2021-22 Term 1 1 STAT1203 PROJECT 2: WASHI DUE: WEEK 13 MON 5PM Like many industrialised nations, Japan has been experiencing the issue of population aging to an unprecedented degree. The country’s population fell by a record of over 480,000 in 2020 and the pace of this decline is predicted to accelerate each year until the 2060s. The hardest hit in the population decline is in the rural areas, where young people left for cities like Tokyo and Osaka to look for jobs. As a result, federal and local governments have numerous iju¯1 programs, with monetary incentives up to millions of yen, to attract young people to move to the rural areas. The Adachis are a young couple drawn by these incentives and the opportunity to live a less stressful life. A few years ago, they moved back to a village near where Glenda and John made their deliveries. They run a farm growing organic produce. One of the crops they grow is ko¯zo2, a main ingredient for making washi paper. (1) The long fibres of kozo create incredibly strong and durable papers. This year, they are trying a new kozo variety and they want to know how it compares to the variety they planted in previous years. A sample of washi papers is made using the new variety. The strength of the papers is tested by applying a particular force to each piece and determining whether a tear (T ) is observed: T T¯ T¯ T¯ T¯ T¯ T¯ T¯ T¯ T¯ T Current Test outcomes Let X be the number of tests (including the current one) between two paper tears (So X = 10 here). They approached Glenda and John for an answer. (a) Glenda and John identified a known probability model (distribution) for the data. What is that distribution? What are the assumptions? What is(are) the meaning(s) of the parameter(s) in the model? (b) State E(X) and var(X) for the model in (a). 1relocation 2a mulberry-like shrub 2021-22 Term 1 2 (c) Based on the Adachis’ experience on the variety they planted in previous years, 1 in 6 pieces of paper under test would result in a tear. Combining this information and the observed data, Glenda and John carried out a 5% significance test. Repeat their workings. (d) Glenda and John were a little surprised at the result, so they ask the Adachis to carry out a few more tests. With more tests, including the first observation, the data look like: (x1, x2, ..., x32) 1, 8, 1, 5, 12, 39, 6, 4, 2, 13, 14, 5, 5, 10, 10, 2, 3, 2, 1, 17 10, 3, 11, 3, 4, 17, 8, 10, 23, 1, 6, 4 Glenda and John repeated the analysis by assuming the sample size n is “large”. It is known that the MLE of the parameter in the model is 1 x¯ where x¯ = 1 n ∑n i=1 xi. So they sent Aloysius a message for help. He comes back with an answer. Repeat the works of Glenda, John and Aloysius. Hint: Aloysius assumes, using the old variety papers, in every 6 tests, one would result in a tear. (2) The kozo the Adachis planted are used by the village’s artisans to produce ultra- thin paper for calligraphy and suiboku-ga, or ink wash painting. The paper is extremely delicate and fragile. The main idea behind suiboku-ga is creating a gradient from light to dark. Objects in the background should be a light gray while subjects in the forefront appear darker. Production of the paper involves a lot of trial and error, so that the paper will soak up the ink in the right way. Over time the artisans discovered that the frequency of paper with flaws, S, in a stack of 200 sheets of papers, follow the distribution: j 0 1 2 3 4 5 6 7 8 9 10 P(S = j) 0.098 0.124 0.138 0.208 0.168 0.112 0.056 0.03 0.031 0.027 0.008 (a) Describe and summarise the distribution of S using methods we learned. (b) The artisans wish to investigate whether a new method would reduce the average number of flaws. To do that, they produce some papers using the new method, and record the following number of flaws in 25 stacks of 200 papers: 4, 2, 2, 3, 2, 3, 4, 2, 0, 3, 0, 3, 2, 3, 6, 0, 0, 1, 3, 3, 0, 1, 0, 2, 4. By making reasonable assumptions, carry out a 5% significance test to determine whether the new method reduces the average number of flaws compared to the old method. 