Question

Study of why EMS workers leave the job.Refer to the Journal of Allied Health(Fall 2011) study of why emergencymedical service (EMS) workers leave the profession,Exercise 3.45 (p. 182). Recall that in a sample of 244former EMS workers, 127 were fully compensated whileon the job, 45 were partially compensated, and 72 hadnon-compensated volunteer positions. Also, the numbersof EMS workers who left because of retirement were 7 forfully compensated workers, 11 for partially compensatedworkers, and 10 for no compensated volunteers.

a.Given that the former EMS worker was fully compensatedwhile on the job, estimate the probability that theworker left the EMS profession due to retirement.

b.Given that the former EMS worker had a non-compensatedvolunteer position, estimate the probabilitythat the worker left the EMS profession due toretirement.

c.Are the events {a former EMS worker was fully compensatedon the job} and {a former EMS worker left thejob due to retirement} independent? Explain.

Short answer

.

Step-by-step solution

Finding the probability that a fully compensated worker left due to retirement

A = Fully compensated worker leaving due to retirement

B = Total number of fully compensated workers

$\begin{array}{rcl}& & \text{P =}\frac{\text{P}\left(\text{A}\right)}{\text{P}\left(\text{B}\right)}\\ & & \text{=}\frac{\text{7}}{\text{127}}\\ & & \text{= 0.055}\\ & & \text{= 5.5\%}\\ & & \end{array}$

The probability of a fully compensated worker leaving due to retirement is 5.5%.

Calculating the probability that a volunteer left due to retirement

P(A) = Volunteer leaving due to retirement

P(B) = Total volunteers

$\begin{array}{rcl}& & \text{P}\left(\text{A|B}\right)\text{=}\frac{P\left(ACB\right)}{P\left(B\right)}\\ & & \text{=}\frac{\text{10}}{\text{72}}\\ & & \text{=}\frac{\text{5}}{\text{36}}\\ & & \text{= 0.1388 = 13.88\%}\\ & & \end{array}$

Determining whether the following events are independent or not

Both events, a fully compensated EMS worker being on job and a fully compensated EMS worker leaving due to retirement, will be independent if the occurrence of retirement does not affect the occurrence of being on job.

P(A) = Fully compensated worker retiring

P(B) = Fully compensated worker on the job

$\begin{array}{rcl}& & \text{P}\left(\text{A|B}\right)\text{= P}\left(\text{A}\right)\\ & & \frac{\text{7}}{\text{127}}\text{=}\frac{\text{7}}{\text{244}}\\ & & \text{But,}\frac{\text{60}}{\text{57}}\frac{\text{57}}{\text{171}}\\ & & \end{array}$

$$\begin{gathered} \text{P}\left(\text{A|B}\right)\text{=P}\left(\text{A}\right) \\ \frac{\text{7}}{\text{127}}\text{=}\frac{\text{7}}{\text{244}} \\ \text{But,}\frac{\text{60}}{\text{57}}\ne \frac{\text{57}}{\text{171}} \end{gathered}$$

Therefore, repairing the car and feeling guilty are not independent events.