Archimedes :: essays papers

Archimedes Few certain details remain about the life of antiquit^s greatest mathematician, Archimedes. We know he was born in 287 B.C. around Syracuse from a report about 1400 years after the fact. Archimedes tells about his father, Pheidias, in his book The Sandreckoner. Pheidias was an astronomer, who was famous for being the author of a treatise on the diameters of the sun and the moon. Historians speculate that Pheidias^ profession explains why Archimedes chose his career. Some scholars have characterized Archimedes as an aristocrat who actively participated in the Syracusan court and may have been related to the ruler of Syracuse, King Hieron II. We also know Archimedes died in 212 B.C. at the age of 75 in Syracuse. It is said that he was killed by a Roman soldier, who was offended by Achimedes, while the Romans seized Syracuse. Archimedes had a wide variety of interests, which included encompassing statics, hydrostatics, optics, astronomy, engineering, geometry, and arithmetic. Archimedes had more stories passed down through history about his clever inventions than his mathematical theorems. This is believed to be so because the average mind of that period would have no interest in the Archimedean spiral, but would pay attention to an invention that could move the earth. Archimedes^? most famous story is attributed to a Roman architect under Emperor Augustus, named Vitruvius. Vitruvius asked Archimedes to devise some way to test the weight of a gold wreath. Archimedes was unsuccessful until one day as he entered a full bath, he noticed that the deeper he submerged into the tub, the more water flowed out of the tub. This made him realize that the amount of water that flowed out of the tub was equal to the volume of the object being submerged. Therefore by putting the wreath into the water, he could tell by the rise in water level the volume of the wreath, despite its irregular shape. This discovery marked the Law of Hydrostatics, which states that a body immersed in fluid loses weight equal to the weight of the amount of fluid it displaces. There are three main mechanical inventions credited to Archimedes. The first one is the Archimedean screw which supposedly could serve as a water pump. The second invention was the compound pulley. The third invention was the way of finding the volume of something by displacement as demonstrated in the story above. Most historians would agree that more important than his great mechanical inventions were his mathematical discoveries. The mathematical works that have been presented to us by Archimedes could be classified into three groups. The first group consists of works that have as their major objective the proof of Archimedes :: essays papers Archimedes Few certain details remain about the life of antiquit^s greatest mathematician, Archimedes. We know he was born in 287 B.C. around Syracuse from a report about 1400 years after the fact. Archimedes tells about his father, Pheidias, in his book The Sandreckoner. Pheidias was an astronomer, who was famous for being the author of a treatise on the diameters of the sun and the moon. Historians speculate that Pheidias^ profession explains why Archimedes chose his career. Some scholars have characterized Archimedes as an aristocrat who actively participated in the Syracusan court and may have been related to the ruler of Syracuse, King Hieron II. We also know Archimedes died in 212 B.C. at the age of 75 in Syracuse. It is said that he was killed by a Roman soldier, who was offended by Achimedes, while the Romans seized Syracuse. Archimedes had a wide variety of interests, which included encompassing statics, hydrostatics, optics, astronomy, engineering, geometry, and arithmetic. Archimedes had more stories passed down through history about his clever inventions than his mathematical theorems. This is believed to be so because the average mind of that period would have no interest in the Archimedean spiral, but would pay attention to an invention that could move the earth. Archimedes^? most famous story is attributed to a Roman architect under Emperor Augustus, named Vitruvius. Vitruvius asked Archimedes to devise some way to test the weight of a gold wreath. Archimedes was unsuccessful until one day as he entered a full bath, he noticed that the deeper he submerged into the tub, the more water flowed out of the tub. This made him realize that the amount of water that flowed out of the tub was equal to the volume of the object being submerged. Therefore by putting the wreath into the water, he could tell by the rise in water level the volume of the wreath, despite its irregular shape. This discovery marked the Law of Hydrostatics, which states that a body immersed in fluid loses weight equal to the weight of the amount of fluid it displaces. There are three main mechanical inventions credited to Archimedes. The first one is the Archimedean screw which supposedly could serve as a water pump. The second invention was the compound pulley. The third invention was the way of finding the volume of something by displacement as demonstrated in the story above. Most historians would agree that more important than his great mechanical inventions were his mathematical discoveries. The mathematical works that have been presented to us by Archimedes could be classified into three groups. The first group consists of works that have as their major objective the proof of

Alfred Tennyson And His Work :: essays research papers fc

Alfred Tennyson and His Work Alfred Tennyson was born on August 6th, 1809, at Somersby, Lincolnshire, fourth of twelve children of George and Elizabeth Tennyson. Tennyson, said to be the best poet of the Victorian era and his poetry will be discussed in this essay. Tennyson had a lifelong fear of mental illness, because several men in his family had a mild form of epilepsy, which then was thought of as a shameful disease. His father and brother Arthur made their epilepsy worse by excessive drinking. His brother Edward had to be put in a mental institution after 1833, and he spent a few weeks himself under doctor's care in 1843. In the late twenties his father's physical and mental condition got worse, and he became paranoid, abusive, and violent. In 1827 Tennyson escaped his troubled home when he followed his two older brothers to Trinity College, Cambridge, where his teacher was William Whewell. Because each of them had won university prizes for poetry the Tennyson brothers became well known at Cambridge. In 1829 The Apostles, an undergraduate club, invited him to join. The members of this group would remain Tennyson's friends all his life. Arthur Hallam was the most important of these friendships. Hallam, a brilliant Victorian young man was recognized by his peers as having unusual promise. He and Tennyson knew each other only four years, but their intense friendship had a major influence on the poet. On a visit to Somersby, Hallam met and later became engaged to Emily Tennyson, and the two friends looked forward to a life-long companionship. Hallam died from illness in 1833 at the age of 22 and shocked Tennyson profoundly. His grief lead to most of his best poetry, including "In Memoriam", "The Passing of Arthur", "Ulysses", and "Tithonus". Since Tennyson was always sensitive to criticism, The bad reviews of his 1832 poems hurt him greatly. Critics in those days took great joy in the harshness of their reviews. John Wilson Croker's harsh criticisms of some of the poems he wrote kept Tennyson from publishing again for another nine years. The success of his 1842 poems made Tennyson a popular poet, and in 1845 he got a government pension of 200 pounds a year, which helped him with his financial difficulties. The success of "The Princess" and "In Memoriam" and his appointment as Poet Laureate in 1850 finally established him as the most popular poet of the Victorian era. By now Tennyson, only 41, had written some of his greatest poetry, but he continued to write and to gain popularity. Prince Albert admired his poetry so much that he would drop by unexpectedly to here some of Tennyson's poetry.

Indian Cuisne :: India Food

India which is also called 'The Republic Of India', is a very large country in South Asia. By size is it the 7th largest country in the world. It also the 2nd most populated democratic country in the world. It is nearly surrounded by water but connected at the north end of the country the the rest of Asia. On it's south is the Indian Ocean. On the west, the Arabian Sea and on the east the Bay Of Bengal. It is connected to countries such as China, Nepal and Pakistan which also have had an affect on India's cuisine. India is also subdivided into 28 States and 7 Union Territories which each have different variations of Indian cuisine . Bhapa This technique is simply steaming, usually in banana leaves or in foil. This is used in Eastern India and other parts of India for Fish and Vegetables. Bhunao/Kasha This is the process of cooking mainly rich meat dishes in a onion spice base on low heat stirring frequently for a very long time. Bagar/Chowk Or Sambara This is the process of tempering foods with the final addition of spices and ghee. It is used for most lentil dishes. Dum-Pukth This is cooking in a sealed steam pot, this method is typical of the cooking of Lucknow and is used for rice dishes such as Biryanis and Pillaus. Talna This is the term for frying both deep and shallow. Tikkis and Kababs are cooked this way. Sekhna Is the process of pan roasting for nuts and griddle breads. Garam Masal Garam Masala is a term for a mixture of spices used in Indian cooking. There are many commercial brands are available, but keen cooks can grind and blend their own. This ensures freshness and means you have control over the quality and quantities of the individual spices. There is no official recipe for Garam Masala, but many Indian families have their own variety. Common ingerdients are Nutmeg, Cumin seeds, toasted Black peppercorns, Whole cloves, Cinnamon sticks, Black or green cardamom. Cumin Cumin from Western Asia, where it has been harvested since Biblical times. People cook with the fruits of the plant, which are usually called the seeds. Cumin is a very weak spice often used in highly spiced cuisines, especially Mexican and Indian. Ginger Ginger is used a lot as a cooking ingredient or spice in Asian cuisine. It is also used in many western dishes such as Gingerbread Men, Ginger Beer and Ginger Snaps.

My Graduation Plan at IUBAT

Every student should have a graduation plan on their mind. A student couldn’t progress without his graduation plan. Such as, a boat without boatman or a computer without processor. Graduation plan refers to what we will do in our future, our educational plan at the organization and what we want to be just like the aim in our life. It’s very important to every student. I’ve also my graduation plan. I’m Shantanu Sarkar.I’m a student of BCSE at IUBAT. IUBAT- International University Of Business Agriculture & Technology. It’s the first non-government university established in Bangladesh. The initial planning began in 1989 and the university established in 1991. Degree programs started in 1992 with agreement with assumption university of Bangkok, Thailand. IUBAT strongly lobbied forth creation of non –government universities and supported the initiative of the government of Bangladesh in passing legislation for formal establishment of non-g overnment universities in the country. IUBAT now operates as a Non-government university Act of 1992. I’ve a graduation plan at IUBAT.I’m a student of BCSE (Bachelor Of Computer Science & Engineering). And I want to be a software engineer. So, the BCSE degree will be conferred only to the student who has fully complied with the graduation plans and has applied for it. The plans are:†¢I’ve to complete all the prescribed courses with a minimum of 141 credit hours plus such courses recommended by the department after reviewing individual background. †¢I’ve to earn ‘A’ grades in each of the core, specialization courses, and English courses. †¢I’ve to achieve the fulfillment of English language requirement, when relevant. †¢I’ve to earn a cumulative grade point average (CGPA) of 3.8. †¢I’ve to earn at least ‘A’ or ‘A-’ grade in the practicum. †¢I’ve to earn at leas t ‘A’ or ‘A-’ grade in the comprehensive examination. †¢I’ve to satisfactory behavior and discipline.So, this is my graduation plan at IUBAT. And I’ve to follow this plan. If I follow this plan carefully and sincerely, I’ll be succeed.