2021-22 Term 1 3 (3) One of the initiatives of the papermakers is to bring old paper back to life. The old paper, koshi is boiled and mixed with kozo, and then beaten to produce the material for making new paper. The quality of the end product, Y (on a scale from 0-10), depends on a number of conditions. Among these conditions are the composition of the mixture of ingredients, water, starch, and weather such as temperature, and humidity. Consider four factors X1, X2, X3, X4. The file washi-regression.csv contains data on Y and X1−X4 on a random sample of the papers. Some relevant summary statistics are given in Table 1 below: Table 1: Selected summary statistics for data in washi-regression.csv Variable ∑n i=1 Xi ∑n i=1 X 2 i ∑n i=1 XiYi ∑n i=1 Yi ∑n i=1 Y 2 i Y 516.030 4846.635 X1 889.9066 18389.0375 8451.1212 X2 740.1446 20683.9492 6794.3747 X3 2342.583 147570.414 21559.777 X4 20331 20743753 188465.6 (a) Based on the data, fit simple linear regressions between Y and each of the predictors. From these regressions, draw scatter plots, fitted lines, and residual plots. Based on these, are there any predictor(s) among X1−X4 that you would not used in a linear model describing Y ? Justify your choice(s). (b) Suppose the following conditions are observed: X1 X2 X3 X4 16 18 55 400 2021-22 Term 1 4 Based on the data, for each the remaining predictors in (a), determine whether the relationship between the predictor and Y is statistically significant, and if so estimate Y , with 95% confidence. There are two kinds of interval estimates in regression analysis. Justify your choice of the interval estimate used. (c) In (b), select only one of the two estimates to use for determining whether there is a good chance that with these conditions the papers would turn out to have an average score of at least 9.3? Justify your choice. (4) The neighbourhood surrounding Glenda and John’s host university has many restaurants. John’s favourite is a sushi restaurant. The sushi are handmade and they are all sold at the same rock-bottom price of 140 yen per dish. More importantly, the restaurant only accepts customers dining by themselves, and is extremely popular among university students who are single and want to meet someone. After a period of time, John has become acquainted with the shop assistant, Arimura, who is a third- year student from the famous Doshisha Women’s College, across the street from the restaurant. Arimura told him that the owner complains it is difficult to predict how much to prepare and since sushi cannot keep, the unsold dishes will have to be discarded. He would like to find a solution but he knows nothing but making sushis. John offers to help. Obviously, the number of dishes each customer orders may differ, and the amount of time they spend at the restaurant also varies. Like many small restaurants in Japan, the orders are placed by buying a ticket from a machine, so with some bookkeeping, Arimura can provide John with a record of exactly how long a customer stays (T ) and the number of dishes served (X). A selection of the data is given in project2-question4.csv (a) John knows that with some assumptions, he can use a well known probability distribution for the different observations of X. What is the distribution and what are the assumptions required? (b) He goes back to his old notes to write down the likelihood and log-likelihood functions carefully based on the observations. He can draw one of the two functions to find the maximum likelihood estimate (MLE). Which one should he use and why? For the graph, he chooses a suitable range of “parameter” values, from which, he finds the MLE of the parameter of his model. Repeat his analysis. (c) He was very happy with his work so he sent the results to Glenda. A few minutes later, Glenda replies with a note teasing him that he must be distracted by Arimura; there is a much simpler solution. She explains the simpler solution is achieved using 2021-22 Term 1 5 a property of the distribution he uses. She attaches detailed justification of the method and the workings, which gives exactly the same results as John’s. Repeat Glenda’s work. (5) Lying in the north of the Kyoto Basin is Midorogaike, a deep muddy pond with a circumference of about 1.5 km. It has been rumoured to be a stage for ghost stories among locals, saying, “A taxi driver picked up a female customer at midnight and was asked to head to Midorogaike. When the taxi arrived at the pond, the driver discovered the passenger had disappeared, leaving behind only ...”. So it is a popular spot for night owls and those who enjoy spooky experience. John signed up for a haunted tour to visit the pond and invited Arimura to join him; Glenda had no appetite for that kind of activity so she stayed in her dorm. The tour began just after sunset. Altogether, including the guide, there were 12 on the tour. Thirty minutes into the tour, the tour guide did a head count, and found that John was missing! So a search started to look for John. (a) After another 30 minutes, John has not been found. Arimura calls Glenda: “ What is the chance of finding John?” Using her knowledge in Statistics, Glenda quickly writes down a probability distribution for the data. What is that distribution and what assumptions are used in the distribution? (b) She learned a method that could be used to estimate the “parameter” in her model. She scribbles down some mathematical expressions and draws a figure accompanying the mathematical expressions. Reproduce Glenda’s analysis. (c) To her disappointment, the analysis yields a surprising result. What is that result and what causes that result to appear? (6) The next day, John recounts his experience to his room mate, Sasaki, an Ecology student. He replied, “Oh yes, by the way, I am working on a project about Midorogaike!”