Finite Mathematics

Introduction and Problem Statement

The population growth among countries around the globe tends to become the problem. Due to the increasing number of population, the needs of the people were also increasing. Aside from this, the available resources to suffice these needs are terribly declining. The current financial crisis of America proves that there is a need to predict and estimate the number of population as the year pass by. In this way, the country can create some measures that will counter the possible problem that is cause by the increasing number of population.

Basically, data or information classified from linear programming and on the basis of intervals of time constitute vital information in the control of an activity, since this is the most effective method of showing the changes that are taking place in the population of United States of America. Closely related to the problem of measuring changes in population growth is the making of forecast of future needs. The management of operation requires a continual making of decisions regarding the future and the basis for such forecasts is the record of the past performance.

Data on population growth is important in determining the possible future of a certain country (Hobbs F. and Stoops N, 19). The said data are of interests chiefly in order that the figures for one period maybe compared with similar figures for other figures. When observations of this kind are arranged in a time sequence and separated by (or represent) more or less regular intervals of time (e.g. years), the progression of values is known as a time series. The concept of trend in economic time series rests in large part upon the secular growth of population, capital and resources. On the other hand, this paper attempts to construct a model that suits to the forecast of the possible future numbers of the population (e.g. 2010 and 2020) of the United States of America in accordance to their previous population records (i.e. from 1900 to 2000). For these reasons, it is necessary, as in so many other fields of experimental, empirical knowledge, for the forecaster to use the behaviour of cycles of population to predict the future whether or not he fully understands the causes of the behaviour we uses. Thus, the main goal of this paper is to identify the predicted number of population in United States of America for the years 2010 and 2020.

Methods and Approach

For a study which seeks future status, the trend analysis will be the most appropriate method to use. As stated in the book of Creswell, “it is employed in studies which aim to project the numbers, demands or needs of the people in the future”(p.12). To help estimate the rate and the direction of changes in the future, surveys repeated at intervals may be used. Basically, this paper will be using the census data of United States of America from 1900 to 2000. The data is also available online through http://www.census.gov. Actually, the census covers a number of population where the variables are concrete. For this reason, the responses are simple and accurate. There is less contradiction in our data since the variable we oath to measure are well defined and clear (i.e. US populations). With regards to methods in trend analysis, this paper will be using “The Method of Least Squares” to predict the population of America in 2010 and 2020.

In “The Method of Least Squares”, the strait line trend is assured and the line trend will have a formula of the type:

In this formula the value of a and b must be determined. The principle of least squares states that a trend best fit of a given set of values when the constants of the equation are chosen so that the sum of the squares of the deviations between the original data and the corresponding trend values are a minimum. If the line fits the data perfectly, each point will lie on the line and the sum of the squares of the deviations will be zero. The farther the points are from the trend, the larger will the deviation, although the line may still be fitted so that these squared deviations will be a minimum.

In order to find these values for a and b by the method of least squares, it is necessary to solve the following normal equations in which time is designated by x and the values in the series by y.

The procedure may be simplified by arbitrarily shifting the origin of the series (in the middle year for an odd number of years and between the two centers for an even number of years) in this way so that, the preceding equations can be reduced to:

After the values of a and b are determined, it then becomes possible with the use of the equation, to compute the actual trend values.

Applications

With respect to previous method of analysis, the following set data are gathered in order to determine the possible population of United States of America in year 2010 and 2020.

Table 1. Data

Year	x	Population (y)	x2	xy
1900	-5	75,994,575	25	-379972875
1910	-4	91,972,266	16	-367889064
1920	-3	105,710,620	9	-317131860
1930	-2	122,775,046	4	-245550092
1940	-1	131,669,275	1	-131669275
1950	0	150,697,361	0	0
1960	1	179,323,175	1	179323175
1970	2	203,211,926	4	406423852
1980	3	226,545,805	9	679637415
1990	4	248,709,873	16	994839492
2000	5	281,421,906	25	1407109530
	0	1,818,031,828	110	2,225,120,298

Source: http://www.census.gov

Using these values, we can now compute for the values of a and b. Thus we have:

The equation will then be read:

Trend of population growth in United States of America 1900-2000

where: y is the number of population

x is the deviation of years

Figure 1. Population Trend

Using the equation, we can now predict the population of United States of America in 2010 and 2020 by setting the values of x to 6 and 7. Thus we have:

For 2010:

For 2020:

Conclusion

Similar to probability, forecasting, by professional and business man alike, is too frequently a guessing game. Even when forecasters agree, they are apt to reach their common conclusion by different methods and for different reasons. And when they happen to be right, they are frequently right because of reasons or conditions they did not anticipate.

These critical observations are written with no disparaging or pharisaical implications. These difficulties are inherent in the art and beset everyone who attempts to move ahead of time and to pierce the veil that shrouds the future. There is no infallible forecasting system. Unorganized forecasting is usually the product of personal judgment or intuition or, sometimes, only a subconscious feeling for the course of future events. It is more art than science, and it will re- main in this unsatisfactory state until its methods can be brought into the realm of the rational and can be based on logical relationships that govern business behaviour and can be stated in measurable terms. General progress in forecasting will come only with the wider understanding and application of the common sense economic principles that govern the fluctuations of aggregate national income, production, and prices. This wider understanding is one of the objectives of this volume. It is hoped that this study will bring to the businessman and to other students of forecasting an understanding of some of these principles and their application to business forecasting. With regards to the findings of the data gathered, the suggested model to be used in order to forecast the population of United States of America is. As stated, the results are based from the least square method.

References:

Creswell, JW Research design. Qualitative and quantitative approaches. Thousand Oaks, California: Sage. 1994.

Hobbs F. and Stoops N. Demographic Trends in the 20th Century. Census 2000 Special Reports, 2002.