2021-22 Term 1 1 STAT1203 PROJECT 1: LETTER LUNCH DUE: WEEK 11 MON 5PM For the past 18 months, Glenda has been admiring the Pacita Abad artworks, made out of handmade paper, on display at the school. So the course description corresponds with Glenda’s expectations of an overseas exchange experience. On this exchange, she also asked her classmate John to join her. POLITICAL THOUGHT This course explores the long history of paper manufacturing and the transfer of East Asian papermaking know how to Europe, especially to Great Britain. The rich history of Japanese paper culture offers a classic example of the fundamentals of the political economy of papermaking, where theoretical knowledge is integrated with technological skills. 1 (1) The exchange will be at a university in the city of Kyoto, Japan. To fund the overseas trip, Glenda and John need to set up a booth to raise money. They plan to sell letter stationery made from washi. They set up a booth on campus. Let N be the number of visitors in an hour. They estimate that on average 15 people would visit the booth every hour with a probability of p = 0.2 that a visitor would purchase a set. Let Y be the number of sets they could sell in an hour, assuming the booth is open for 6 hours a day and that every visitor acts independently of others. (a) Find a suitable distribution for N . State all assumptions. (b) Are N and Y related? Write down expressions for the joint PDF of N and Y , and marginal PDF of Y (c) Find a suitable distribution for Y and then combine with (a) and (b) to write down an expression for P(Y = k). Do not evaluate this expression. (d) Find E(Y ) and var(Y ). (e) Write an expression for P(Y ≥ k), and comment. (f) Let W be the total number of sets sold in 10 days, sketch the distribution of W and comment on the shape of the distribution. (g) Based on (f), find an approximation of P(W > 576). Compare this to (e) and comment. 2021-22 Term 1 2 (2) The cost of each set is 12 dollars from the supplier. The sets are also available in cartons of 6 dozens, at a discounted price of 10 dollars per set. (a) They rationalise that it would be unlikely to sell more than 624 sets. What is their justification? (b) They want to sell each set at 20 dollars. Which of the following two strategies, A or B, gives a higher expected profit? A: Order 8 cartons for a total of 576 sets B: Order 8 cartons and another 48 individual sets for a total of 624 sets (3) A reason they chose the course is the hands on experience in papermaking that the course offers. Forty percent of the classes would be accompanied by workshops(s) on papermaking; with probability 3/4 there would be one workshop and probability 1/4 there would be two. The scheduled class time runs between 1130am - 1pm each day of the week. But the actual length varies. It depends on whether there are workshops and if there are, the number of workshops (X). The number of hours (T ) the class would run beyond (or before, if T < 0) 1pm has following PDF: f(t) = { k[1 + (X − 1)t], −1 < t < 1, 0 ≤ X ≤ 2 0, otherwise . (a) Find the value of k. (b) Not far from the university is a ramen shop called Yorokobi. It sells ramen as well as take out bentos. The bentos are cheap and tasty. Glenda and John also like to read the washi paper note attached to each bento, that is handwritten by the shop girl. However, the bentos are popular, so they must reach the shop by 2pm; 2021-22 Term 1 3 otherwise the bentos might be sold out. Assuming it takes 20 minutes to reach the shop from the university, find the probability that the bentos would definitely be available when they arrive at the shop. (c) The number of bentos made and sold each day depends on the supply and demand on that particular day. Assume that the actual time (in minutes) for the bentos to be sold out, measured from 2pm, follows a Uniform(−20, 20) distribution, write an expression to find the chance they would be able to get their bentos (THERE IS NO NEED TO EVALUATE THIS EXPRESSION). (4) Glenda and John often have to wait for the bentos after the orders are placed. During the wait, they usually wander in the neighbourhood. Next to the ramen shop is a community center. Outside the community center is a notice board where community news are posted. One day while waiting, they notice a call for volunteers to deliver letter lunches to the elderly. It says: “Letter lunch is an activity which delivers letters and lunches with health exercises, to elderly who live alone or who do not have much opportunity to go out due to the pandemic.” Each bento box is prepared with a variety of food, eg., eggplant and pepper meat miso, new potato mustard mayonnaise salad, curry-flavoured boiled eggs, etc.. The community center wants to promote a healthy diet for the elderly, so the number (N) of meat-based items is limited to no more than two in a bento. When the lunch is finished, the elderly would call the organiser to pick the boxes up, and place their next orders. Along with their next orders, the community center also encourages the elderly to return a note. A note either contains a praise of how delicious the food is (S = 2) or no mention of the food (S = 1). Sometimes, no note is found (S = 0). Over time, the community center established the following information: P(S = 0|N = 0) = 0, P(S = 1|N = 0) = 0.3; P(S = 0|N = 1) = 0.3, P(S = 1|N = 1) = 0.3; P(S = 0|N = 2) = 0.5, P(S = 1|N = 2) = 0.4. For the sake of organisation and fairness to all, on any given day, the same type of bentos are made. The menu rotates so that over time, N = 0 in 50% of the days, N = 1 in 30% of the days and N = 2 for the remaining days. (a) Based on the given information, draw a probability tree and use it to construct the joint PDF between S and N . 