Question

the accompanying table describes the random variable x, the numbers of adults in groups of five who reported sleepwalking. complete parts (a) through (d) below.

click the icon to view the table.

a. find the probability of getting exactly 4 sleepwalkers among 5 adults.
0.033 (type an integer or a decimal. do not round.)

b. find the probability of getting 4 or more sleepwalkers among 5 adults.
0.041 (type an integer or a decimal. do not round.)

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

oa. since the probability of getting 4 sleepwalkers is the result from part (a), this is the relevant probability.
ob. since the probability of getting 5 sleepwalkers is less likely than getting 4 sleepwalkers, the result from part (a) is the relevant probability.
oc. since the probability of getting 4 or more sleepwalkers is the probability of the given or more extreme result, the result from part (b) is the relevant probability.
od. since the probability of getting fewer than 4 sleepwalkers is the complement of the result from part (b), this is the relevant probability.

probability distribution for *

x
0
1
2
3
4
5
p(x)
0.167
0.388
0.289
0.115
0.033
0.008

print
done

Explanation:

Step1: Find the probability of getting exactly 4 sleepwalkers

From the probability distribution table, when \(x = 4\), \(P(x)=0.033\).

Step2: Find the probability of getting 4 or more sleepwalkers

The probability of getting 4 or more sleepwalkers is \(P(x = 4)+P(x = 5)\).
Since \(P(x = 4)=0.033\) and \(P(x = 5)=0.008\), then \(P(x\geq4)=0.033 + 0.008=0.041\).

Step3: Determine the relevant probability for part c

A probability is considered significantly high if it is greater than \(0.05\). Since \(P(x = 4)=0.033<0.05\) and \(P(x\geq4)=0.041<0.05\), we are looking for the probability relevant to check if \(4\) is a significantly high number. The formula for the complement of \(P(x\geq4)\) is \(P(x < 4)=1 - P(x\geq4)\). But when checking if a single - value \(x = 4\) is significantly high, we compare \(P(x = 4)\) with the significance level. However, in the context of the problem (using the given options), since we are asked to pick from the options based on the probability values from the table:

• Option A: Since the probability of getting 4 sleepwalkers (\(P(x = 4)=0.033\)) is not the result from part (a) (part (a) is \(P(x = 4)\) and part (b) is \(P(x\geq4)\)).
• Option B: Since \(P(x = 5)=0.008\) and \(P(x = 4)=0.033\), \(P(x = 5)<P(x = 4)\), so getting 5 is less likely than getting 4, but this is not relevant for checking if 4 is significantly high.
• Option C: The probability of getting 4 or more (\(P(x\geq4)=0.041\)) is the complement of getting fewer than 4 (\(P(x < 4)=1 - 0.041 = 0.959\)), and we use \(P(x\geq4)\) to check if 4 is a significantly high number (because a significantly high number \(x\) for a random variable \(X\) is determined by \(P(X\geq x)\)).
• Option D: The probability of getting fewer than 4 is the complement of getting 4 or more, but it is not relevant for checking if 4 is a significantly high number.

Answer:

a. \(0.033\)
b. \(0.041\)
c. C. Since the probability of getting 4 or more sleepwalkers is the probability of the given or more extreme result, the result from part (b) is the relevant probability.