In Exercises 21–25, refer to the accompanying table, 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).

Using Probabilities for Identifying Significant Events

a.Find the probability of getting exactly 1 sleepwalker among 5 adults.

b. Find the probability of getting 1 or fewer sleepwalkers among 5 adults.

c. Which probability is relevant for determining whether 1 is a significantly lownumber of sleepwalkers among 5 adults: the result from part (a) or part (b)?

d. Is 1 a significantly low number of sleepwalkers among 5 adults? Why or why not?

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.

a.

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

Using the probability distribution table, the probability corresponding to 1 sleepwalker among 5 adults is 0.363.

Thus, the probability of getting exactly 1 sleepwalker among 5 adults is 0.363.

b.

Using the probability distribution table, the probability corresponding to 1 sleepwalker is 0.363.

The probability corresponding 0 sleepwalkers is 0.172.

The probability of getting 1 or fewer sleepwalkers among 5 adults is computed as:

$$\begin{gathered} P\left(x⩽1\right)=P\left(x=0\right)+P\left(x=1\right) \\ =0.172+0.363 \\ =0.535 \end{gathered}$$

Thus, the probability of getting 1 or fewer sleepwalkers among 5 adults is 0.535.

c.

The following probability expression is used to determine if the given sample value is significantly low or not:

$P\left(x\text{or}\text{fewer}\right)⩽0.05$

If the above expression holds true, then the number of successes for that event can be considered significantly low.

Here, part (b) computes the probability of 1 or fewer sleepwalkers in a sample of 5 adults.

Therefore, the probability computed in part (b) is relevant for concluding whether 1 sleepwalker is significantly low or not.

d.

Since the probability of 1 or fewer number of sleepwalkers in a sample of 5 adults is 0.535, which is not less than or equal to 0.05, thus, the number of sleepwalkers equal to 1 cannot be considered significantly low.