My Belief

Jounal What is My Belief? Belief is something inspirational that encourages others to progress and work towards self-improvement. That is especially important for today’s youth and can play an important role in shaping our society. Beliefs can influence a young one’s values, conviction, and attitudes, which will shape the person that one will grow to be. That has the ability to focus that efforts on others rather than on themselves and inspire others not only by their words, but more so by it actions that move us to do the same.Beliefs help others by offering good examples, by inspiring others realize the endless possibilities to reach our goals, and by moving others to be the best that I can be. In my opinion, parents are the most effective and influential role models in the lives of their children. The youth of society can learn from, and aspire to be like their parents as they are being reared in their childhood years. Fathers and mothers strive to teach their childr en important values and beliefs as well as demonstrate attitudes and behaviors considered appropriate and beneficial for society’s well-being.Something that has been the strong belief in my life is love of my parents as family. Because my role models are my parents and they possess the quality of good parents. I know they are not perfect, but they carry the virtue and quality of a parent that everyone could wish for. My parents are very responsible, good providers, committed to their duties and responsibilities, loving and determined in pursuing their plans to nurture us with good values and norms so that we will became responsible, well-disciplined, and God-loving. And they also taught us to love others, as we grow old.Now I want to explain several lessons about why the loves of my parents are strong belief to me. First of all, I remember when I was young, my parents made sure that they provided us with what we needed and they do their best, even in the midst of hardships. E specially, my dad has always taught me that I have to work my hardest for whatever I want in life. My dad had me with his wife, my mother, when he was twenty-five. My parents had very little money to raise me on their own, so my dad went to work right away to help support his family.He gave up all his freedom and the fun on a young’s life to help support his family. My dad has been working at least two jobs at a time since the age of twenty-five. They didn’t stop or even pause in fulfilling their duties and responsibilities as parents they always seek a solution or find ways to keep our life better and meaningful. Second, my parents always show and teach us the good values so that we can past it on to the next generation. They are always there to support us in our endeavor and in pursuing our dreams.They are very supportive especially when I need to decide on my own, they are always there on my side, not to be hindered in my plans but instead to give advice and support that I needed most. At a point when I was failing in a subject of 12th standard, third, as I was not very good in academics. My parent then had a long conversation with me about life; they told me that it would be smarter if I will study hard because it would be easier for me to get a job and earn more money, and that this was the best way I could help the family.They always tell us about what they went through because of not having the chance to concentrate on studies at all and â€Å"Only you can change your life. No one can do it for you†. By telling us this, they want us to take advantage of what we have to have a better future. But what got my parents to where they is today was not from being lazy by no means but working hard and driving himself to become something better than what they was and to achieve them dreams. My parents don’t only give advice about having a good future but also about morals.They always tell us that if I order for people to respect us, we have to respect them at all times. To him having good morals is really important because not only does it make us do the right thing but also they will help us succeed in life. Therefore, I should thank God for met my parents. Fourth, my parents are a real understanding person. they gives me the best advice I can get and that is why I think my parents have given me the best emotional support I can receive. My parents have also been the best role model for me.They have thought me not to worry about what others might think or say about me, as long as I’m doing the right thing. They have shown me that being humble and warm hearted could win more love then being filled with superficial materials. They also teach me to stand up for what I believe in no matter what the case is. They have though me that no mountain is high enough. Those are a few reasons why they are an excellent role model in my life. In conclusion, here are some of the reasons on how my parents have impacted my l ife and how the love made an impact in society too.My parents are my role model because I want to be like them, the way they nurtured me. And my parents have also taught me that to be a person in the future, it will give other the hope and desire by saying it ‘I can do it so you can’, my parents make sure that they are on my side in every step of the way of our life. To me the love of my parents are my strongest beliefs because I want to be like them, the way they nurtured us. Furthermore, they always tell me that if I order for people to respect us we have to respect them at all times. This saying always makes me strong.

Analysis of 2012/2013 Botswana Budget Speech

UNIVERSITY OF BOTSWANA NAME: MATILDAH TILLY KAUNDA COURSE: MGT 306 LECTURER: DR MAROBELA TOPIC: ANALYSIS OF THE BUDGET SPEECH ? INTRODUCTION Neo-liberalism is a set of economic policies that have become widespread during the last 25 years or so. Neo-liberalism is the case where the rich grow richer and the poor grow poorer. These are a set of policies that are under the influence exerted by the International Monetary Fund (IMF), the World Bank and the Inter- American Development Bank.They include frameworks of free market trade and no government intervention as well as elements of privatization. This report explores the theories of Managerialism, New public Management and finally neoliberalism implementation in Botswana, to assess whether such a change is constructive to the development of Botswana and Batswana by referring to the budget speech to evaluate whether Botswana is prepared for such a change Managerialism is the pursuit of goals by managers other that for profit maximizati on.According to Locke & Spender (2011) Managerialism is what occurs when a special group called managers ensconces itself systematically in a n organization and deprives owners and employees in their decision making power including the distribution of emoluments and justifies that takeover on the grounds of the managing group’ education and exclusive possession of codified knowledge and know- how necessary to the efficient running of the organization. Application of managerialism to the public sector involves privatization profit motive, incentives for managers and delegation of power.According to Pollitt & Bonkaert (2011) New Public Management refers to deliberate changes to the structures and processes of public sector organization with the objective of getting them to run better. It basically means changes in the way things are done to make them more efficient, more responsive to those who use them; their main focus is on achieving objectives like reduction of poverty. The main emphasis of New Public management is the need to change inefficient, money losing state enterprises into competitive, profit making, taxpaying businesses that provide quality goods services to consumers.This is greatly in line with the words that Honorable Math ambo said in the recent budget speech. He talked of the need for transfer of state owned enterprises to private ones. In the budget Speech Mr Mathambo stated that â€Å" a Privatization Master Plan adopted in 2005 and aimed at identifying all public enterprises suitable for privatization is being revised to among others, identify services and Public Enterprises that are suitable for outsourcing and divestiture during the period 2012 to 2017.In Botswana as said by the minister a new master plan or NDP 10 set out a prudent strategy for ensuring fiscal sustainability