What this assignment is about The offensive team scores and often gets the attention. However. without a defense and cooperative play no team can win a Championship. In this assignment, we'll explore the concept ot independence, dependence, and exclusivity in terms of Statistics. I hope that this assignment will shed light not only on the Statistical calculations, but how these words can give meaning to contexts and situations where independence, \"success". and teamwork connect. Background Story - Us at WFS: You and I are now midlevel management at the travel agency Whatcom Friendly Skies 14B{WFS146}. Throughout the Covid19 Pandemic, the travel industry was hit hard and we were forced to merge with our competitor, Skagit Skies. Skagit Skies, incidentally, also runs a smaller freight and delivery subdivision in cooperation with local enterprises such as WhARF Inc, Starsh Ice Cream inc, and Starsh Trucking LLC. Although we at WFS kept our company name and our jobs. many coworkers who were once employees at Skagit Skies were restructured or hired by competitors in Seattie. In order for our merger to be successful and to survive as an independent company based in IWhatcom County, we need to be fair to our new and old employees alike and create a system that serves our customers better than competitors in Seattle and Vancouver. We have a di'itiwltjob. but our CEO, who graduated from Whatcom College, firmly believes this: that the big cities with old money don't always win! She's asking us to show them our genuine WhatmmSkagit hospitality: she's asking us to win. and she has put you and me on the Workforce Integration Task-Force to help our workers come together and create a new company representing the Pacic Northwest! Lucette's informal response to the \"Killer Whale" Caring and customer service is powerful. The Skagit boys shouldn't forget that it was us who purchased them and not the other way around! Background Story - Them: The new Skagit Skies "Boys, I'm gunna say it tough, but it's always been this way. In sales, there are prey and there are predators. There are alphas and betas". The weak become prey, and the strong become predators. Get those sales no matter what!! Be like a Wolf: be independent and be free! Don't be like those dependent "Friendlies" up in Bellingham! Lock down the deals and fight for the sale: remember that it's every man for himself out there!!! Win! Win! Win! That's all that matters. Either that; or get out!" - The "Killer Whale", manager at Skagit Skies "Note that this contradicts actual scientific research. The author who wrote about "alpha males" redacted the term in follow-up research because evidence did not support those assertions. Background Events/Situation - A corporate Merger Due to our recent merger, WFS-146 has two accounting divisions, two computer systems, and two service teams that oversee customer service. In particular, the Former Skagit Skies Service Team (Team A) and the traditional Whatcom Friendly Skies 146 Service Team (Team B) still work in two different physical locations. Teleworking is something that is often discussed from time-to-time; but the split locations will be a management challenge that we need to eventually address. But there is a more pressing issue. Regarding our merger, the CEO is particularly concerned that - despite being one company- our service teams aren't cooperating enough. The transition has been particularly difficult for Team A. In fact, we lost many former Skagit Skies employees, including large numbers of employees who lived or commuted from Snohomish County and King County. Many who stayed are unhappy that they must now commute to Bellingham for work. On the other hand, many Bellingham employees received pay cuts during the pandemic, which has led many long-time WFS employees to be less than nice to the former Skagit Skies employees! We have workplace integration challenges to overcome! You and I often speak with Human Resources to address ongoing difficulties such as how some employees must now commute further, making it hard to balance child care with work. The challenges notwithstanding, we don't want to be mutually independent anymore: WFS-146 and Skagit Skies are now one company! Problem 1 Computer System Overlap - Uncovering Hidden Information using Statistical Reasoning Our computer system shows that 5300 customers received services from "us" in our current service period. 3101 received help from Team A (the larger, former Skagit Skies team) and 2102 received help from Team B (the smaller, original WFS 146 serviceteam). 