In Exercises 21–25, refer to the accompanyingtable, which describes the numbers of adults in groups of fivewho reported sleepwalking (based on data from “Prevalence andComorbidity of Nocturnal Wandering In the U.S. Adult GeneralPopulation,” by Ohayon et al., Neurology, Vol. 78, No. 20).

Use the range rule of thumb to determine whether 3 is a significantly high number of sleepwalkers in a group of 5 adults.

x	P(x)
0	0.172
1	0.363
2	0.306
3	0.129
4	0.027
5	0.002

The probability distribution for the number of adults in groups who reported sleepwalking is provided.

Letx be the number of sleepwalkers in a group of five.

The requirements are as follows:

1)The variable x is anumerical random variable as the values are of a quantitative nature.

2)The sum of the probabilities is computed as:

$$\begin{gathered} \sum P\left(x\right)=0.172+0.363+0.306+0.129+0.027+0.002 \\ =0.999 \end{gathered}$$

Therefore,the sum of the probabilities is approximately equal to 1 with a round of error as 0.001.

3) Each value of P(x) is between 0 and 1.

Thus, there are no requirements that are not satisfied.

The mean for a random variable is computed as:

$$\begin{gathered} \mu =\sum \left[x\times P\left(x\right)\right] \\ =\left(0\times 0.172\right)+\left(1\times 0.363\right)+\left(2\times 0.306\right)+...+\left(5\times 0.002\right) \\ =1.48 \\ \approx 1.5 \end{gathered}$$

Thus, the mean number of sleepwalkers is 1.5.

The standard deviation of the random variable x is computed as:

$\sigma =\sqrt{\sum \left[x^2\times P\left(x\right)\right]-{\mu }^2}$

The calculations are as follows:

$$\begin{gathered} \sum \left[x^2·P\left(x\right)\right]=\left(0^2\times 0.172\right)+\left(1^2\times 0.363\right)+\left(2^2\times 0.306\right)+...+\left(5^2\times 0.002\right) \\ =3.23 \end{gathered}$$

The standard deviation is given as:

$$\begin{gathered} \sigma =\sqrt{\sum \left[x^2·P\left(x\right)\right]-{\mu }^2} \\ =\sqrt{3.23-{1.5}^2} \\ =0.98 \\ \approx 1 \end{gathered}$$

Thus, the standard deviation of x is 1.

Significant high values are the values greater than or equal to $\mu +2\sigma$.

The calculations are computed as:

$$\begin{gathered} \mu +2\sigma =1.5+\left(2\times 1\right) \\ =3.5 \end{gathered}$$

Since 3 is less than 3.5, this implies that it is not a significantly high number of sleepwalkers in a group of 5 adults.