Final answer:
The question involves applying mathematical principles to compute probabilities in a healthcare setting and to perform hypothesis testing regarding the prevalence of a disease in a nursing home.
Step-by-step explanation:
The question relates to probability and statistics which are core topics within mathematics, particularly useful in health and nursing settings. Given the provided scenarios, mathematical techniques would be applied to compute the probabilities and conduct hypothesis testing.
Probability Calculations
For example, we would use the exponential distribution to calculate the likelihood of patients arriving within a certain time frame at an urgent care facility. The properties of this distribution allow for the calculation of various probabilities, such as the chance of a patient arriving within two minutes or taking more than 15 minutes between successive visits.
Hypothesis Testing
In the case of the nursing home study, statistical hypothesis testing, specifically a one-sample proportion test, would be conducted to compare the local incidence rate of an infectious disease to the national rate, using a significance level of 0.05.