569 of our customers received help from neither Team: this usually represents the situation when a customer gets help at one of our partner airlines or travel agencies through our help desk. Indeed, this is the era of international air alliances! a) To better understand the significance of the data above, draw a Venn Diagram (see Ch. 3.5) representing this situation. Clearly include the number of customers in A, B, A and B, the "neither team" group, and the Total Sample S. Do the numbers "add up" in a way that you would expect for two separate companies that share no customers? Key Conceptual Point1: If Team A and Team B worked in different corporations that shared no customers, customer service would be Mutually Exclusive. This is of course a vocabulary word in this Chapter. Key Conceptual Point2: Notice that the numbers cannot possibly add up to 5300 under the assumption of mutual exclusivity. Based on the Mathematics, this implies the existence of certain customers who received help from both Team A and Team B. The Venn Diagram is therefore a great tool for uncovering such "secrets" that may not be evident at first glance. b) Moving away from the top-down view represented by the Venn diagram, let's see things from the perspective of the customer. We can do this by computing a statistic!! Let P(A) be the probability that a customer received help from Team A and let P(B) be the probability that a customer received help from Team B. Compute P(A) and P(B). Key Conceptual Point: Note that in calculating P(A), you must divide by the size of the population. Just like an average income, a voter approval rating, or a per-capita gap of a country, the division operation creates the P(A) statistic. More generally, when this kind of division operation occurs, the resulting statistic is often easier to relate to an individual. This is one way in which Statistics (the field of study) is used: to transform numbers into forms that better relate to personal choices and decisions. C) A certain mean-spirited manager, called "the Killer Whale (KW)", is the former leader of restructuring and layoffs at Skagit Skies. The KW happens to be a friend of the current CEO. KW claims that customers are "either with Team A or with Team B" and that the two probabilities "pretty much add up to 100%, so that's everyone". Determine if this is actually true. That is, Determine whether A and B are mutually exclusive using the techniques shown in your textbook. If A and B are not mutually exclusive, how close are they to being mutually exclusive?Key Conceptual Point: Note the Black-And-White thinking and the fixation on "winners" and "losers" in KW's way of thinking. Note also that such divisions can undermine the success of a corporate merger. In terms of mathematics, this type of thinking may sometimes be captured in the formulas for mutual exclusivity. d) Based on your numbers in c, do you agree or disagree with the KW: are we really not working together at all? Some employees feel that he might use "mutually exclusive" as a pretext to propose further job cuts in the merger/integration team. You are on the management team. You can either agree or disagree; but you need to justify your reasoning! Post your response on the Discussion board. e) If you haven't already done so in part c, determine P(A and B) and P(A or B). Write to the graded discussion board what you think P(A and B) and P(A or B) mean for customers. Hint: P(A and B) should be much smaller than P(A or B). If you get the (incorrect) result that P(A and B) is larger than P(A or B), then you might be using the "General English" definition of the word rather than the correct, Statistics definition of "A and B" and "A or B". Be careful! Do ask questions if you feel confused: I'd be happy to help! f) Are A and B mutually independent? If A and B were mutually independent, what should P(A and B) be? What is the correct, actual value of P(A and B)? Is being mutually independent different from being mutually exclusive? Key Conceptual Point: In everyday usage "independent" and "mutually exclusive" are often synonyms. This is false in Statistics. "Mutually Exclusive" means that A and B cannot both occur at the same time. "Mutually Independent" roughly means that the occurrence of A does not influence the occurrence of B. If the results produced by teams A and B are mutually independent then it could mean multiple things. One possibility is that the teams may be working in geographically separate locations. Another possibility is that the teams may not be cooperating with one another! g) Calculate P(A|B). This is the probability that a customer got help from team A even though they also previously got help from B. Now; Calculate P(B|A). This is the probability that a customer got help from B even though they also got help from A. Compare these numbers. Key Conceptual Point: The statistic P(A|B) can be used to argue that team A is better than team B. This is how the argument works: let's say that Team B is terrible but that team A is excellent. In the case that a customer has worked with both teams separately,it becomes likely that there are many customers who "fail" to