Chapter 12, Problem 43SGR

To determine

The probability for the given statement.

Answer to Problem 43SGR

The probability is 97.5%.

Explanation of Solution

Given:

From the online poll, 28% of adult feels standard 40 hour work week should be increased.

Sheila thinks the number should be lower, so she conducts the survey of the 250 random adults.

Probability of more than 55 of the surveyed adults feel that the standard 40 hour week should be increased.

Considering x as the random variable which shows the number of adults feel that the standard 40 hour week should be increased.

Here, x has binomial distribution with the parameters n=250 and s=0.28.

Using the normal distribution:

  $\begin{array}{l}\overline{x}=ns\\ =250\left(0.28\right)\\ =70\end{array}$

Evaluating the standard deviation:

  $\begin{array}{l}\sigma =\sqrt{nsf}\\ =\sqrt{250\left(0.28\right)\left(0.72\right)}\\ \text{where,}\\ f=1-s=0.72\\ \text{Hence,}\\ \sigma =7.1\end{array}$

Now, 55 is about 2 standard deviation less than the mean:

Hence, the probability that more than 55 of the surveyed adults feel that the standard 40 hour weel should be increased:

  $\begin{array}{l}P\left(x>55\right)=1-P\left(x\le 55\right)\\ =1-\left(2\%+0.5\%\right)\\ =97.5\%\end{array}$

Therefore, the probability is 97.5%