. Midorogaike is thought to have been formed during the last ice age. Northern flora and fauna that have survived from that period still grow in various parts of the pond. In the middle of the pond are a number of “floating islands”, formed by decomposing organic materials. These islands appear depending on the season, surface in the summer and increase in size, up to 1/3 of the total surface area of Midorogaike. In the winter they become flooded and are submerged. These floating islands are extremely important for the insects, plants, and birds that depend on them. 2021-22 Term 1 6 Sasaki first identified 8 different islands at the end of last summer. Then starting on the day of the following Spring Solstice, he returned to observe whether any of the islands had re-emerged. He stopped his experiment after 31 days. By then only six of the islands had re- emerged. The following are the time (T ) it took various floating islands to re-emerged (counted from the day of the Spring Solstice): 12,9,25,11,11,8,31,31 Sasaki wants to estimate the average time for the whole colony of islands to re-emerged. He knows John has taken a course in Statistics so he approached John for advice. (a) Using his knowledge, John easily comes up with a probability distribution for the data. What is that distribution and what assumptions are used in the distribution? (b) John argues, since Sasaki is interested on islands that re-emerged, then the data about the two that did not re-emerge can be discarded. Using the remaining data, he derives the solution carefully and draws a figure accompanying the derivations. Reproduce John’s analysis. (c) John is pleased with his work so at dinner, he shares his work with Glenda. Glenda took a glance and says: “I think you are doubly distracted by Arimura and your experience last night”. She then works out an alternate solution and draws another figure. Repeat Glenda’s work. (d) Which one, John or Glenda, or are both correct? Explain. 

2021-22 Term 1 1 STAT1203 PROJECT 2: WASHI DUE: WEEK 13 MON 5PM Like many industrialised nations, Japan has been experiencing the issue of population aging to an unprecedented degree. The country’s population fell by a record of over 480,000 in 2020 and the pace of this decline is predicted to accelerate each year until the 2060s. The hardest hit in the population decline is in the rural areas, where young people left for cities like Tokyo and Osaka to look for jobs. As a result, federal and local governments have numerous iju¯1 programs, with monetary incentives up to millions of yen, to attract young people to move to the rural areas. The Adachis are a young couple drawn by these incentives and the opportunity to live a less stressful life. A few years ago, they moved back to a village near where Glenda and John made their deliveries. They run a farm growing organic produce. One of the crops they grow is ko¯zo2, a main ingredient for making washi paper. (1) The long fibres of kozo create incredibly strong and durable papers. This year, they are trying a new kozo variety and they want to know how it compares to the variety they planted in previous years. A sample of washi papers is made using the new variety. The strength of the papers is tested by applying a particular force to each piece and determining whether a tear (T ) is observed: T T¯ T¯ T¯ T¯ T¯ T¯ T¯ T¯ T¯ T Current Test outcomes Let X be the number of tests (including the current one) between two paper tears (So X = 10 here). They approached Glenda and John for an answer. (a) Glenda and John identified a known probability model (distribution) for the data. What is that distribution? What are the assumptions? What is(are) the meaning(s) of the parameter(s) in the model? (b) State E(X) and var(X) for the model in (a). 