while supporting private sector development. A central feature of the strategy is for Government to reduce its dominance by cutting Government expenditure as a share of GDP from 40 to 30 percent Neoliberalism explains the state of no government intervention, free market trade and privatization.According to a journal of sociology (2007) this term broadly means the project of economic and social transformation under the sign of the free market and needs which are formerly met by public agencies in communities and families met by companies selling services in the market. Basically the policy recommendations of neoliberalism are concerned mainly with dismantling what remains of the regulations welfare state. These recommendations include deregulation of business; privatization of public activities and assets; elimination of, or cutbacks in, social welfare programs; and reduction of taxes on businesses and the investing class.The theory of neoliberals advocates for no or rather reduced government spending in the economy. Locke, R (2011) stated that International monetary fund which was created to administer the international monetary system is a strong supporter of neoliberalism or rather privatization. According to an article by Victor Baatweeng dated 12 January 2011,The international Monetary fund (IMF) has advised Botswana and other Southern African Customs Union (SACU) member states to slash their expenditure in order to ensure fiscal and debt sustainability.As a result, the IMF has recommended an appropriate mix of revenue and recurrent expenditure measures, with particular emphasis on reducing the wage bill. The Minister of Finance and Development Planning, Kenneth Matambo announced during his budget speech on Wednesday that the government is concerned that due to its dominance in Economic activities, the public sector wage bill has escalated over time and is high compared to that of other comparable middle income countries and that it needs to be reduced.According to Matambo, this can be achieved by reducing the size of the public sector, with functions and activities which are better carried out in a commercial environment being provided by the private sector. It looks like finally the government is responding to the pressures of IMF. However this is a drastic move that will only cause social upheavals and throw some people into the labor market. This move that can be taken by developed countries rather than developmental states like Botswana.Considering the fact that unemployment as mentioned in the recent budget speech continues to be high, and the government planning to cut wage bill and freeze posts, this initiative will solve nothing but rather lead the county into a downhill. Also the introduction of privatization will cause people who were employed by the government to lose their jobs when the private companies take over. The reason for this is that cost cutting has become the main strategy companies adopt in response to the liberalization of markets.These companies will come with short term contracts, part time positions, minimum wages and no job security. Instead of finding ways to curb the issue the government plans to add fuel to the fire. H0 This move to privatization and outsourcing of services most importantly essential services like water and electricity will only have a negative impact on the poor . According to Steger and Roy (2010), there should be state ownership of crucial national enterprises like energy and railroads.They further stated that Keynes in particular advocates for massive government spending in time of economic crises to create new jobs and lift consumer spending. The paramount objective of the capitalist’s economy is to make profit rather than enhancing wellbeing in economically efficient ways. Prices in the private sector tend to be high unlike in the public sector where they are subsidized to cater for those below the poverty line. In the budget speech the Minister stated that individuals below Botswana Poverty Datum Line declined from 30. 6% of population in 2002/03 to 20. % in 2009/10 but taking this initiative of privat ization will negatively affect those living below poverty datum line and may even end up increasing their number. State intervention is important as well as crucial to the economy as well as to the welfare of its citizens. It ensures social efficiency and fair and just allocation of resources and prevents market failure. According to monopolistic competition economy is a further reason for intervention by the government because it has the potential for the misallocation of resources through fixing wrong prices and making the customers worse off.Hughes(2003)’ s view on privatization was that market systems does not necessarily bring high employment , price stability and the socially desired rate of economic growth and thus the essentiality of public policy to secure such objectives. There are a number of reasons for the need of neoliberalism and why less state intervention in a state may be appropriate and viable. According to a journal by Wendy Larner† Liberalization is essentially about the introduction of competition. The main argument for avocation of liberalization is that competition forces alternative providers to improve productivity and service quality. She further stated that though there is reduction of employment as a consequence of liberalization and privatization, in many cases, lower staff levels result in work intensification. According to Locke (2001) Private sector practices and technologies are superior to those used in the public sector, thus there is high efficiency and effectiveness. He further stated that this idea has a long tradition that can be traced back to 1868. Privatization comes with competition and thus considered a viable strategy for improving performance of public bureaucracies because it lowers costs and increases efficiency.A further argument for privatization is to reduce cross subsidies and charges for services in accordance with their true cost. The argument is that subsidies are economically undesirable as true costs and inefficiencies are can be hidden. There are other mechanisms preferable like direct funding from the budget or giving cash to those to be given assistance. Neoliberasation also reduces government borrowing. The government should only borrow for long term assets such as power stations. A lower government borrowing has lower interest rates and thus helping the economy. Hughes 2003 pg 104). CONCLUSION Government intervention in the market is mainly aimed at reducing injustices and inequalities. While state intervention should be reduced to a minimum to promote efficiency, government should always keep an eye out for situations that only government intervention will regulate in everyone’s best interest against the interest of just a firm. It is difficult to choose a stance between state interventions in the economy because it comes with both advantages and disadvantages for the country.However for a developmental state like Botswana I believe that it is advisable n ot to undergo this change as it not well developed and the welfare of Batswana depends on it. REFERENCES Baatweng, V. (2011, January 12). IMF SLASHES WAGE BILL. Larner W, Neo- liberalism:policy, ideology Governmentality. studies in political economy . Journal of Sociology  © 2009 the Australian Sociological Association, Volume 45(4): 331–338 Locke, R. R. (2011). confronting managerialism. New York: Zed Book Ltd. Kotz M (2002). Globalisarion and Neoliberalism. Rethinking Marxism, Volume 12, Number 2, Summer 2002, pp. 64-79. , 64-79.Matambo, O. (2012). BUDGET SPEECH 2012. Gaborone: Government Printing and Publishing Services,. Steger& Roy (2010) NEO LIBERISALISM- A VERY SHORT INTRODUCTION. New York, USA: Oxford University Press. Locke & Spender (2011). Confronting Managerialism: How the Business Elite and Their Schools Threw Our Lives Out of Balance (Economic Controversies) by. new york, USA: zed Books Ltd. Hughes E (2003). public management and administration. An introductio n. NEW YORK, USA: Palgrave macmillan. Pollit et al (2011) Comparative Analysis- New Public Management, Governance and †¦. , New York, USA, XFORD University Press

Om Heizer Om10 Ism 04

