You are Engaged by Cinquieme Generation Technology as a Business Operations Consultant: Mathematics and Statistic Assignment, SIMGE, Singapore

The Case

Cinquieme Generation Technology is the leading provider of Information and Communication Technology (ICT) to service providers, businesses, and commercial users. Cinquieme Generation Technology enables the full value of connectivity by offering solutions to help their customers maintaining productivity and business continuity.

In view of the rapid growth in digital solutions for remote working, Cinquieme Generation Technology set up a new division to develop cutting-edge tools, Remote Working Solutions (RWS), including digital hardware, software, and services to their customers. With managed services, Cinquieme Generation Technology creates real value and business continuity by taking an integrated approach covering all aspects of their customers’ businesses.

You are engaged by Cinquieme Generation Technology as a business operations consultant and they would like you to perform the following analyses for them.

Project Requirement

Part 1 (Functions & Graphs)
The research department in Cinquieme Generation Technology established the following price-demand, cost, and revenue functions for their Remote Working Solutions digital hardware:
𝑝(𝑥) = 50 − 1.25𝑥 Price-demand function
𝐶(𝑥) = 160 + 10𝑥 Cost function
𝑅(𝑥) = 𝑥 ∙ 𝑝(𝑥) Revenue function
where x is in thousands of units, and C(x) and R(x) are in thousands of dollars. All three functions have domain 1 ≤ 𝑥 ≤ 40.
(a) Graph the cost function and the revenue function simultaneously in the same coordinate system.

(b) Write a profit function.
(c) Determine algebraically when P = 0, P < 0 and P > 0, to the nearest
unit.

(c) Determine algebraically the maximum profit (to the nearest thousand dollars) and the output (to the nearest unit) that produces the maximum profit. What is the wholesale price of the Remote Working Solutions digital hardware (to the nearest dollar) at this output?

Part 2 (Data Description & Probability Distribution)
The weekly record of average reported working hours from one of Cinquieme Generation Technology customers in a random sample of 35 weeks from the past 1 year is listed below:
34 33 36 35 37 31 37
39 34 35 37 35 32 35
33 35 32 34 32 32 39
34 31 35 33 31 38 34
36 34 37 34 36 39 34
(a) Construct a frequency and relative frequency table using class intervals of width 2, starting at 29.5.

(b) Construct a histogram and frequency polygon.

(c) Find the mean and standard deviation for the grouped data

(d) What is the probability that an employee chosen at random from the sample will have a reported working hour of at least 35 hours per week?

(e) Given that the daily working hours are 8 hours with a 5-day workweek. Comment on whether Cinquieme Generation Technology should promote Remote Working Solutions to the customer.

Part 3 (Logic, Sets, and Counting)
Cinquieme Generation Technology software development department consists of 6 programmers working remotely and 4 programmers working onsite.
(a) How many ways can the department select a chief programmer, a backup programmer, and a programming librarian for the Remote Working Solutions project?
(b) How many of the selections in part (a) consist entirely of remote programmers?
(c) How many ways can the department select a team of 3 programmers to work on a particular project?
(d) How many ways can the department select a team of 3 programmers with at least one working onsite?

Part 4 (Probability)
By testing a large number of individuals, it has been determined that 82% of the population are not infected by COVID-19, 11% are infected with mild symptoms and 7% are infected with severe problems. After vaccination,
95% of the persons not infected by COVID-19, 30% of those infected with mild symptoms, and 5% of those infected with severe problems will successfully develop immunity to COVID-19.
(a) Construct a probability table and probability tree based on the information provided.
(b) What is the probability that after vaccination, a person who successfully developed immunity are those not infected by COVID19?
(c) Comment on whether remote working is still preferred given the availability of the vaccine?