1relocation 2a mulberry-like shrub 2021-22 Term 1 2 (c) Based on the Adachis’ experience on the variety they planted in previous years, 1 in 6 pieces of paper under test would result in a tear. Combining this information and the observed data, Glenda and John carried out a 5% significance test. Repeat their workings. (d) Glenda and John were a little surprised at the result, so they ask the Adachis to carry out a few more tests. With more tests, including the first observation, the data look like: (x1, x2, ..., x32) 1, 8, 1, 5, 12, 39, 6, 4, 2, 13, 14, 5, 5, 10, 10, 2, 3, 2, 1, 17 10, 3, 11, 3, 4, 17, 8, 10, 23, 1, 6, 4 Glenda and John repeated the analysis by assuming the sample size n is “large”. It is known that the MLE of the parameter in the model is 1 x¯ where x¯ = 1 n ∑n i=1 xi. So they sent Aloysius a message for help. He comes back with an answer. Repeat the works of Glenda, John and Aloysius. Hint: Aloysius assumes, using the old variety papers, in every 6 tests, one would result in a tear. (2) The kozo the Adachis planted are used by the village’s artisans to produce ultra- thin paper for calligraphy and suiboku-ga, or ink wash painting. The paper is extremely delicate and fragile. The main idea behind suiboku-ga is creating a gradient from light to dark. Objects in the background should be a light gray while subjects in the forefront appear darker. Production of the paper involves a lot of trial and error, so that the paper will soak up the ink in the right way. Over time the artisans discovered that the frequency of paper with flaws, S, in a stack of 200 sheets of papers, follow the distribution: j 0 1 2 3 4 5 6 7 8 9 10 P(S = j) 0.098 0.124 0.138 0.208 0.168 0.112 0.056 0.03 0.031 0.027 0.008 (a) Describe and summarise the distribution of S using methods we learned. (b) The artisans wish to investigate whether a new method would reduce the average number of flaws. To do that, they produce some papers using the new method, and record the following number of flaws in 25 stacks of 200 papers: 4, 2, 2, 3, 2, 3, 4, 2, 0, 3, 0, 3, 2, 3, 6, 0, 0, 1, 3, 3, 0, 1, 0, 2, 4. By making reasonable assumptions, carry out a 5% significance test to determine whether the new method reduces the average number of flaws compared to the old method. 2021-22 Term 1 3 (3) One of the initiatives of the papermakers is to bring old paper back to life. The old paper, koshi is boiled and mixed with kozo, and then beaten to produce the material for making new paper. The quality of the end product, Y (on a scale from 0-10), depends on a number of conditions. Among these conditions are the composition of the mixture of ingredients, water, starch, and weather such as temperature, and humidity. Consider four factors X1, X2, X3, X4. The file washi-regression.csv contains data on Y and X1−X4 on a random sample of the papers. Some relevant summary statistics are given in Table 1 below: Table 1: Selected summary statistics for data in washi-regression.csv Variable ∑n i=1 Xi ∑n i=1 X 2 i ∑n i=1 XiYi ∑n i=1 Yi ∑n i=1 Y 2 i Y 516.030 4846.635 X1 889.9066 18389.0375 8451.1212 X2 740.1446 20683.9492 6794.3747 X3 2342.583 147570.414 21559.777 X4 20331 20743753 188465.6 (a) Based on the data, fit simple linear regressions between Y and each of the predictors. From these regressions, draw scatter plots, fitted lines, and residual plots. Based on these, are there any predictor(s) among X1−X4 that you would not used in a linear model describing Y ? Justify your choice(s). (b) Suppose the following conditions are observed: X1 X2 X3 X4 16 18 55 400 2021-22 Term 1 4 Based on the data, for each the remaining predictors in (a), determine whether the relationship between the predictor and Y is statistically significant, and if so estimate Y , with 95% confidence. There are two kinds of interval estimates in regression analysis. Justify your choice of the interval estimate used. (c) In (b), select only one of the two estimates to use for determining whether there is a good chance that with these conditions the papers would turn out to have an average score of at least 9.3? Justify your choice. (4) The neighbourhood surrounding Glenda and John’s host university has many restaurants. John’s favourite is a sushi restaurant. The sushi are handmade and they are all sold at the same rock-bottom price of 140 yen per dish. More importantly, the restaurant only accepts customers dining by themselves, and is extremely popular among university students who are single and want to meet someone. After a period of time, John has become acquainted with the shop assistant, Arimura, who is a third- year student from the famous Doshisha Women’s College, across the street from the restaurant. Arimura told him that the owner complains it is difficult to predict how much to prepare and since sushi cannot keep, the unsold dishes will have to be discarded. He would like to find a solution but he knows nothing but making sushis. John offers to help. Obviously, the number of dishes each customer orders may differ, and the amount of time they spend at the restaurant also varies. Like many small restaurants in Japan, the orders are placed by buying a ticket from a machine, so with some bookkeeping, Arimura can provide John with a record of exactly how long a customer stays (T ) and the number of dishes served (X). A selection of the data is given in project2-question4.csv (a) John knows that with some assumptions, he can use a well known probability distribution for the different observations of X. What is the distribution and what are the assumptions required? (b) He goes back to his old notes to write down the likelihood and log-likelihood functions carefully based on the observations. He can draw one of the two functions to find the maximum likelihood estimate (MLE). Which one should he use and why? For the graph, he chooses a suitable range of “parameter” values, from which, he finds the MLE of the parameter of his model. Repeat his analysis. (c) He was very happy with his work so he sent the results to Glenda. A few minutes later, Glenda replies with a note teasing him that he must be distracted by Arimura; there is a much simpler solution. She explains the simpler solution is achieved using 2021-22 Term 1 5 a property of the distribution he uses. She attaches detailed justification of the method and the workings, which gives exactly the same results as John’s. Repeat Glenda’s work. (5) Lying in the north of the Kyoto Basin is Midorogaike, a deep muddy pond with a circumference of about 1.5 km. It has been rumoured to be a stage for ghost stories among locals, saying, “A taxi driver picked up a female customer at midnight and was asked to head to Midorogaike. When the taxi arrived at the pond, the driver discovered the passenger had disappeared, leaving behind only ...”. So it is a popular spot for night owls and those who enjoy spooky experience. John signed up for a haunted tour to visit the pond and invited Arimura to join him; Glenda had no appetite for that kind of activity so she stayed in her dorm. The tour began just after sunset. Altogether, including the guide, there were 12 on the tour. Thirty minutes into the tour, the tour guide did a head count, and found that John was missing! So a search started to look for John. (a) After another 30 minutes, John has not been found. Arimura calls Glenda: “ What is the chance of finding John?” Using her knowledge in Statistics, Glenda quickly writes down a probability distribution for the data. What is that distribution and what assumptions are used in the distribution? (b) She learned a method that could be used to estimate the “parameter” in her model. She scribbles down some mathematical expressions and draws a figure accompanying the mathematical expressions. Reproduce Glenda’s analysis. (c) To her disappointment, the analysis yields a surprising result. What is that result and what causes that result to appear? (6) The next day, John recounts his experience to his room mate, Sasaki, an Ecology student. He replied, “Oh yes, by the way, I am working on a project about Midorogaike!”. Midorogaike is thought to have been formed during the last ice age. Northern flora and fauna that have survived from that period still grow in various parts of the pond. In the middle of the pond are a number of “floating islands”, formed by decomposing organic materials. These islands appear depending on the season, surface in the summer and increase in size, up to 1/3 of the total surface area of Midorogaike. In the winter they become flooded and are submerged. These floating islands are extremely important for the insects, plants, and birds that depend on them. 2021-22 Term 1 6 Sasaki first identified 8 different islands at the end of last summer. Then starting on the day of the following Spring Solstice, he returned to observe whether any of the islands had re-emerged. He stopped his experiment after 31 days. By then only six of the islands had re- emerged. The following are the time (T ) it took various floating islands to re-emerged (counted from the day of the Spring Solstice): 12,9,25,11,11,8,31,31 Sasaki wants to estimate the average time for the whole colony of islands to re-emerged. He knows John has taken a course in Statistics so he approached John for advice. (a) Using his knowledge, John easily comes up with a probability distribution for the data. What is that distribution and what assumptions are used in the distribution? (b) John argues, since Sasaki is interested on islands that re-emerged, then the data about the two that did not re-emerge can be discarded. Using the remaining data, he derives the solution carefully and draws a figure accompanying the derivations. Reproduce John’s analysis. (c) John is pleased with his work so at dinner, he shares his work with Glenda. Glenda took a glance and says: “I think you are doubly distracted by Arimura and your experience last night”. She then works out an alternate solution and draws another figure. Repeat Glenda’s work. (d) Which one, John or Glenda, or are both correct? Explain.