Chapter FORECASTING Discussion Questions 1.? Qualitative models incorporate subjective factors into the forecasting model. Qualitative models are useful when subjective factors are important. When quantitative data are difficult to obtain, qualitative models may be appropriate. 2.? Approaches are qualitative and quantitative. Qualitative is relatively subjective; quantitative uses numeric models. 3.? Short-range (under 3 months), medium-range (3 months to 3 years), and long-range (over 3 years). 4.? The steps that should be used to develop a forecasting system are: (a)?Determine the purpose and use of the forecast (b)? Select the item or quantities that are to be forecasted (c)? Determine the time horizon of the forecast (d)? Select the type of forecasting model to be used (e)? Gather the necessary data (f)? Validate the forecasting model (g)? Make the forecast (h)? Implement and evaluate the results 5.? Any three of: sales planning, production planning and budgeting, cash budgeting, analyzing various operating plans. 6.? There is no mechanism for growth in these models; they are built exclusively from historical demand values. Such methods will always lag trends. .? Exponential smoothing is a weighted moving average where all previous values are weighted with a set of weights that decline exponentially. 8.? MAD, MSE, and MAPE are common measures of forecast accuracy. To find the more accurate forecasting model, forecast with each tool for several periods where the demand outcome is known, and calculate MSE, MAPE, or MAD for each. The smaller error indicates the better forecast. 9.? The Delphi technique involves: (a)? Assembling a group of experts in such a manner as to preclude direct communication between identifiable members of the group (b)?Assembling the responses of each expert to the questions or problems of interest (c)? Summarizing these responses (d)? Providing each expert with the summary of all responses (e)? Asking each expert to study the summary of the responses and respond again to the questions or problems of interest. (f)? Repeating steps (b) through (e) several times as necessary to obtain convergence in responses. If convergence has not been obtained by the end of the fourth cycle, the responses at that time should probably be accepted and the process terminated—little additional convergence is likely if the process is continued. 0.? A time series model predicts on the basis of the assumption that the future is a function of the past, whereas an associative model incorporates into the model the variables of factors that might influence the quantity being forecast. 11.? A time series is a sequence of evenly spaced data points with the four components of trend, seasonality, cyclical, and random variation. 12.? When the smoothing constant, (, is large (close to 1. 0), more weight is given to recent data; when ( is low (close to 0. 0), more weight is given to past data. 13.? Seasonal patterns are of fixed duration a nd repeat regularly.Cycles vary in length and regularity. Seasonal indices allow â€Å"generic† forecasts to be made specific to the month, week, etc. , of the application. 14.? Exponential smoothing weighs all previous values with a set of weights that decline exponentially. It can place a full weight on the most recent period (with an alpha of 1. 0). This, in effect, is the naive approach, which places all its emphasis on last period’s actual demand. 15.? Adaptive forecasting refers to computer monitoring of tracking signals and self-adjustment if a signal passes its present limit. 16.?Tracking signals alert the user of a forecasting tool to periods in which the forecast was in significant error. 17.? The correlation coefficient measures the degree to which the independent and dependent variables move together. A negative value would mean that as X increases, Y tends to fall. The variables move together, but move in opposite directions. 18.? Independent variable (x) is said to explain variations in the dependent variable (y). 19.? Nearly every industry has seasonality. The seasonality must be filtered out for good medium-range planning (of production and inventory) and performance evaluation. 20.? There are many examples.Demand for raw materials and component parts such as steel or tires is a function of demand for goods such as automobiles. 21.? Obviously, as we go farther into the future, it becomes more difficult to make forecasts, and we must diminish our reliance on the forecasts. Ethical Dilemma This exercise, derived from an actual situation, deals as much with ethics as with forecasting. Here are a few points to consider:  ¦ No one likes a system they don’t understand, and most college presidents would feel uncomfortable with this one. It does offer the advantage of depoliticizing the funds al- location if used wisely and fairly.But to do so means all parties must have input to the process (such as smoothing constants) and all data need to be open to everyone.  ¦ The smoothing constants could be selected by an agreed-upon criteria (such as lowest MAD) or could be based on input from experts on the board as well as the college.  ¦ Abuse of the system is tied to assigning alphas based on what results they yield, rather than what alphas make the most sense.  ¦ Regression is open to abuse as well. Models can use many years of data yielding one result or few years yielding a totally different forecast.Selection of associative variables can have a major impact on results as well. Active Model Exercises* ACTIVE MODEL 4. 1: Moving Averages 1.? What does the graph look like when n = 1? The forecast graph mirrors the data graph but one period later. 2.? What happens to the graph as the number of periods in the moving average increases? The forecast graph becomes shorter and smoother. 3.? What value for n minimizes the MAD for this data? n = 1 (a naive forecast) ACTIVE MODEL 4. 2: Exponential Smoothing 1.? Wha t happens to the graph when alpha equals zero? The graph is a straight line.The forecast is the same in each period. 2.? What happens to the graph when alpha equals one? The forecast follows the same pattern as the demand (except for the first forecast) but is offset by one period. This is a naive forecast. 3.? Generalize what happens to a forecast as alpha increases. As alpha increases the forecast is more sensitive to changes in demand. *Active Models 4. 1, 4. 2, 4. 3, and 4. 4 appear on our Web site, www. pearsonhighered. com/heizer. 4.? At what level of alpha is the mean absolute deviation (MAD) minimized? alpha = . 16 ACTIVE MODEL 4. 3: Exponential Smoothing with Trend Adjustment .? Scroll through different values for alpha and beta. Which smoothing constant appears to have the greater effect on the graph? alpha 2.? With beta set to zero, find the best alpha and observe the MAD. Now find the best beta. Observe the MAD. Does the addition of a trend improve the forecast? alpha = . 11, MAD = 2. 59; beta above . 6 changes the MAD (by a little) to 2. 54. ACTIVE MODEL 4. 4: Trend Projections 1.? What is the annual trend in the data? 10. 54 2.? Use the scrollbars for the slope and intercept to determine the values that minimize the MAD. Are these the same values that regression yields?No, they are not the same values. For example, an intercept of 57. 81 with a slope of 9. 44 yields a MAD of 7. 17. End-of-Chapter Problems [pic] (b) | | |Weighted | |Week of |Pints Used |Moving Average | |August 31 |360 | | |September 7 |389 |381 ( . 1 = ? 38. 1 | |September 14 |410 |368 ( . 3 = 110. 4 | |September 21 |381 |374 ( . 6 = 224. 4 | |September 28 |368 |372. | |October 5 |374 | | | |Forecast 372. 9 | | (c) | | | |Forecasting | Error | | |Week of |Pints |Forecast |Error |( . 20 |Forecast| |August 31 |360 |360 |0 |0 |360 | |September 7 |389 |360 |29 |5. 8 |365. 8 | |September 14 |410 |365. 8 |44. 2 |8. 84 |374. 64 | |September 21 |381 |374. 64 |6. 36 |1. 272 |375. 12 | |Se ptember 28 |368 |375. 912 |–7. 912 |–1. 5824 |374. 3296| |October 5 |374 |374. 3296 |–. 3296 |–. 06592 |374. 2636| The forecast is 374. 26. (d)? The three-year moving average appears to give better results. [pic] [pic] Naive tracks the ups and downs best but lags the data by one period. Exponential smoothing is probably better because it smoothes the data and does not have as much variation. TEACHING NOTE: Notice how well exponential smoothing forecasts the naive. [pic] (c)? The banking industry has a great deal of seasonality in its processing requirements [pic] b) | | |Two-Year | | | |Year |Mileage |Moving Average |Error ||Error| | |1 |3,000 | | | | | |2 |4,000 | | | | | |3 |3,400 |3,500 |–100 | |100 | |4 |3,800 |3,700 |100 | |100 | |5 |3,700 |3,600 |100 | |100 | | | |Totals| |100 | | |300 | | [pic] 4. 