purchase a product with Team B then go on to purchase a product with fantastic team A. In this situation, one would expect a relatively high number for P(A|B) but a low number for P(B|A). However! The same statistic can be used to argue the opposite! Suppose that B is the scoring done by defensive members in a Soccer team and A is the scoring done by the offensive members of the same Soccer team. While it may be rare for a member of the defensive team B to score, team B may indirectly contribute to winning, resulting in a high P(A|B) [assists] while having a low P(A|B) [direct scoring] !! Similarly, if B is a second year Math class and A is a first year Math class, team B is dependent on team A for their success. Note how the subjective emotions of "Freedom" and "Independence" can cause logical fallacies if they are taken in isolation. Both the Statistics and the situation do indeed matter when making an argument! h) [Read and think, but don't write an answer. Use these ideas in problem 3f] Based on P(A|B) and P(BIA) alone, the Killer Whale argues that the Skagit Skies team is superior to the WFS 146 team. He claims that customers get "poor service" from Team B. In his (mean) words "I hate to say it, but WFS has so many screw-ups, and that's why their customers need to also get help from us. It's because, sorry to say, they suck and we're clearly better than them. " I will ask you to reconsider this proposition in Problem 3. We will need to reflect on some lurking variables (Ref: Chapter 1 for the definition of a lurking variable). Key Conceptual Point: Because WFS 146 has access to all of its own service records, one could argue that problem 1 really involves Censuses rather than Samples. This is normal in business when it comes to analyzing employee performance. However; the key point here is that the same kind of data can be hard to find if we rely on customers. This is because customers do not work for a company and have no obligation to help or cooperate in compiling statistics. Therefore, gathering data about customer satisfaction can be harder than gathering data about employee motivation. Whether employees tell the boss the full truth, however, can further complicate the accuracy of such data. Background to Problem 2: In problem 2, I will ask you to calculate a "traditional" probability problem. In problem 3, we will reflect on the results of Problem 2 to better understand the teamwork and relationship between Team A and Team B. The Mathematical Model in problem 2 represents what would happen if people were accurately represented by inanimate balls with no emotional capability of human interaction. Although decontextualized, the Mathematical Model forms a basis of comparison to see how different people are from unemotional, disconnected objects.Comment: Incidentally; in Criminal Justice and Law it's often seen to be desirable that everyone is "equal in the eyes of the law". In theory, decisions in the Justice System should align with statistical predictions of neutrality. However; the Justice system isn't fair: many statistical studies of bias have shown this. Consider studying Criminal Justice or Sociology for more information! Problem 2 Revisiting Urns and Balls: The Math Model found in Textbooks Four Red, "Team B" balls and six Blue "Team A" balls are placed into an um labeled "WFS 146". Key Conceptual Point: I'm going to point this out so that it is clear: note that the since the team B total in problem #1 is about 2000 and the team A total in problem #1 is about 3000, these are approximately in a 4 to 6 ratio. Therefore, the choice of four red and six blue balls in problem #2 is a direct attempt to approximate results using a mathematical model. In other words, the goal of this analysis is to expand upon the context of problem #1 (the previous problem) and use it to figure out what's going on in problem #3 (to follow); diving deeply into understanding "independence" in order to argue for protecting the jobs and futures of the employees on our team (an argumentative purpose to the Statistics). a) The CEO draws two marbles, with replacement. Draw a tree diagram, labeling all probabilities. Refer to example 3.24 and 3.25, but please use the team names "A" and "B" instead of the color names "B" (Blue) and R" (red) to label your tree. Key Conceptual Point: Note that in the case of sampling with replacement, the probability of drawing a red ball and the probability of drawing a blue ball are mutually independent. This is analogous to the case of two teams of people who don't really work together: it's as if there were no cooperation! b) The CEO draws two marbles, without replacement. Draw a tree diagram, labeling all probabilities. Refer to example 3.24 and 3.25, but please use the team names "A" and "B" instead of the color names "B" (Blue) and R" (Red) to label your tree. Key Conceptual