Question

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 4 sleepwalkers among 5 adults.

b. Find the probability of getting 4 or more sleepwalkers among 5 adults.

c. Which probability is relevant for determining whether 4 is a significantly highnumber of sleepwalkers among 5 adults: the result from part (a) or part (b)?

d. Is 4 a significantly high 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

Short answer

a. The probability of getting exactly 4 sleepwalkers among 5 adults is 0.027.

b. The probability of getting 4 or more sleepwalkers among 5 adults is 0.029.

c. The probability used in (b) part.

d. Yes, 4 is a significantly high number of sleepwalkers among 5 adults.

Step-by-step solution

Given information

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

Calculate the probability of getting exactly 4 sleepwalkers among 5 adults

a.

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

Using the probability distribution table, the probability corresponding to 4 sleepwalkers among 5 adults is 0.027.

Thus, the probability of getting exactly 4 sleepwalkers among 5 adults is 0.027.

Calculate the probability of getting 4 or more sleepwalkers among 5 adults

b.

Using the probability distribution table, the probability corresponding to 4 sleepwalkers is 0.027.

The probability corresponding 5 sleepwalkers is 0.002.

The probability of getting 4 or more sleepwalkers among 5 adults is computed as:

$$\begin{gathered} P\left(x⩾4\right)=P\left(x=4\right)+P\left(x=5\right) \\ =0.027+0.002 \\ =0.029 \end{gathered}$$

Thus, the probability of getting 4 or more sleepwalkers among 5 adults is 0.029.

State the probability relevant to determine whether 4 is a significantly high number of sleepwalkers among 5 adults

c.

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

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

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

Here, part (b) computed the probability of 4 or more sleepwalkers in a sample of 5 adults.

Therefore, the probability computed in part(b) is relevant for concluding whether 4 sleepwalkers is significantly high or not.

Check whether 4 is a significantly high number of sleepwalkers among 5 adults

d.

Since the probability of 4 or more number of sleepwalkers in a sample of 5 adults is 0.029, which is less than 0.05, thus, the number of sleepwalkers equal to 4 can be considered significantly high.