5? (c)? Weighted 2 year M. A. ith . 6 weight for most recent year. |Year |Mileage |Forecast |Error ||Error| | |1 |3,000 | | | | |2 |4,000 | | | | |3 |3,400 |3,600 |–200 |200 | |4 |3,800 |3,640 |160 |160 | |5 |3,700 |3,640 |60 |60 | | | | | | | 420 | | Forecast for year 6 is 3,740 miles. [pic] 4. 5? (d) | | |Forecast |Error ( |New | |Year |Mileage |Forecast |Error |( = . 50 |Forecast | |1 |3,000 |3,000 | ?0 | 0 |3,000 | |2 |4,000 |3,000 |1,000 |500 |3,500 | |3 |3,400 |3,500 | –100 |–50 |3,450 | |4 |3,800 |3,450 | 350 |175 |3,625 | |5 |3,700 |3,625 | 75 |? 38 |3,663 | | | |Total |1,325| | | | The forecast is 3,663 miles. 4. 6 |Y Sales |X Period |X2 |XY | |January |20 |1 |1 |20 | |February |21 |2 |4 |42 | |March |15 |3 |9 |45 | |April |14 |4 |16 |56 | |May |13 |5 |25 |65 | |June |16 |6 |36 |96 | |July |17 |7 |49 |119 | |August |18 |8 |64 |144 | |September |20 |9 |81 |180 | |October |20 |10 |100 |200 | |November |21 |11 |121 |231 | |December |23 |12 |144 |276 | |Sum | 18 |78 |650 |1,474 | |Average |? 18. 2 | 6. 5 | | | (a) [pic] (b)? [i]? NaiveThe coming January = December = 23 [ii]? 3-month moving (20 + 21 + 23)/3 = 21. 33 [iii]? 6-month weighted [(0. 1 ( 17) + (. 1 ( 18) + (0. 1 ( 20) + (0. 2 ( 20) + (0. 2 ( 21) + (0. 3 ( 23)]/1. 0 = 20. 6 [iv]? Exponential smoothing with alpha = 0. 3 [pic] [v]? Trend? [pic] [pic] Forecast = 15. 73? +?. 38(13) = 20. 67, where next January is the 13th month. (c)? Only trend provides an equation that can extend beyond one month 4. 7? Present = Period (week) 6. a) So: where [pic] )If the weights are 20, 15, 15, and 10, there will be no change in the forecast because these are the same relative weights as in part (a), i. e. , 20/60, 15/60, 15/60, and 10/60. c)If the weights are 0. 4, 0. 3, 0. 2, and 0. 1, then the forecast becomes 56. 3, or 56 patients. [pic] [pic] |Temperature |2 day M. A. | |Error||(Error)2| Absolute |% Error | |93 |— | — |— |— | |94 |— | — |— |— | |93 |93. 5 | 0. 5 |? 0. 25| 100(. 5/93) | = 0. 54% | |95 |93. 5 | 1. 5 | ? 2. 25| 100(1. 5/95) | = 1. 58% | |96 |94. 0 | 2. 0 |? 4. 0 0| 100(2/96) | = 2. 08% | |88 |95. 5 | 7. | 56. 25| 100(7. 5/88) | = 8. 52% | |90 |92. 0 | 2. 0 |? 4. 00| 100(2/90) | = 2. 22% | | | | |13. 5| | | 66. 75 | | |14. 94% | MAD = 13. 5/5 = 2. 7 (d)? MSE = 66. 75/5 = 13. 35 (e)? MAPE = 14. 94%/5 = 2. 99% 4. 9? (a, b) The computations for both the two- and three-month averages appear in the table; the results appear in the figure below. [pic] (c)? MAD (two-month moving average) = . 750/10 = . 075 MAD (three-month moving average) = . 793/9 = . 088 Therefore, the two-month moving average seems to have performed better. [pic] (c)? The forecasts are about the same. [pic] 4. 12? t |Day |Actual |Forecast | | | | |Demand |Demand | | |1 |Monday |88 |88 | | |2 |Tuesday |72 |88 | | |3 |Wednesday |68 |84 | | |4 |Thursday |48 |80 | | |5 |Friday | |72 |( Answer | Ft = Ft–1 + ((At–1 – Ft–1) Let ( = . 25. Let Monday forecast demand = 88 F2 = 88 + . 25(88 – 88) = 88 + 0 = 88 F3 = 88 + . 25(72 – 88) = 88 – 4 = 84 F4 = 84 + . 25(68 – 84) = 84 – 4 = 80 F5 = 80 + . 25(48 – 80) = 80 – 8 = 72 4. 13? (a)? Exponential smoothing, ( = 0. 6: | | |Exponential |Absolute | |Year |Demand |Smoothing ( = 0. |Deviation | |1 |45 |41 |4. 0 | |2 |50 |41. 0 + 0. 6(45–41) = 43. 4 |6. 6 | |3 |52 |43. 4 + 0. 6(50–43. 4) = 47. 4 |4. 6 | |4 |56 |47. 4 + 0. 6(52–47. 4) = 50. 2 |5. 8 | |5 |58 |50. 2 + 0. 6(56–50. 2) = 53. 7 |4. 3 | |6 |? |53. 7 + 0. 6(58–53. 7) = 56. 3 | | ( = 25. 3 MAD = 5. 06 Exponential smoothing, ( = 0. 9: | | |Exponential |Absolute | |Year |Demand |Smoothing ( = 0. |Deviation | |1 |45 |41 |4. 0 | |2 |50 |41. 0 + 0. 9(45–41) = 44. 6 |5. 4 | |3 |52 |44. 6 + 0. 9(50–44. 6 ) = 49. 5 |2. 5 | |4 |56 |49. 5 + 0. 9(52–49. 5) = 51. 8 |4. 2 | |5 |58 |51. 8 + 0. 9(56–51. 8) = 55. 6 |2. 4 | |6 |? |55. 6 + 0. 9(58–55. 6) = 57. 8 | | ( = 18. 5 MAD = 3. 7 (b)? 3-year moving average: | | |Three-Year |Absolute | |Year |Demand |Moving Average |Deviation | |1 45 | | | |2 |50 | | | |3 |52 | | | |4 |56 |(45 + 50 + 52)/3 = 49 |7 | |5 |58 | (50 + 52 + 56)/3 = 52. 7 |5. 3 | |6 |? | (52 + 56 + 58)/3 = 55. 3 | | ( = 12. 3 MAD = 6. 2 (c)? Trend projection: | | | |Absolute | |Year |Demand |Trend Projection |Deviation | |1 |45 |42. 6 + 3. 2 ( 1 = 45. 8 |0. 8 | |2 |50 |42. 6 + 3. 2 ( 2 = 49. 0 |1. 0 | |3 |52 |42. 6 + 3. 2 ( 3 = 52. 2 |0. 2 | |4 |56 |42. 6 + 3. 2 ( 4 = 55. 4 |0. | |5 |58 |42. 6 + 3. 2 ( 5 = 58. 6 |0. 6 | |6 |? |42. 6 + 3. 2 ( 6 = 61. 8 | | ( = 3. 2 MAD = 0. 64 [pic] | X |Y |XY |X2 | | 1 |45 | 45 | 1 | | 2 |50 |100 | 4 | | 3 |52 |156 | 9 | | 4 |56 |224 |16 | | 5 |58 |290 |25 | Then: (X = 15, (Y = 261, (XY = 815, (X2 = 55, [pic]= 3, [pic]= 52. 2 Therefore: [pic] (d)? Comparing the results of the forecasting methodologies for parts (a), (b), and (c). |Forecast Methodology |MAD | |Exponential smoothing, ( = 0. |5. 06 | |Exponential smoothing, ( = 0. 9 |3. 7 | |3-year moving average |6. 2 | |Trend projection |0. 64 | Based on a mean absolute deviation criterion, the trend projection is to be preferred over the exponential smoothing with ( = 0. 6, exponential smoothing with ( = 0. 9, or the 3-year moving average forecast methodologies. 4. 14 Method 1:MAD: (0. 20 + 0. 05 + 0. 05 + 0. 20)/4 = . 125 ( better MSE : (0. 04 + 0. 0025 + 0. 0025 + 0. 04)/4 = . 021 Method 2:MAD: (0. 1 + 0. 20 + 0. 10 + 0. 11) / 4 = . 1275 MSE : (0. 01 + 0. 04 + 0. 01 + 0. 0121) / 4 = . 018 ( better 4. 15 | |Forecast Three-Year |Absolute | |Year |Sales |Moving Average |Deviation | |2005 |450 | | | |2006 |495 | | | |2007 |518 | | | |2008 |563 |(450 + 495 + 518)/3 = 487. 7 |75. 3 | |2009 |584 |(495 + 518 + 563)/3 = 525. 3 |58. 7 | |2010 | |(518 + 563 + 584)/3 = 555. 0 | | | | | ( = 134 | | | | MAD = 67 | 4. 16 Year |Time Period X |Sales Y |X2 |XY | |2005 |1 |450 | 1 |450 | |2006 |2 |495 | 4 |990 | |2007 |3 |518 | 9 |1554 | |2008 |4 |563 |16 |2252 | |2009 |5 |584 |25 |2920 | | | | ( = 2610| |( = 55 | |( = 8166 | [pic] [pic] |Year |Sales |Forecast Trend |Absolute Deviation | |2005 |450 |454. 8 |4. 8 | |2006 |495 |488. 4 |6. | |2007 |518 |522. 0 |4. 0 | |2008 |563 |555. 6 |7. 4 | |2009 |584 |589. 2 |5. 2 | |2010 | |622. 8 | | | | | | ( = 28 | | | | | MAD = 5. 6 | 4. 17 | | |Forecast Exponential |Absolute | |Year |Sales |Smoothing ( = 0. 6 |Deviation | |2005 |450 |410. 0 |40. | |2006 |495 |410 + 0. 6(450 – 410) = 434. 0 |61. 0 | |2007 |518 |434 + 0. 6(495 – 434) = 470. 6 |47. 4 | |2008 |563 |470. 6 + 0. 6(518 – 470. 6) = 499. 0 |64. 0 | |2009 |584 |499 + 0. 6(563 – 499) = 537. 4 |46. 6 | |2010 | |537. 4 + 0. 6(584 – 537. 4) = 565. 6 | | | | | ( = 259 | | | | MAD = 51. 8 | | | |Forecast Exponential |Absolute | |Year |Sales |Smoothing ( = 0. |Deviation | |2005 |450 |410. 0 |40. 0 | |2006 |495 |410 + 0. 9(450 – 410) = 446. 0 |49. 0 | |2007 |518 |446 + 0. 9(495 – 446) = 490. 1 |27. 9 | |2008 |563 |490. 1 + 0. 9(518 – 490. 1) = 515. 2 |47. 8 | |2009 |584 |515. 2 + 0. 9(563 – 515. 2) = 558. 2 |25. 8 | |2010 | |558. 2 + 0. 9(584 – 558. 2) = 581. 4 | | | | |( = 190. 5 | | | |MAD = 38. 1 | (Refer to Solved Problem 4. 1)For ( = 0. 3, absolute deviations for 2005–2009 are 40. 0, 73. 0, 74. 1, 96. 9, 88. 8, respectively. So the MAD = 372. 8/5 = 74. 6. [pic] Because it gives the lowest MAD, the smoothing constant of ( = 0. 9 gives the most accurate forecast. 4. 18? We need to find the smoothing constant (. We know in general that Ft = Ft–1 + ((At–1 – Ft–1); t = 2, 3, 4. Choose either t = 3 or t = 4 (t = 2 won’t let us find ( because F2 = 50 = 50 + ((50 – 50) holds for any (). Let’s pick t = 3. Then F3 = 48 = 50 + ((42 – 50) or 48 = 50 + 42( – 50( or –2 = –8( So, . 25 = ( Now we can find F5 : F5 = 50 + ((46 – 50)F5 = 50 + 46( – 50( = 50 – 4( For ( = . 25, F5 = 50 – 4(. 25) = 49 The forecast for time period 5 = 49 units. 4. 19? Trend adjusted exponential smoothing: ( = 0. 1, ( = 0. 2 | | |Unadjusted | |Adjusted | | | |Month |Income |Forecast |Trend |Forecast ||Error||Error2 | |February |70. 0 | 65. 0 | 0. 0 | 65 |? 5. 0 |? 25. 0 | |March |68. 5 | 65. 5 | 0. 1 | 65. 6 |? 2. 9 |? 8. 4 | |April |64. 8 | 65. 9 | 0. 16 |66. 05 |? 1. 2 |? 1. 6 | |May |71. 7 | 65. 92 | 0. 13 |66. 06 |? 5. 6 |? 31. 9 | |June |71. | 66. 62 | 0. 25 |66. 87 |? 4. 4 |? 19. 7 | |July |72. 8 | 67. 31 | 0. 33 |67. 64 |? 5. 2 |? 26. 6 | |August | | 68. 16 | |68. 60 | |24. 3| | |113. 2| | MAD = 24. 