Problem #1 Background \"The Difcult Customer" (ls Paul's behavior \"Normal\"] We have a really difficult customer. A customer named Paul has a very long beard and he insists on traveling with his blue ox. His ox is sometimes hard to work with but he travels with us and hes recently been booking tours in British Columbia and Whatcom County. He absolutely loves nature and the forests! We want to make our tours work for Paul: and we think that he does exceptionally well with our tour leader and manager, Lucette. (Rumors are that Paul is secretly in love with Lucette._ and that Lucette may also be in love with Paul.) To investigate this. we looked at statistics for different tours that Paul went on. But there's a problem: average customer satisfaction scores are different for all of our tours! Also. when Paul went on Lucette's tour in the late Winter of 2022. the nationwide airline meltdown made the experience tern'ble overall. How can we check to see if Paul really likes going on tours with Lucette? Comment: These names are a reference to American and Canadian legends of Paul Bunyan. Comment2: Bebe is the name of Paul's Blue 0):. En Problem 1 Paul went on the following three tours and reported the following customer satisfaction numbers. The survey was conducted on a likert scale: 1 means \"Totally dissatisfied": 2 means "dissatised\" 3 means neutral 4 means :liked the tour" and 5 means "strongly liked the tour". Here are the results: n=25 customers, Whistler, British Columbia, Canadian Cross-Country Skiing with Mountaineer and Ornithologist Sindbad. Supervisor: Ali Baba aka (also known as) "Papa Ali" Average Satisfaction Score: 3.8, Standard Deviation=0.2 Paul's Score: 4.1 Comment: "Many people felt it was too cold, but I didn't mind it too much. I personally loved the beautiful snow-capped mountains and the forest! Ali even helped me set Bebe up with Ski's! In the valley, a very big bird came after Bebe but we managed to drive it away." n=20 customers, Blaine Salish Sea Seaplane Flight with Captain Sita Average Satisfaction Score: 4.4, Standard Deviation 0.4 Paul's Score 4.1 Comment: "This was my second tour, following the warm recommendation of Sindbad, to see the beautiful Salish Sea. Captain Sita was very nice to us, and the plane, the "Whatcom Bigfoot". The plane had a neat picture of Bigfoot wearing sunglasses, carrying Mount Baker, and flying all at the same time! I was surprised to learn that it is the most advanced Seaplane in the entire Pacific Northwest! The amazing plane, even with its experienced pilot, broke while trying to pick up Bebe. I should have known better than to insist on carrying Bebe with us! Luckily- in this modern, 21st century story- Captain Sita contacted her employee, lead engineer Rama, who re-stringed the internal cables -by hand- and fixed the airplane! This was a very nice tour! Rama and Sita truly make great partners! That said, I wish that Bebe could have joined me! " n=30 customers, Bellingham Tour with L.D. Kensack (Lucette D. Kensack) Average Satisfaction Score 1.7, Standard Deviation 0.3 Paul's Score: 2.6 Comment: "Awful! How could we be made to wait for that long? We had a good tour guide, however. " a) What was Paul's favorite tour? b) Calculate the Z-Scores for Paul's rating on all three tours. c) Which tour has Paul's highest and lowest z-scores? These scores represent tours where Paul's score deviates the most from the average person's satisfaction score. d) On the discussion board, write down an imagined Satisfaction score and Z-score for yourself, a friend, or someone that you imagine. Assume that you or the person of your imagination was on one of the tours above together with Paul Bunyan. Then; also on the discussion board explain the reasoning behind your z-score. Remember that z-scores can be positive or negative and they are comparisons to the rest of the (Customer) Population given a "Normal" distribution. Post your solution to the discussion board. #Background: Airplane Safety and Average Weights: Problems 2 through 4 Air Midwest Flight 5481, which seated 21 passengers, tragically crashed in 2003 due to overloading and inappropriate maintenance. Results from the investigation of the crash by the United States National Transportation Safety Board (NTSB) led to the revision of the averageweight used in airplane weighting calculations from 170 pounds to today's values. This accident furthermore raised a debate within the US aviation industry about the practice of using average weights instead of real weights to calculate loading for small passenger aircraft. The Statistics of means, as we have studied them. is at the heart of this debate. in fact, many experts now call for using exact weights rather than average weights for small-sized aircraft. WFS-MB is exactly the type of airline which may y such smaller planes! These problems will explore the Mathematics of why. The consequences ofthe Central Limit Theorem play a central role in this understanding. Background for Problem 2 We are a small company, out we dream big! For many years we at WFS-M have dreamed of owning our own, Edseat jet airplane. We may nally have this opportunity! As part of our preparations for making this dream a reality, carry out the following calculations relating to airplane safety: Em Problem 2 Our Company Dream Suppose a large, jet-airplane seats 200 passengers. Assume that the airplane is completely