Chapter 12, Problem 44SGR

To determine

The probability that at least 160 of the adults surveyed wear a watch.

Answer to Problem 44SGR

The probability is 2.5%.

Explanation of Solution

Given:

From the online poll, 74% of adults wear watches.

Timmy surveyed 200 random adults.

Using the normal distribution to approximate the probability as follows:

Now, evaluating the mean of the normal approximation as follows:

  $\begin{array}{c}\overline{x}=ns\\ =200\left(0.74\right)\\ =148\end{array}$

Evaluating the standard deviation of the normal approximation as follows:

  $\begin{array}{l}\sigma =\sqrt{nsf}\\ =\sqrt{200.\left(0.26\right).\left(0.74\right)}\left\{f=1-s=0.26\right\}\\ =6.20\end{array}$

Here, 160 is about 2 standard deviation greater than the mean.

Therefore, the probability that at least 160 of the adults surveyed wear a watch is:

  $\begin{array}{l}P\left(x\ge 160\right)=2\%+0.5\%\\ =2.5\%\end{array}$

Therefore, the required probability is 2.5%.