3/6 = 4. 05, MSE = 113. 2/6 = 18. 87. Note that all numbers are rounded. Note: To use POM for Windows to solve this problem, a period 0, which contains the initial forecast and initial trend, must be added. 4. 20? Trend adjusted exponential smoothing: ( = 0. 1, ( = 0. 8 [pic] [pic] [pic] [pic] [pic] [pic] [pic] [pic] [pic] [pic] [pic] [pic] 4. 23? Students must determine the naive forecast for the four months .The naive forecast for March is the February actual of 83, etc. |(a) | |Actual |Forecast ||Error| ||% Error| | | |March |101 |120 |19 |100 (19/101) = 18. 81% | | |April |? 96 |114 |18 |100 (18/96) ? = 18. 75% | | |May |? 89 |110 |21 |100 (21/89) ? = 23. 60% | | |June |108 |108 |? 0 |100 (0/108) ? = 0% | | | | | | |58 | | | 61. 16% | [pic] |(b)| |Actual |Naive ||Error| ||% Error| | | |March |101 |? 83 |18 |100 (18/101) = 17. 82% | | |April |? 96 |101 |? |100 (5/96) ? = 5. 21% | | |May |? 89 |? 96 |? 7 |100 (7/89) ? =? 7. 87% | | |June |108 |? 89 |19 |100 (19/108) = 17. 59% | | | | | | |49| | |48. 49% | | [pic] Naive outperforms management. (c)? MAD for the manager’s technique is 14. 5, while MAD for the naive forecast is only 12. 25. MAPEs are 15. 29% and 12. 12%, respectively. So the naive method is better. 4. 24? (a)? Graph of demand The observations obviously do not form a straight line but do tend to cluster about a straight line over the range shown. (b)? Least-squares regression: [pic] Assume Appearances X |Demand Y |X2 |Y2 |XY | |3 | 3 | 9 | 9 | 9 | |4 | 6 |16 | 36 |24 | |7 | 7 |49 | 49 |49 | |6 | 5 |36 | 25 |30 | |8 |10 |64 |100 |80 | |5 | 7 |25 | 49 |35 | |9 | ? | | | | (X = 33, (Y = 38, (XY = 227, (X2 = 199, [pic]= 5. 5, [pic]= 6. 33. Therefore: [pic] The following figure shows both the data and the resulting equation: [pic] (c) If there are nine performances by Stone Temple Pilots, the estimated sales are: (d) R = . 82 is the correlation coefficient, and R2 = . 68 means 68% of the variation in sales can be explained by TV appearances. 4. 25? |Number of | | | | | |Accidents | | | | |Month |(y) |x |xy |x2 | |January | 30 | 1 | 30 | 1 | |February | 40 | 2 | 80 | 4 | |March | 60 | 3 |180 | 9 | |April | 90 | 4 |360 |16 | |? Totals | |220 | | | [pic] The regression line is y = 5 + 20x. The forecast for May (x = 5) is y = 5 + 20(5) = 105. 4. 26 |Season |Year1 |Year2 |Average |Average |Seasonal |Year3 | | |Demand |Demand |Year1(Year2 |Season |Index |Demand | | | | |Demand |Demand | | | |Fall |200 |250 |225. 0 |250 |0. 90 |270 | |Winter |350 |300 |325. |250 |1. 30 |390 | |Spring |150 |165 |157. 5 |250 |0. 63 |189 | |Summer |300 |285 |292. 5 |250 |1. 17 |351 | 4. 27 | | Winter |Spring |Summer |Fall | |2006 |1,400 |1,500 |1,000 |600 | |2007 |1,200 |1,400 |2,100 |750 | |2008 |1,000 |1,600 |2,000 |650 | |2009 | 900 |1,500 |1,900 | 500 | | |4,500 |6,000 |7,000 |2,500 | 4. 28 | | | | |Average | | | | | | |Average |Quarterly |Seasonal | |Quarter |2007 |2008 |2009 |Demand |Demand |Index | |Winter | 73 | 65 | 89 | 75. 67 |106. 67 |0. 709 | |Spring |104 | 82 |146 |110. 67 |106. 67 |1. 037 | |Summer |168 |124 |205 |165. 67 |106. 67 |1. 553 | |Fall | 74 | 52 | 98 | 74. 67 |106. 67 |0. 700 | 4. 29? 2011 is 25 years beyond 1986. Therefore, the 2011 quarter numbers are 101 through 104. | | | | |(5) | | |(2) |(3) |(4) |Adjusted | |(1) |Quarter |Forecast |Seasonal |Forecast | |Quarter |Number |(77 + . 3Q) |Factor |[(3) ( (4)] | |Winter |101 |12 0. 43 | . 8 | 96. 344 | |Spring |102 |120. 86 |1. 1 |132. 946 | |Summer |103 |121. 29 |1. 4 |169. 806 | |Fall |104 |121. 72 | . 7 | 85. 204 | 4. 30? Given Y = 36 + 4. 3X (a) Y = 36 + 4. 3(70) = 337 (b) Y = 36 + 4. 3(80) = 380 (c) Y = 36 + 4. 3(90) = 423 4. 31 4. 33? (a)? See the table below. For next year (x = 6), the number of transistors (in millions) is forecasted as y = 126 + 18(6) = 126 + 108 = 234. Then y = a + bx, where y = number sold, x = price, and |4. 32? a) | x |y |xy |x2 | | | 16 | 330 | 5,280 |256 | | | 12 | 270 | 3,240 |144 | | | 18 | 380 | 6,840 |324 | | | 14 | 300 | 4,200 |196 | | | 60 |1,280 |19,560 |920 | So at x = 2. 80, y = 1,454. 6 – 277. 6($2. 80) = 677. 32. Now round to the nearest integer: Answer: 677 lattes. [pic] (b)? If the forecast is for 20 guests, the bar sales forecast is 50 + 18(20) = $410. Each guest accounts for an additional $18 in bar sales. |Table for Problem 4. 33 | | | | | |Year |Transistors | | | | | | | |(x) |(y) |xy |x2 |126 + 18x |E rror |Error2 ||% Error| | | |? 1 |140 |? 140 |? 1 |144 |–4 |? 16 |100 (4/140)? = 2. 86% | | |? 2 |160 |? 320 |? 4 |162 |–2 | 4 |100 (2/160)? = 1. 25% | | |? 3 |190 |? 570 |? 9 |180 |10 |100 |100 (10/190) = 5. 26% | | |? 4 |200 |? 800 |16 |198 |? 2 | 4 |100 (2/200) = 1. 00% | | |? |210 |1,050 |25 |216 |–6 |? 36 |100 (6/210)? = 2. 86% | |Totals |15 | | |900 | | |2,800 | | (b)? MSE = 160/5 = 32 (c)? MAPE = 13. 23%/5 = 2. 65% 4. 34? Y = 7. 5 + 3. 5X1 + 4. 5X2 + 2. 5X3 (a)? 28 (b)? 43 (c)? 58 4. 35? (a)? [pic] = 13,473 + 37. 65(1860) = 83,502 (b)? The predicted selling price is $83,502, but this is the average price for a house of this size. There are other factors besides square footage that will impact the selling price of a house. If such a house sold for $95,000, then these other factors could be contributing to the additional value. (c)?Some other quantitative variables would be age of the house, number of bedrooms, size of the lot, and size of the garage, etc. (d)? Coefficient of determination = (0. 63)2 = 0. 397. This means that only about 39. 7% of the variability in the sales price of a house is explained by this regression model that only includes square footage as the explanatory variable. 4. 36? (a)? Given: Y = 90 + 48. 5X1 + 0. 4X2 where: [pic] If: Number of days on the road ( X1 = 5 and distance traveled ( X2 = 300 then: Y = 90 + 48. 5 ( 5 + 0. 4 ( 300 = 90 + 242. 5 + 120 = 452. 5 Therefore, the expected cost of the trip is $452. 50. (b)? The reimbursement request is much higher than predicted by the model. This request should probably be questioned by the accountant. (c)?A number of other variables should be included, such as: 1.? the type of travel (air or car) 2.? conference fees, if any 3.? costs of entertaining customers 4.? other transportation costs—cab, limousine, special tolls, or parking In addition, the correlation coefficient of 0. 68 is not exceptionally high. It indicates that the model explains approximately 46% of the overall variation in trip cost. This correlation coefficient would suggest that the model is not a particularly good one. 4. 37? (a, b) |Period |Demand |Forecast |Error |Running sum ||error| | | 1 |20 |20 |0. 00 |0. 00 |0. 00 | | 2 |21 |20 |1. 00 |1. 0 |1. 00 | | 3 |28 |20. 5 |7. 50 |8. 50 |7. 50 | | 4 |37 |24. 25 |12. 75 |21. 25 |12. 75 | | 5 |25 |30. 63 |–5. 63 |15. 63 |5. 63 | | 6 |29 |27. 81 |1. 19 |16. 82 |1. 19 | | 7 |36 |28. 41 |7. 59 |24. 41 |7. 59 | | 8 |22 |32. 20 |–10. 20 |14. 21 |10. 20 | | 9 |25 |27. 11 |–2. 10 |12. 10 |2. 10 | |10 |28 |26. 05 | 1. 95 |14. 05 | | | | | | |1. 95 | | | | | | | | | | | | | | | |MAD[pic]5. 00 | Cumulative error = 14. 05; MAD = 5? Tracking = 14. 05/5 ( 2. 82 4. 38? (a)? least squares equation: Y = –0. 158 + 0. 1308X (b)? Y = –0. 158 + 0. 1308(22) = 2. 719 million (c)? coefficient of correlation = r = 0. 966 coefficient of determination = r2 = 0. 934 4. 39 |Year X |Patients Y |X2 |Y2 |XY | |? 1 |? 36 | 1 |? 1,296 | 36 | |? 2 |? 33 | |? 1,089 | 66 | |? 3 |? 40 | 9 |? 1,600 |? 120 | |? 4 |? 41 |? 16 |? 1,681 |? 164 | |? 5 |? 40 |? 25 |? 1,600 |? 200 | |? 6 |? 55 |? 36 |? 3,025 |? 330 | |? 7 |? 60 |? 49 |? 3,600 |? 420 | |? 8 |? 54 |? 64 |? 2,916 |? 432 | |? 9 |? 58 |? 81 |? 3,364 |? 522 | |10 |? 61 |100 |? 3,721 |? 10 | |55 | | |478 | | |X |Y |Forecast |Deviation |Deviation | |? 1 |36 |29. 8 + 3. 28 ( ? 1 = 33. 1 |? 2. 9 |2. 9 | |? 2 |33 |29. 8 + 3. 28 ( ? 2 = 36. 3 |–3. 3 |3. 3 | |? 3 |40 |29. 8 + 3. 28 ( ? 3 = 39. 6 |? 0. 4 |0. 4 | |? 4 |41 |29. 8 + 3. 28 ( ? 4 = 42. 9 |–1. 9 |1. 9 | |? 5 |40 |29. 8 + 3. 28 ( ? 5 = 46. 2 |–6. 2 |6. 2 | |? 6 |55 |29. 8 + 3. 28 ( ? 6 = 49. 4 |? 5. 6 |5. 6 | |? 7 |60 |29. 8 + 3. 28 ( ? 7 = 52. 7 |? 7. 3 |7. 3 | |? |54 |29. 8 + 3. 28 ( ? 8 = 56. 1 |–2. 1 |2. 1 | |? 9 |58 |29. 8 + 3. 28 ( ? 9 = 59. 3 |–1. 3 |1. 3 | |10 |61 |29. 8 + 3. 28 ( 10 = 62. 6 |–1. 6 |1. 6 | | | | | | ( = | | | | | |32. 6 | | | | | |MAD = 3. 26 | The MAD is 3. 26—this is approximately 7% of the average number of patients and 10% of the minimum number of patients. We also see absolute deviations, for years 5, 6, and 7 in the range 5. 