full due to overbooking and that the passengers are chosen randomly from the customer shepopulation. indeed. when the plane is overcooked, it is the Corporate policy that passengers be randomly selected to be "reseated" to another ight. Assume that the average weight of passengers in the customer population is 195 pounds in the Winter with a standard deviation of 30 pounds. Note that the number 195 pounds is drawn from a study of the total population of the United States. aJCalculate the expected total weight of the passengers. bJApplying the Central Limit Theorem; determine the standard error of the \"mean passenger weight\" tor this airplane. c}Determine the probability that the average weight of the passengers on any given ight is 5% or more greater than the average weight in the customer population. You should expect a small number. Express your answer as a percent. d} Post your answers to problems 1a, 1b, and 1: to the Discussion Board. You can also write \"Help!\" or "Can someone help me with. or alternatively you may ask another specic question if you're not sure. To make this discussion more authentic and similar to a project-based discussion with coworkers at an actual business, i plan not to participate in your discussion ct problem #2. i will only step in ifwhat I see is very, very wrong; however, I'm rather condent in your classmates' ability to work things out in a group discussion. Good luck! Background for Problem 3 Now, let's compare the big, new airplane with a much smaller airplane that seats 20 passengers. Note that this is about the same Size as Air Midwest Flight 5481, which had 21 people on board when it crashed. '3'\". Problem 3 Our Biggest Airplane The large businessjet owned by WFS-146 seats 20 passengers. Assume that the airplane is completer full and that the passengers are chosen randomly from the customer population. Assume that the average weight of passengers in the customer population is 195 pounds in the Winter and has a standard deviation of 30 pounds. a]:Calculate the expected total weight of the passengers. b]: Applying the Central Limit Theorem: determine the standard error of the \"mean passenger weight\" for this airplane. c} Determine the probability that the average weight of the passengers on any given ight is 5% or more greater than the average weight in the customer population. Express your answer as a percent. d]: Compare your answer to 2c and 3c. (Note.- This is a consequence of the Central Limit Theorem). Does this result surprise you? Write down your reaction in the discussion board. Background for Problem 4 Tour companies often use small airplanes tor individualized tour packages. Let's imagine that we have a very special package to deliver; let's imagine that Paul and Lucette have since fallen in love! Lucette's co-worlter, Captain Site. has decided to retum an old favor and give Lucette a free tour on her seaplane. There'siust one problem: Paul weighs 400 pounds. Let's take a look at some calculations related to passenger weight and safety. These problems relate to why Captain Sita cannot make random guesses and must carefully consider the actual weight of passengers on her airplane! Small airplanes do not use average weight calculations: they use exact weights! I hope this problem demonstrates what we should NOT do: that is, what happens to the \"law of large numbers" when the number is small. Specically. we will look at the case n=4.. E Problem 4 Small Airplanes Mark individually!) Suppose a small, propellered seaplane own by Captain Sita seats 4 passengers. Assume that the airplane is completely full and that the passengers are chosen randomly from the customer population. Assume that the average weight of passengers in the customer population is 195 pounds in the winter with a standard deviation of 30 pounds. Note that if the airplane is overweight some of the passengers' luggage might be left behind. a]: Deten'nine the probability that the actual average weight of the passengers is between 5 and 15 percent greater than the average weight in the customer population. Be sure to show your work! b]: Determine the probability that the actual average weight of the passengers is between 15 and 25 percent greater than the average weight in the customer population. Be sure to show your work! c} Suppose that Paul weighs 400 pounds and the other passengers, including Lucette, weigh 550 pounds total. First, calculate the average weight of the four passengers, including Paul. Then, determine the percentile of this average. Be sure to show your work'l d]: Compare your answers in problem 4 to problem 3 and 2. Are you surprised by the result? Comment on why it may be safe for a large airplane (n=200) to use average weights in its calculations but not for smaller airplanes (n=20, n=4]i to do so. Cite and discuss the numbers that you have calculated in this assignment in your response. Post your response to the discussion board so that students who are not attempting the 3 problem can learn from your results and debate the importance or your calculation. That said; do not answer questions about how you got your solutions. I would like every student attempting the 33 problem to come to a solution in their own way. ~~Thank you so much for your dedication to learning Statistics! -Kourosh Ghaderim ~~This is the end of the assignment! l hope that you will all enjoy it and find it to he informativel