Point: Note that in the case of sampling with replacement, the probability of drawing a red ball and the probability of drawing a blue ball are mutually dependent. It might feel counterintuitive for something like drawing balls -which arguably have no emotions or feelings- can result in some kind of dependence. However, the process in 2b is the process of resource depletion. Whether it is mining, overfishing,hunting, oil extraction, or global warming, the number of "balls" that have been extracted from the Environment can have an impact on our probability ofsuccess! The Story Advances: The Leadup to Problem 3 The data in Problem 1 showed that there exists a group \"A and B\" of customers who got help from both Team A and Team B. To further study our merger process. the CEO asked us to analyze this "A and B" data further. She feels that further study might reveal the key to cooperation between ourtwo teams. The Killer Whale, who has been lecturing again about the (unscientic) notion of an \"Alpha Male Attitude", is rumored to be preparing a presentation to the CEO where restructuring 'replaoeable" employees on the Bellingham Team while giving raises to the \"superior" employees in the former Skagit Skies team. Meanwhile, the Computer Records manager has kindly provided all of the needed passenger data that we need. Lucette thinks that the CEO secretly disagrees with KW's view because she has tasked you with additional data analysis. Incidentally, interviews with customers suggest that Team A does a fantastic job with sales generation and selling tour packages. Likewise, interviews with customers of Team B suggest that Team B does a fantastic job with multilingual support, family care, customer service, and cooperation with our international ight partners. However; this is selfreport data based on interviews: does it agree with our statistics? Let's see what we can nd! New Data: Missing Data Computer Records reports that there were a total of 961 logs 01 passengers who made a purchase with WFS atter receiving help at least twice. The data is subdivided into the following categories. AA - This is a case where a customer made a purchase after visiting team A rst and \"closing the deal" with team A at a different time. AB- This is a case where a customer made a purchase after visiting team A rst but then completing the sale with team B at a later date. in a sense, team B \"closed the deal" in the AB case. BA This is a case where a customer made a purchase after visiting team B rst but then completing the sale with team A at a later date. in a sense, team A \"closed the deal" in the BA case. BB - This is a case where a customer made a purchase after visiting team B once and \"closing the deal" with team B at a ditferent time. Key Conceptual Point: Note that the AB and BA categories happen to add up to the same total as the \"A and B" category of problem #'1. These numbers are related! This data needed for the analysis is not complete because Team A. while willing to present numbers that showcase \"their work" {AA data}, was uncooperative when it came to the other data that they have. More specifically, they refused to help us get access to the "AB" and 'BA" data. Because we are on Team B, the \"BB\" data was easy to get. However, the AA data and the BB data alone is far from enough to get a full picture of what is going on. Evidently, the KW doesn't want us to know! Background to Problem 3 After a frustrating exchange of emails with Team A, Lucette went directly to our Company Headquarters and got the combined Team A and Team B sales data from the Records Manager {the number 961). Also; crossreferencing the sales invoices in corporate HQ we were able to get the \"BA+EIB" subtotal (533: see chart)'. Team A was not able to interfere with the release of this information because the collective data is managed by our Corporate HQ. which operates at a higher level of access. This data isn't enough to fully complete our contingency table; however, instead of emailing the {uncooperative} KW back and waiting fora possible non-response we have decided to use algebra to calculate the unknown data. KW may be trying to limit access to our information so that our proposal to the CEO will fail. Let's complete the table and move forward with our \"independent\" analysis! Key Conceptual Point: Note that by symmetry, Team A can be expected to have the \"AB+AA subtotal" if they are to contact HQ. Problem 3 a] Complete the contingency table. _ Totals 96'! b] Use this table to compute probabilities PILAA). PiAB}. PEA), PiBE}, PAA or AB] and PiBA or BB]. Then, place these on a Tree Diagram similar to Problem #2. Your solution should look similar to Example 3.25; however, I would like you to use decimals for the probability rounded to the tenth of a percent. Example: "22.43%" In the next step, we will compare our gures to