6–7. 3.The comparison of the MAD with the average and minimum number of patients and the comparatively large deviations during the middle years indicate that the forecast model is not exceptionally accurate. It is more useful for predicting general trends than the actual number of patients to be seen in a specific year. 4. 40 | |Crime |Patients | | | | |Year |Rate X |Y |X2 |Y2 |XY | |? 1 |? 58. 3 |? 36 |? 3,398. 9 |? 1,296 |? 2,098. 8 | |? 2 |? 61. 1 |? 33 |? 3,733. 2 |? 1,089 |? 2,016. 3 | |? 3 |? 73. |? 40 |? 5,387. 6 |? 1,600 |? 2,936. 0 | |? 4 |? 75. 7 |? 41 |? 5,730. 5 |? 1,681 |? 3,103. 7 | |? 5 |? 81. 1 |? 40 |? 6,577. 2 |? 1,600 |? 3,244. 0 | |? 6 |? 89. 0 |? 55 |? 7,921. 0 |? 3,025 |? 4,895. 0 | |? 7 |101. 1 |? 60 |10,221. 2 |? 3,600 |? 6,066. 0 | |? 8 |? 94 . 8 |? 54 |? 8,987. 0 |? 2,916 |? 5,119. 2 | |? 9 |103. 3 |? 58 |10,670. 9 |? 3,364 |? 5,991. 4 | |10 |116. 2 |? 61 |13,502. 4 |? 3,721 |? 7,088. 2 | |Column | |854. | | |478 | |Totals | | | | | | |months) |(Millions) |(1,000,000s) | | | | |Year |(X) |(Y) |X2 |Y2 |XY | |? 1 |? 7 |1. 5 |? 49 |? 2. 25 |10. 5 | |? 2 |? 2 |1. 0 | 4 |? 1. 00 |? 2. 0 | |? 3 |? 6 |1. 3 |? 36 |? 1. 69 |? 7. 8 | |? 4 |? 4 |1. 5 |? 16 |? 2. 25 |? 6. 0 | |? 5 |14 |2. 5 |196 |? 6. 25 |35. 0 | |? 6 |15 |2. 7 |225 |? 7. 9 |40. 5 | |? 7 |16 |2. 4 |256 |? 5. 76 |38. 4 | |? 8 |12 |2. 0 |144 |? 4. 00 |24. 0 | |? 9 |14 |2. 7 |196 |? 7. 29 |37. 8 | |10 |20 |4. 4 |400 |19. 36 |88. 0 | |11 |15 |3. 4 |225 |11. 56 |51. 0 | |12 |? 7 |1. 7 |? 49 |? 2. 89 |11. 9 | Given: Y = a + bX where: [pic] and (X = 132, (Y = 27. 1, (XY = 352. 9, (X2 = 1796, (Y2 = 71. 59, [pic] = 11, [pic]= 2. 26. Then: [pic] andY = 0. 511 + 0. 159X (c)?Given a tourist population of 10,000,000, the model predicts a ridership of: Y = 0. 511 + 0. 159 ( 10 = 2. 101, or 2,101,000 persons. (d)? If there are no tourists at all, the model predicts a ridership of 0. 511, or 511,000 persons. One would not place much confidence in this forecast, however, because the number of tourists (zero) is outside the range of data used to develop the model. (e)? The standard error of the estimate is given by: (f)? The correlation coefficient and the coefficient of determination are given by: [pic] 4. 42? (a)? This problem gives students a chance to tackle a realistic problem in business, i. e. , not enough data to make a good forecast.As can be seen in the accompanying figure, the data contains both seasonal and trend factors. [pic] Averaging methods are not appropriate with trend, seasonal, or other patterns in the data. Moving averages smooth out seasonality. Exponential smoothing can forecast January next year, but not farther. Because seasonality is strong, a naive model that students create on their own might be best. (b) One model might be: Ft+1 = At–11 That is forecastnext period = actualone year earlier to account for seasonality. But this ignores the trend. One very good approach would be to calculate the increase from each month last year to each month this year, sum all 12 increases, and divide by 12.The forecast for next year would equal the value for the same month this year plus the average increase over the 12 months of last year. (c) Using this model, the January forecast for next year becomes: [pic] where 148 = total monthly increases from last year to this year. The forecasts for each of the months of next year then become: |Jan. |29 | |July. |56 | |Feb. |26 | |Aug. |53 | |Mar. |32 | |Sep. |45 | |Apr. |35 | |Oct. |35 | |May. |42 | |Nov. |38 | |Jun. |50 | |Dec. |29 | Both history and forecast for the next year are shown in the accompanying figure: [pic] 4. 3? (a) and (b) See the following table: | |Actual |Smoothed | |Smoothed | | |Week |Value |Value |Forecast |Value |Forecast | |t |A(t) |Ft (( = 0. 2) |Err or |Ft (( = 0. 6)|Error | | 1 |50 |+50. 0 |? +0. 0 |+50. 0 |? +0. 0 | | 2 |35 |+50. 0 |–15. 0 |+50. 0 |–15. 0 | | 3 |25 |+47. 0 |–22. 0 |+41. 0 |–16. 0 | | 4 |40 |+42. 6 |? –2. 6 |+31. 4 |? +8. 6 | | 5 |45 |+42. 1 |? –2. 9 |+36. 6 |? +8. | | 6 |35 |+42. 7 |? –7. 7 |+41. 6 |? –6. 6 | | 7 |20 |+41. 1 |–21. 1 |+37. 6 |–17. 6 | | 8 |30 |+36. 9 |? –6. 9 |+27. 1 |? +2. 9 | | 9 |35 |+35. 5 |? –0. 5 |+28. 8 |? +6. 2 | |10 |20 |+35. 4 |–15. 4 |+32. 5 |–12. 5 | |11 |15 |+32. 3 |–17. 3 |+25. 0 |–10. 0 | |12 |40 |+28. 9 |+11. 1 |+19. 0 |+21. 0 | |13 |55 |+31. 1 |+23. 9 |+31. 6 |+23. 4 | |14 |35 |+35. 9 |? 0. 9 |+45. 6 |–10. 6 | |15 |25 |+36. 7 |–10. 7 |+39. 3 |–14. 3 | |16 |55 |+33. 6 |+21. 4 |+30. 7 |+24. 3 | |17 |55 |+37. 8 |+17. 2 |+45. 3 |? +9. 7 | |18 |40 |+41. 3 |? –1. 3 |+51. 1 |–11. 1 | |19 |35 |+41. 0 |? –6. 0 |+44. 4 |? –9. 4 | |20 |60 |+39. 8 |+20. 2 |+38. 8 |+21. 2 | |21 |75 |+43. 9 |+31. 1 |+51. 5 |+23. 5 | |22 |50 |+50. 1 |? –0. 1 |+65. 6 |–15. | |23 |40 |+50. 1 |–10. 1 |+56. 2 |–16. 2 | |24 |65 |+48. 1 |+16. 9 |+46. 5 |+18. 5 | |25 | |+51. 4 | |+57. 6 | | | | |MAD = 11. 8 |MAD = 13. 45 | (c)? Students should note how stable the smoothed values are for ( = 0. 2. When compared to actual week 25 calls of 85, the smoothing constant, ( = 0. 6, appears to do a slightly better job. On the basis of the standard error of the estimate and the MAD, the 0. 2 constant is better. However, other smoothing constants need to be examined. |4. 4 | | | | | | |Week |Actual Value |Smoothed Value |Trend Estimate |Forecast |Forecast | |t |At |Ft (( = 0. 3) |Tt (( = 0. 2) |FITt |Error | |? 1 |50. 000 |50. 000 |? 0. 000 |50. 000 | 0. 000 | |? 2 |35. 000 |50. 000 |? 0. 000 |50. 000 |–15. 000 | |? 3 |25. 000 |45. 500 |–0. 900 |44. 600 |–19. 600 | |? 4 |40. 000 |38. 720 |– 2. 076 |36. 644 | 3. 56 | |? 5 |45. 000 |37. 651 |–1. 875 |35. 776 | 9. 224 | |? 6 |35. 000 |38. 543 |–1. 321 |37. 222 |? –2. 222 | |? 7 |20. 000 |36. 555 |–1. 455 |35. 101 |–15. 101 | |? 8 |30. 000 |30. 571 |–2. 361 |28. 210 | 1. 790 | |? 9 |35. 000 |28. 747 |–2. 253 |26. 494 | 8. 506 | |10 |20. 000 |29. 046 |–1. 743 |27. 03 |? –7. 303 | |11 |15. 000 |25. 112 |–2. 181 |22. 931 |? –7. 931 | |12 |40. 000 |20. 552 |–2. 657 |17. 895 |? 22. 105 | |13 |55. 000 |24. 526 |–1. 331 |23. 196 |? 31. 804 | |14 |35. 000 |32. 737 |? 0. 578 |33. 315 | 1. 685 | |15 |25. 000 |33. 820 |? 0. 679 |34. 499 |? –9. 499 | |16 |55. 000 |31. 649 |? 0. 109 |31. 58 |? 23. 242 | |17 |55. 000 |38. 731 |? 1. 503 |40. 234 |? 14. 766 | |18 |40. 000 |44. 664 |? 2. 389 |47. 053 |? –7. 053 | |19 |35. 000 |44. 937 |? 1. 966 |46. 903 |–11. 903 | |20 |60. 000 |43. 332 |? 1. 252 |44. 584 |? 15. 416 | |21 |75. 00 0 |49. 209 |? 2. 177 |51. 386 |? 23. 614 | |22 |50. 000 |58. 470 |? 3. 94 |62. 064 |–12. 064 | |23 |40. 000 |58. 445 |? 2. 870 |61. 315 |–21. 315 | |24 |65. 000 |54. 920 |? 1. 591 |56. 511 | 8. 489 | |25 | |59. 058 |? 2. 100 |61. 158 | | To evaluate the trend adjusted exponential smoothing model, actual week 25 calls are compared to the forecasted value. The model appears to be producing a forecast approximately mid-range between that given by simple exponential smoothing using ( = 0. 2 and ( = 0. 6.Trend adjustment does not appear to give any significant improvement. 4. 45 |Month |At |Ft ||At – Ft | |(At – Ft) | |May |100 |100 | 0 | 0 | |June | 80 |104 |24 |–24 | |July |110 | 99 |11 |11 | |August |115 |101 |14 |14 | |September |105 |104 | 1 | 1 | |October |110 |104 |6 |6 | |November |125 |105 |20 |20 | December |120 |109 |11 |11 | | | | |Sum: 87 |Sum: 39 | |4. 46 (a) | |X |Y |X2 |Y2 |XY | | |? 421 |? 2. 90 |? 177241 | 8. 41 |? 1220. 9 | | |? 377 | ? 2. 93 |? 142129 | 8. 58 |? 1104. 6 | | |? 585 |? 3. 00 |? 342225 | 9. 00 |? 1755. 0 | | |? 690 |? 3. 45 |? 476100 |? 11. 90 |? 2380. 5 | | |? 608 |? 3. 66 |? 369664 |? 13. 40 |? 2225. 3 | | |? 390 |? 2. 88 |? 52100 | 8. 29 |? 1123. 2 | | |? 415 |? 2. 15 |? 172225 | 4. 62 | 892. 3 | | |? 481 |? 2. 53 |? 231361 | 6. 40 |? 1216. 9 | | |? 729 |? 3. 22 |? 531441 |? 10. 37 |? 2347. 4 | | |? 501 |? 1. 99 |? 251001 | 3. 96 | 997. 0 | | |? 613 |? 2. 75 |? 375769 | 7. 56 |? 1685. 8 | | |? 709 |? 3. 90 |? 502681 |? 15. 21 |? 2765. 1 | | |? 366 |? 1. 60 |? 133956 | 2. 56 | 585. 6 | | |Column |6885 | |36. 6 | | | |totals | | | | | |January |400 |— |— | — |— | |February |380 |400 |— |20. 0 |— | |March |410 |398 |— |12. 0 |— | |April |375 | 399. 2 |396. 67 |24. 2 |21. 67 | |May |405 | 396. 8 |388. 33 |8. 22 |16. 67 | | | | |MAD = | |16. 11| | |19. 17| | (d)Note that Amit has more forecast observations, while Barbara’s moving average does not start until month 4. Also note that the MAD for Amit is an average of 4 numbers, while Barbara’s is only 2. Amit’s MAD for exponential smoothing (16. 