to go to theme park is 1.7%.