the Mathematical model used in problem #2. Key Conceptual Point: Note how PiAA}. RAB), PiBA}, HEB) correspond to the inner squares of the contingency table and how PiAA or AB] and PiBA or BB} correspond to subtotal squares. Keyr Conceptual PointZ: One of your colleagues in the records department said it's \"almost reversed" between Team A and E in the rst row and \"the second row... what's going on here?" as compared to what might be expected based on the ball model in problem #2. Analysis at the Tree Diagram- c} As previously explained, the tree diagram in #2 represents what may be expected if A and B were independent events. Compare cells AA and BB to your answer in #2a. How close are these to the prediction? Key Conceptual Point: This is the data that the Killer Whale wanted you to have. d] Using the methods shown in your textbook, use the full data to determine if A and B are independent events. Key Conceptual Point: If A and El are independent, then the two teams would behave like marbles in urns or "Alpha-Male Lone Wolves" that are \"lndividualists\". According to the data. is this really the case?r That is, do our numbers match what we predicted in problem #2? Note that this analysis should include data that the Killer Whale didn't want you to have! Key Conceptual Question: Try to predict what the Killer Whale might argue. You'll see the answer soon, in part i. e] Calculate HEW}. This represents the probability that a customer completes a sale after receiving help from A rst and then El second. Also calculate P{A|B}. This represents the probability that a customer completes a sale after receiving help from El first and then A second. Based on these numbers, does the order of help given matter? Why do you think this might be given the strengths of Team A and Team B? Post your answer to the discussion board. Key Conceptual Point: I hope this result -and the entirety of this assignment- will help you to answer f, which is next. f) It has happened! The Killer Whale has argued that "fiercely independent" Team A "closes about three-quarters of all deals" [AA+BA] and that "Whenever Team A starts a deal, Team B screws it up and often can't close" [AB is low]. The KW argues that team A is clearly the "superior team". The KW concludes that "weak" members of Team B should be restructured to "cut unnecessary losses". Lucette strongly disagrees, arguing: (1) That Team A underperforms in many key ways as compared to what should be expected from a truly independent operation. (2)That the characterization of "Lone Wolf" behavior is an ideology of independence rather than something which is backed up by Statistics. Lucette arguest that it is a false narrative that discounts the important teamwork that team B has to offer. (3)That Team A is overly aggressive; therefore, the real reason why the AB number is low is because many customers who work with team A first don't come back at all! That is; it's not all because team B can't close these deals: customers don't even come back due to team A's poor attitude. Moreover, this kind of mean-spirited sales strategy will negatively affect our operations in the long term. Caring and warm customer support is very important, even when it concerns shipping freight! (4)That team B, despite being much smaller, contributes significantly to team A's success. Thereby, it is not fair for team A to take credit for what is at least partly team B's contribution to each final sale. (5)That; the CEO was right to acquire Skagit Skies: the two teams complement each other nicely and that further cooperation could yield even better outcomes. (6)However; the hostile attitude is not conducive to this cooperation. Answer these questions in response to Lucette's argument: -Explain which of these assertions can be supported, in your view, using Statistics based on the data given. -For each point that is supported, write how convincing you feel the statistical evidence is for that point. -Note, in your view, which unsupported or weakly supported points should be investigated further. Finally, note any points for which there might be weak or no statistical support but which you believe is true for other reasons. You can cite any reason, includingsomething that you have learned outside of this class. Write your answer in the discussion board. g) Answer this question: Do you agree or disagree with Lucette and the Killer Whale? If you disagree with both of them, you can write your own independent opinion. If you prefer, you can combine your answer to item f and item g. h) Short answer: If you were a traveler flying with WFS146, which Team would you want to receive help from first before making a purchase? You don't need to explain the reason for your choice if you don't want to, but do share your pick with your classmates! "This is the end of the formal assignment. Thank you for your hard work