Part 1. Monopoly
For details about Monopoly, see
STA 9708 LN10.B InClass Monopoly Insurance
In the following questions, you will be rolling one four-sided fair die and one six-sided fair die. Thus, the largest sum you can get is 10.
(i) Explain in your own words how the probability distribution of the sum of the roll of a fair four-sided die and fair six-sided die would be computed. Write this up as if you are explaining it to a colleague not in this class.
(ii) Select your own starting position on the Monopoly board that is within reach of a Community Chest or Chance square. Do not start on Pacific Ave. Place houses on one color-set of properties and hotels on a another color-set of properties within reach. The rents you must pay is shown on the Deed cards. Assume you do not own any of the properties in that neighborhood. For landing on Chance, designate one card in advance as the outcome; likewise for Community Chest. State (in sentences) the choices you have made. You will encounter additional choices to be made, such as the choice of the number of houses: make those additional choices as you see fit – and state them.
Include a paragraph in which you summarize the layout you have created.
(iii) Construct the payout distribution according to the value rolled. Display that in four columns: (a) the value rolled – values 2 to 10, (b) the name of the outcome square (“Park Place,” “Go”), (c) probability of the outcome, and (d) the payout (or “rent”).
(iv) Show the computation of the expected value and the variance of the payout from the payout distribution you created in the previous question.
(v) Suppose you wanted to buy insurance for the payout of the above next roll. Explain how your computed expected value is related to the pricing of that insurance.
(vi) In terms of pricing the insurance contract, the insurer needs to consider the list of possible payouts, not just the expected value (and/or the variance). Why? It might help you to imagine that the amounts involved are large, say, in the thousands of dollars. For example, landing on Boardwalk with a hotel would cost $2,000,000 instead of $2,000.
(vii) Recall that in question (e) of LN10.B In-Class Monopoly Exercise you were asked to compute the probability of reaching Boardwalk by the following path: first, roll a five to land on Chance and then out of the deck of 16 cards randomly select “Go to Boardwalk.” I will call that an example of a compound path. In this question, you are asked to define your own compound path, from the same starting position you chose, above. Your compound path will begin with a die roll that lands either on Chance or on Community Chest. If both are within reach, pick one, only. From the 16 cards in that deck, decide on one card that sends you to some other location as the second leg of the compound path. This could “Nearest railroad,” “Go,” or any other card that compels you to move. The chances of picking that card are obviously 1/16.
Now, compute the probability of reaching the destination on your chosen card by your compound path. Then, try to explain the reasoning behind the computation.
Part 2. Regression
Refer to the data, scatterplot, and regression analysis in the Excel file STA 9710 Final Project Q4 Excel XXXXXXXXXXRaisinets. The Count variable includes fractional values because some bags contain one or more small beads of pure chocolate with no raisin.
(i) For your choice of Count, compute an approximate 95% prediction interval. Show your work in your Word document.
(ii) Explain what it is that you hope to capture with that interval.
(iii) Add that prediction interval to the regression scatterplot I provided. Insert your name to the title of the plot. Show an image of that modified scatterplot.
(iv) Does the interval you added make sense relative to the data? Explain.
(v) A mischievous child is given two additional bags of Raisinets from that same production run. One bag contains Count=48 raisinets and the other Count=43. She opens both bags. From the bag with 48, she randomly selects and eats five raisinets, so both bags now contain 43 raisinets.
Your job is to decide from which bag she ate five. You weigh the contents of both bags: one has net weight of 43.2 grams; the other of 44.3 grams.
(a) Which of those bags do you think originally contained 48 Raisinets? Explain your reasoning.
(b) Estimate the original net weight of the bag that held 48.