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

Use the range rule of thumb to determine whether 4 is a significantly high number of sleepwalkersin 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.

Let x 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 countable.

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:

$$\begin{gathered} \sigma =\sqrt{\sum \left[x^2·P\left(x\right)\right]-{\mu }^2} \\ =\sqrt{3.23-{1.5}^2} \\ =0.98 \\ \approx 1.0 \end{gathered}$$

Thus, the standard deviation of x is 1.0.

Significant high values are the values greater than or equal to .

The calculations are computed as:

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

Since 4 is greater than 3.5, this implies that it is a significantly high number of sleepwalkers in groups of five adults.