Chapter 4, Problem 46SE

A survey of adults aged 18 and older conducted by Princess Cruises asked how many days into your vacation does it take until you feel truly relaxed (USA Today, August 24, 2011). The responses were as follows: 422—a day or less; 181—2 days; 80—3 days; 121—4 or more days; and 201—never feel relaxed.

1. a. How many adults participated in the Princess Cruises survey?
2. b. What response has the highest probability? What is the probability of this response?
3. c. What is the probability a respondent never feels truly relaxed on a vacation?
4. d. What is the probability it takes a respondent 2 or more days to feel truly relaxed?

To determine

Compute the number of adults participated in the Princes Cruises survey.

Answer to Problem 46SE

There is about 1,005 participates in the Princes C survey.

Explanation of Solution

Calculation:

The given data shows that 422 of adults take only one day or less, 181 of the adults take 2days, 80 take three days, 121 take four or more days, and 201 never take feel relaxed.

The number of adults Participate in the survey can be obtained as follows:

$\text{Total adults=}\left[\begin{array}{l}N\left(1\text{ days or less}\right)+N\left(\text{2 days }\right)+N\left(\text{3 days }\right)+\\ N\left(\text{4 days or more}\right)+N\left(\text{never relaxed}\right)\end{array}\right]$

From the data, 422 of adults take only one day or less, 181 of the adults take 2days, 80 take three days, 121 take four or more days, and 201 never take feel relaxed. Substitute these values in the above equation.

Therefore,

$\begin{array}{c}\text{Total adults}=422+181+80+121+201\\ =1,005\end{array}$

Thus, there is about 1,005 participates in the Princes C survey.

To determine

Identify what respond has the highest probability and compute its probability value.

Answer to Problem 46SE

The responses who take a day or less to feel truly relaxed, has the highest probability is a day or less. The probability of the event is$P\left(\text{a day or less}\right)=0.4199$.

Explanation of Solution

Calculation:

The probability of the event can be obtained as follows:

$P\left(E\right)=\frac{\text{Number of ways event occurs}}{\text{Total frequency}}$

Here, the most frequent response is a day or less. Therefore highest probability response is a day or less. The frequency of the event is 422 and total number of adults is 1,005.

Therefore,

$\begin{array}{c}P\left(\text{a day or less}\right)=\frac{422}{1,005}\\ =0.4199\end{array}$

Hence, the probability that selected adult relaxed in a day or less is 0.4199.

To determine

Compute its probability that a respondent never feels truly relaxed on a vocation.

Answer to Problem 46SE

The probability that a respondent never feels truly relaxed on a vocation is 0.20.

Explanation of Solution

Calculation:

There is a total of 1,005 adults and 201 never feel relaxed. Substitute this values in the probability formula.

Therefore,

$\begin{array}{c}P\left(\text{never feel relaxed}\right)=\frac{201}{1,005}\\ =0.2\end{array}$

Hence, the probability that a respondent never feels truly relaxed on a vocation is 0.20.

To determine

Compute its probability that it take a respondent 2 or more days to feel truly relaxed.

Answer to Problem 46SE

The probability for respondent 2 or more days to feel truly relaxed is 0.3801.

Explanation of Solution

Calculation:

There are total of 1,005 adults, 181 of the adults take 2 days, 80 take 3 days, 121 take for or more days to feel relaxed. That is 382 $\left(=181+80+121\right)$of the respondent take 2 or more days to feel relaxed.

Substitute these values in the probability formula.

Therefore,

$\begin{array}{c}P\left(2\text{or more days}\right)=\frac{382}{1,005}\\ =0.38009\\ =0.3801\end{array}$

Hence, the probability for respondent 2 or more days to feel truly relaxed is 0.3801.