2021-22 Term 1 4 (b) An elderly returns a bento with a note praising the food. By using the following methods: (i) probability tree (ii) joint PDF table (iii) Bayes Theorem, find the probability that the bento had one meat item. (c) Two elderly who live in different parts of the neighbourhood order these bentos every day. Let S1 and S2 be defined as S, for the the two elderly. Are S1, S2 independent? Justify. (d) Suppose S1, S2 are independent given N , i.e., P(S1, S2|N) = P(S1|N)P(S2|N), this situation is called conditional independence. Find the probability that the bento had 1 meat item if S1 = S2 = 2. (5) The community center also organises similar activities in remote areas of the prefecture. Since it would be infeasible to bring bentos to remote areas, the center has called upon villagers from nearby villages to contribute to the making of the letter lunch bentos. The center then activates volunteers to deliver the bentos to the elderly in these remote areas. Since their weekends are free, Glenda and John decided to sign up as volunteers. Their first assignment is located in an area about 1.5 hours from the city. They are told that villagers who have made bentos would place their bentos outside the front door of their house. There are two nearby villages, A and B; there are 198 families in Village A and 222 families in Village B. Glenda and John decide to split their effort. Their goal is to collect 15 bentos altogether, and then deliver them to their assigned locations. They would go from house to house, and look for signs that a villager has placed a bento outside the front door. Some of the houses are near to each other, but some are quite far apart. After a while, Glenda called John: “I looked at 25 houses and found one bento”. John is a little luckier, he has found 2 bentos after passing by 30 houses. (a) Make reasonable assumptions, estimate the probability a house in village A would have a bento at the front door. (b) Would the answer to (a) be different had Glenda reported: “I need to go around 25 houses to find the first bento”? Explain why or why not. There is no need to come up with an estimate of the probability this way. 2021-22 Term 1 5 (c) Find two ways to estimate the probability that there are more than 10 houses in Village A with a bento at their front door. (d) Make reasonable assumptions, estimate the probability a house in village B would have a bento at the front door. (e) Estimate the probability that they would be able to collect 15 bentos altogether. Once again, there are two ways to solve this problem. Write down both ways but solve the problem using one. Compare and contrast between these two ways. (6) Before they left for the trip, they wanted to estimate the times they needed to make the deliveries. The houses where the elderly resided in are scattered across the remote areas, some close to the villages, but some further away. The community center did ask the elderly to call in after they received their letter lunch, so there is a good record of the times it took to make these deliveries. (a) Glenda and John browsed through one of these records. Twenty random samples of n = 6 observations of the time it took to make a delivery from previous activities are stored in the file letterlunch-question6a.csv. Based on the data: (i) find the sample mean, X¯, of each sample (ii) calculate the variance of the statistic in (i) based on the 20 random samples (iii) justify that the variance in (ii) is an estimate of the sampling variation of X¯. Thereby, estimate SE(X¯) (iv) in (i)-(iii), we are able to understand the meaning of the sampling variation and standard error of a statistic through repeated sampling (20 samples). In practice, repeated sampling is not possible. Use any of the 20 samples, illustrate another way in which SE(X¯) can be estimated. Compare and contrast between (iv) and (iii) (b) Moving on, they saw another set of, more recent, records. They noticed that the pattern of data was very different from that in (a). The attached file letterlunch-question6b.csv contains 20 random samples of n = 12 observations from these more recent deliveries. They were puzzled, so they asked the community center for the reason. The woman who works in the community center said that, up 2021-22 Term 1 6 until 6 months ago, the remote areas had only been occupied by the locals. Recently, a German architect called Karl Bengs, who moved to Japan in 1993, had completed reviving a number of derelict kominkas1. As a result, there has been an influx of retirees from the cities. However, these kominkas are in locations that are further away from the other houses. Use the new data: (i) find the sample mean, X¯, of each sample (ii) calculate the variance of the statistic in (i) based on the 20 random samples and thereby estimate SE(X¯) (c) Select one sample from each of letterlunch-question6a.csv and letterlunch-question6b.csv (i) estimate the population variance (standard deviation) of X in (a) and (b) and then articulate why these variances are related to the performance of X¯ as an estimator in (a) and (b) (ii) the relevance of the sample size, n, in X¯ as an estimator 1Literally means “old folk-houses”