Question

do males or females feel more tense or stressed out at work? a survey of employed adults conducted online by a company on behalf of a research organization revealed the data in the contingency table shown to the right. complete parts (a) through (d) below.

gender | felt tense or stressed out at work

yes

no

total

male | 305 | 200 | 505
female | 315 | 100 | 415
total | 620 | 300 | 920

a. what is the probability that a randomly selected person’s gender is female? 0.451 (type an integer or a decimal. round to three decimal places as needed.)

b. what is the probability that a randomly selected person feels tense or stressed out at work and is female? 0.342 (type an integer or a decimal. round to three decimal places as needed.)

c. what is the probability that a randomly selected person feels tense or stressed out at work or is female? 0.783 (type an integer or a decimal. round to three decimal places as needed.)

d. explain the difference in the results in (b) and (c).

a. the joint event in part (c) includes the possibility that just one of the two characteristics occurs.

b. the two characteristics in part (b) are mutually exclusive, while those in part (c) are not.

c. the two characteristics in part (b) are collectively exhaustive, while those in part (c) are not.

(options a, b, c with radio buttons.)

Explanation:

Part (a)

Step1: Find total number of people

Total number of people is the sum of all totals in the table, which is \(505 + 415 = 920\) (or from the "Total" row, it's 920). The number of females is 415.

Step2: Calculate probability

Probability that a randomly selected person's gender is female is \(\frac{\text{Number of females}}{\text{Total number of people}}=\frac{415}{920}\approx0.451\) (rounded to three decimal places).

Step1: Find number of females who felt tense or stressed

From the table, number of females who felt tense or stressed (Yes) is 315. Total number of people is 920.

Step2: Calculate probability

Probability is \(\frac{\text{Number of females who felt tense or stressed}}{\text{Total number of people}}=\frac{315}{920}\approx0.342\) (rounded to three decimal places).

Step1: Find number of people who felt tense or stressed or are female

Using the principle of inclusion - exclusion: \(n(\text{tense or stressed or female})=n(\text{tense or stressed})+n(\text{female})-n(\text{tense or stressed and female})\). \(n(\text{tense or stressed}) = 620\), \(n(\text{female}) = 415\), \(n(\text{tense or stressed and female}) = 315\). So \(620 + 415-315 = 720\).

Step2: Calculate probability

Probability is \(\frac{720}{920}\approx0.783\) (rounded to three decimal places).

Answer:

\(0.451\)

Part (b)