1) is lower than that of Barbara’s moving average (19. 17). So his forecast seems to be better. 4. 48? (a) |Quarter |Contracts X |Sales Y |X2 |Y2 |XY | |1 |? 153 |? 8 |? 23,409 |? 64 |? 1,224 | |2 |? 172 |10 |? 29,584 |100 |? 1,720 | |3 |? 197 |15 |? 38,809 |225 |? 2,955 | |4 |? 178 |? 9 |? 31,684 |? 81 |? 1,602 | |5 |? 185 |12 |? 34,225 |144 |? 2,220 | |6 |? 199 |13 |? 39,601 |169 |? 2,587 | |7 |? 205 |12 |? 42,025 |144 |? ,460 | |8 |? 226 |16 |? 51,076 |256 |? 3,616 | |Totals | | 1,515 | | |95 | b = (18384 – 8 ( 189. 375 ( 11. 875)/(290,413 – 8 ( 189. 375 ( 189. 375) = 0. 1121 a = 11. 875 – 0. 1121 ( 189. 375 = –9. 3495 Sales ( y) = –9. 349 + 0. 1121 (Contracts) (b) [pic] 4. 49? (a) |Method ( Exponential Smoothing | | | |0. 6 = ( | | | |Year |Deposits (Y) |Forecast ||E rror| |Error2 | | 1 |? 0. 25 |0. 25 |0. 00 |? 0. 00 | | 2 |? . 24 |0. 25 |0. 01 |? 0. 0001 | | 3 |? 0. 24 |0. 244 |0. 004 |? 0. 0000 | | 4 |? 0. 26 |0. 241 |0. 018 |? 0. 0003 | | 5 |? 0. 25 |0. 252 |0. 002 |? 0. 00 | | 6 |? 0. 30 |0. 251 |0. 048 |? 0. 0023 | | 7 |? 0. 31 |0. 280 |0. 029 |? 0. 0008 | | 8 |? 0. 32 |0. 298 |0. 021 |? 0. 0004 | | 9 |? 0. 24 |0. 311 |0. 071 |? 0. 0051 | |10 |? 0. 26 |0. 68 |0. 008 |? 0. 0000 | |11 |? 0. 25 |0. 263 |0. 013 |? 0. 0002 | |12 |? 0. 33 |0. 255 |0. 074 |? 0. 0055 | |13 |? 0. 50 |0. 300 |0. 199 |? 0. 0399 | |14 |? 0. 95 |0. 420 |0. 529 |? 0. 2808 | |15 |? 1. 70 |0. 738 |0. 961 |? 0. 925 | |16 |? 2. 30 |1. 315 |0. 984 |? 0. 9698 | |17 |? 2. 80 |1. 906 |0. 893 |? 0. 7990 | |18 |? 2. 80 |2. 442 |0. 357 |? 0. 278 | |19 |? 2. 70 |2. 656 |0. 043 |? 0. 0018 | |20 |? 3. 90 |2. 682 |1. 217 |? 1. 4816 | |21 |? 4. 90 |3. 413 |1. 486 |? 2. 2108 | |22 |? 5. 30 |4. 305 |0. 994 |? 0. 9895 | |23 |? 6. 20 |4. 90 |1. 297 |? 1. 6845 | |24 |? 4. 10 |5. 680 |1. 580 |? 2. 499 | |25 |? 4. 50 |4. 732 |0. 232 |? 0. 0540 | |26 |? 6. 10 |4. 592 |1. 507 |? 2. 2712 | |27 |? 7. 0 |5. 497 |2. 202 |? 4. 8524 | |28 |10. 10 |6. 818 |3. 281 |10. 7658 | |29 |15. 20 |8. 787 |6. 412 |41. 1195 | (Continued) 4. 49? (a)? (Continued) |Method ( Exponential Smoothing | | | |0. 6 = ( | | | |Year |Deposits (Y) |Forecast ||Error| |Error2 | |30 |? 18. 10 |12. 6350 | 5. 46498 |29. 8660 | |31 |? 24. 10 |15. 9140 |8. 19 |67. 01 | |32 |? 25. 0 |20. 8256 |4. 774 |22. 7949 | |33 |? 30. 30 |23. 69 | 6. 60976 |43. 69 | |34 |? 36. 00 |27. 6561 | 8. 34390 |69. 62 | |35 |? 31. 10 |32. 6624 | 1. 56244 | 2. 44121 | |36 |? 31. 70 |31. 72 | 0. 024975 | 0. 000624 | |37 |? 38. 50 |31. 71 |6. 79 |? 46. 1042 | |38 |? 47. 90 |35. 784 |12. 116 |146. 798 | |39 |? 49. 10 |43. 0536 |6. 046 |36. 56 | |40 |? 55. 80 |46. 814 | 9. 11856 | 83. 1481 | |41 |? 70. 10 |52. 1526 |17. 9474 |322. 11 | |42 |? 70. 90 |62. 9210 | 7. 97897 |63. 66 | |43 |? 79. 10 |67. 7084 |11. 3916 |129. 768 | |44 |? 94. 0 0 |74. 5434 | 19. 4566 | 378. 561 | |TOTALS | |787. 30 | | | |150. 3 | | |1,513. 22 | |AVERAGE | 17. 8932 | | 3. 416 | 34. 39 | | | | |(MAD) |(MSE) | |Next period forecast = 86. 2173 |Standard error = 6. 07519 | Method ( Linear Regression (Trend Analysis) | |Year |Period (X) |Deposits (Y) |Forecast |Error2 | |? 1 |? 1 |0. 25 |–17. 330 |309. 061 | |? 2 |? 2 |0. 24 |–15. 692 |253. 823 | |? 3 |? 3 |0. 24 |–14. 054 |204. 31 | |? 4 |? 4 |0. 26 |–12. 415 |160. 662 | |? 5 |? 5 |0. 25 |–10. 777 |121. 594 | |? 6 |? 6 |0. 30 |? –9. 1387 |89. 0883 | |? 7 |? 7 |0. 31 |? –7. 50 |61. 0019 | |? 8 |? 8 |0. 32 |? –5. 8621 |38. 2181 | |? |? 9 |0. 24 |? –4. 2238 |19. 9254 | |10 |10 |0. 26 |? –2. 5855 |8. 09681 | |11 |11 |0. 25 |? –0. 947 |1. 43328 | |12 |12 |0. 33 |? 0. 691098 |0. 130392 | |13 |13 |0. 50 |? 2. 329 |3. 34667 | |14 |14 |0. 95 |? 3. 96769 |9. 10642 | |15 |15 |1. 70 |? 5. 60598 |15. 2567 | |16 |16 |2. 30 |? 7. 24 427 |24. 4458 | |17 |17 |2. 0 |? 8. 88257 |36. 9976 | |18 |18 |2. 80 |? 10. 52 |59. 6117 | |19 |19 |2. 70 |? 12. 1592 |89. 4756 | |20 |20 |3. 90 |? 13. 7974 |97. 9594 | |21 |21 |4. 90 |? 15. 4357 |111. 0 | |22 |22 |5. 30 |? 17. 0740 |138. 628 | |23 |23 |6. 20 |? 18. 7123 |156. 558 | |24 |24 |4. 10 |? 20. 35 |264. 083 | |25 |25 |4. 50 |? 21. 99 |305. 62 | |26 |26 |6. 10 |? 23. 6272 |307. 203 | |27 |27 |7. 70 |? 25. 2655 |308. 547 | |28 |28 |10. 10 |? 26. 9038 |282. 367 | |29 |29 |15. 20 |? 28. 5421 |178. 011 | |30 |30 |18. 10 |? 30. 18 |145. 936 | |31 |31 |24. 10 |? 31. 8187 |59. 58 | |32 |32 |25. 60 |? 33. 46 |61. 73 | |33 |33 |30. 30 |? 35. 0953 |22. 9945 | |34 |34 |36. 0 |? 36. 7336 |0. 5381 | |35 |35 |31. 10 |? 38. 3718 |52. 8798 | |36 |36 |31. 70 |? 40. 01 |69. 0585 | |37 |37 |38. 50 |? 41. 6484 |9. 91266 | |38 |38 | 47. 90 |? 43. 2867 |21. 2823 | |39 | 39 |49. 10 |? 44. 9250 |17. 43 | |40 | 40 |55. 80 |? 46. 5633 |? ? 85. 3163 | |41 | 41 |70. 10 |? 48. 2016 |? 479. 54 | |42 | 4 2 |70. 90 |? 49. 84 |? 443. 28 | |43 | 43 |79. 10 |? 51. 4782 |? 762. 964 | |44 | 44 |94. 00 |? 53. 1165 | 1,671. 46 | |TOTALS | |990. 00 | | |787. 30 | | | | | | | | | | | | | |7,559. 95 | | |AVERAGE |22. 50 | 17. 893 | |171. 817 | | | | | |(MSE) | |Method ( Least squares–Simple Regression on GSP | | |a |b | | | | |–17. 636 |13. 936 | | | | |Coefficients: |GSP |Deposits | | | | |Year |(X) |(Y) |Forecast ||Error| |Error2 | |? 1 |0. 40 |? 0. 25 |–12. 198 |? 12. 4482 |? 154. 957 | |? 2 |0. 40 |? 0. 24 |–12. 198 |? 12. 4382 |? 154. 71 | |? 3 |0. 50 |? 0. 24 |–10. 839 |? 11. 0788 |? 122. 740 | |? 4 |0. 70 |? 0. 26 |–8. 12 | 8. 38 | 70. 226 | |? 5 |0. 90 |? 0. 25 |–5. 4014 | 5. 65137 | 31. 94 | |? 6 |1. 00 |? 0. 30 |–4. 0420 | 4. 342 | 18. 8530 | |? 7 |1. 40 |? 0. 31 |? 1. 39545 | 1. 08545 | 1. 17820 | |? 8 |1. 70 |? 0. 32 |? 5. 47354 | 5. 5354 | 26. 56 | |? 9 |1. 30 |? 0. 24 |? 0. 036086 | 0. 203914 | 0. 041581 | |10 |1. 20 |? 0. 2 6 |–1. 3233 | 1. 58328 | 2. 50676 | |11 |1. 10 |? 0. 25 |–2. 6826 | 2. 93264 | 8. 60038 | |12 |0. 90 |? 0. 33 |–5. 4014 | 5. 73137 | 32. 8486 | |13 |1. 20 |? 0. 50 |–1. 3233 | 1. 82328 | 3. 32434 | |14 |1. 20 |? 0. 95 |–1. 3233 | 2. 27328 | 5. 16779 | |15 |1. 20 |? 1. 70 |–1. 3233 | 3. 02328 | 9. 14020 | |16 |1. 60 |? 2. 30 |? 4. 11418 | 1. 81418 | 3. 9124 | |17 |1. 50 |? 2. 80 |? 2. 75481 | 0. 045186 | 0. 002042 | |18 |1. 60 |? 2. 80 |? 4. 11418 | 1. 31418 | 1. 727 | |19 |1. 70 |? 2. 70 |? 5. 47354 | 2. 77354 | 7. 69253 | |20 |1. 90 |? 3. 90 |? 8. 19227 | 4. 29227 | 18. 4236 | |21 |1. 90 |? 4. 90 |? 8. 19227 | 3. 29227 | 10. 8390 | |22 |2. 30 |? 5. 30 |13. 6297 | 8. 32972 | 69. 3843 | |23 |2. 50 |? 6. 20 |16. 3484 |? 10. 1484 |? 102. 991 | |24 |2. 80 |? 4. 10 |20. 4265 |? 16. 3265 |? 266. 56 | |25 |2. 90 |? 4. 50 |21. 79 |? 17. 29 |? 298. 80 | |26 |3. 40 |? 6. 10 |28. 5827 |? 22. 4827 |? 505. 473 | |27 |3. 80 |? 7. 70 |34. 02 |? 26. 32 |? 6 92. 752 | |28 |4. 10 |10. 10 |38. 0983 |? 27. 9983 |? 783. 90 | |29 |4. 00 |15. 20 |36. 74 |? 21. 54 |? 463. 924 | |30 |4. 00 |18. 10 |36. 74 |? 18. 64 |? 347. 41 | |31 |3. 90 |24. 10 |35. 3795 |? 11. 2795 |? 127. 228 | |32 |3. 80 |25. 60 |34. 02 | 8. 42018 | 70. 8994 | |33 |3. 0 |30. 30 |34. 02 | 3. 72018 | 13. 8397 | |34 |3. 70 |36. 00 |32. 66 | 3. 33918 | 11. 15 | |35 |4. 10 |31. 10 |38. 0983 | 6. 99827 | 48. 9757 | |36 |4. 10 |31. 70 |38. 0983 | 6. 39827 |? 40. 9378 | |37 |4. 00 |38. 50 |36. 74 | 1. 76 | 3. 10146 | |38 |4. 50 |47. 90 |43. 5357 | 4. 36428 | 19. 05 | |39 |4. 60 |49. 10 |44. 8951 | 4. 20491 | 17. 6813 | |40 |4. 50 |55. 80 |43. 5357 |? 12. 2643 |? 150. 412 | |41 |4. 60 |70. 10 |44. 951 |? 25. 20 |? 635. 288 | |42 |4. 60 |70. 90 |44. 8951 |? 26. 00 |? 676. 256 | |43 |4. 70 |79. 10 |46. 2544 |? 32. 8456 |1,078. 83 | |44 |5. 00 |94. 00 |50. 3325 |? 43. 6675 |1,906. 85 | |TOTALS | | | |451. 223 |9,016. 45 | |AVERAGE | | | |? 10. 2551 |? 204. 92 | | | | | |? (MAD) |? (MS E) | Given that one wishes to develop a five-year forecast, trend analysis is the appropriate choice. Measures of error and goodness-of-fit are really irrelevant.Exponential smoothing provides a forecast only of deposits for the next year—and thus does not address the five-year forecast problem. In order to use the regression model based upon GSP, one must first develop a model to forecast GSP, and then use the forecast of GSP in the model to forecast deposits. This requires the development of two models—one of which (the model for GSP) must be based solely on time as the independent variable (time is the only other variable we are given). (b)? One could make a case for exclusion of the older data. Were we to exclude data from roughly the first 25 years, the forecasts for the later year