A computer salesperson sold an average of 3 computers per week.

(a) Calculate the probability that in a given week, he will sell 2 or more but less than 5 computers.

(b) Assuming there are 5 working days per week, what is the probability that in a given day he will sell 1 computer?

(c) At the beginning of one particular week, the salesperson decides to have 4 computers in stock. Find the probability that this stock will not be enough for that week.

It is found that the mean number of patients arriving at the emergency ward at a hospital between 12 a.m. to 6 a.m. is 8.5.

(a) What is the probability that on a given Sunday, exactly 8 patients arrive at the emergency ward between 12 a.m. to 6 a.m.?

(b) What is the probability that 4 or fewer patients arrive at the emergency ward between 12 a.m. to 6 a.m.?

(c) How many patients would you expect to arrive on Tuesday between 12 a.m. to 6 a.m.?

(d) What is the standard deviation?

The provided scenarios involve calculating probabilities using Poisson and exponential distributions, as well as interpreting a box plot for car salesperson data.

Explanation:

Probability distribution, exponential distribution, and Poisson distribution are the main mathematical concepts required to answer questions from the provided dataset concerning sales, computer sales, emergency ward patients, website traffic, and urgent care patient arrivals.

For instance, to solve the probability questions related to computer sales or emergency ward patient arrivals, we utilize a Poisson distribution as these events occur independently and the average rate (mean) of occurrence is known. The exponential distribution, characterized by the average time between events, is suitable for answering questions about website traffic and urgent care arrivals, as these involve continuous time between random independent events. Detailed calculations would include using the appropriate probability mass function for the Poisson distribution, and the cumulative density function for the exponential distribution, tailored to the specific question at hand.

A box plot can be constructed to visualize the data on car salespersons, and its interpretation informs about data concentration and spread. For calculations such as finding the probability of at most a certain number of events or the time between events, corresponding probabilities can be calculated using the respective distribution formulas or tables.