Determining and Forecasting The Growth Rate of Beef Cattle

**Introduction **

** **Mathematical models and formulas play a vital role in determining the growth in a given set of data. By using these models firms operating the business of rearing beef cattle forecast the growth rate of beef cattle to enable them plan for the resources essential for the rearing. The preceding chapters will focus on mathematical formulas to predict the growth rate of beef cattle such as linear regression modeling and Lagrange interpolation formula.

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The purpose of choosing this topic is that; it is essential for beef rearing firms to forecast the growth of beef cattle. Forecasting is necessary for the business because it will enable the firm estimate the financial requirements, correctly lay out the plan formulation and have control of the firm’s operations. The preceding chapters also use a hypothetical data of growth of Angus- which is a commonly known type of beef cattle.

This paper will enlighten on methods used for forecasting, how to use these methods, by effectively doing the accurate calculation and also how to interpret the results. Also, to enlighten on disadvantages and advantages of using each method and at what instance is suitable to use each method.

**Linear regression model **

Linear regression is the process of using a statistical approach to find an association between two or more variables, in which one variable is dependent on other independent variables. Regression analysis is the method to discover the relationship between one or more response variables (also called dependent variables, explained variables, predicted variables, or regressands, usually denoted by y), and the predictors (also called independent variables, explanatory variables, control variables, or regressors, usually denoted by x_{1}, x_{2},…x_{p}), (Yan and Su).

y= β_{0 + }β_{1 + }ε

The parameters β_{0} and β_{1 }are usually called regression coefficients. These coefficients have a simple and often useful interpretation. The slope β_{1} is the change in the mean of the distribution of y produced by a unit change in x, (Montgomery).

The intercept of the line is the parameter β_{0 }while the parameter ε is the error associated with the estimation.

Just like a straight line whose gradient and the intercept is known, someone can find the value of y- axis having any point of x. Linear regression model forecasts the data if the intercept (β_{0}) and the gradient (β_{1} ) of the model are known. The equation of least square regression model is given by

.

whereby

=

=

For example, the table below shows the growth rate Angus ( a type of beef cattle).

Age (months) | 0 | 6 | 12 | 18 |

Weight(Kilograms) | 40 | 150 | 280 | 430 |

The growth of the cattle in the twenty-seventh month can be estimated using linear regression model. To solve the problem above;

One, identify the response variable (y) and predictor variables (x). In the above case, age is the predictor variable and weight are the response variable. Two obtain the mean of the variables

Mean = where x_{i} is the individual value of the variable x, and n is the total number of observations.

Mean of age = = 9 months, mean weight = = 225

Three, find the parameters of ^{2.} Using a table to find the following results

X (age in months) | Y (weight in Kilograms) | ^{2} |
|||

0 | 40 | -9 | -185 | 1665 | 81 |

6 | 150 | -3 | -75 | 225 | 9 |

12 | 280 | 3 | 55 | 165 | 9 |

18 | 430 | 9 | 205 | 1845 | 81 |

Total |
0 |
0 |
3900 |
180 |

= = =

= = =

The linear regression equation is given by

=

To find the growth in weight of the Angus beef cattle in the twenty-seventh month, substitute the x in the linear regression equation. For example

= kilograms.

This method helps the beef cattle rearing firm to predict the weight of the cattle. Enabling the firm to determine when it is best to slaughter the cow based on the cost incurred when bringing up it. Linear regression analysis enables investigators to predict the value of y based on the value that x take, (Jekel). Hence useful for forecasting of a given set of quantitative data.

However, linear regression is limiting only to data that is linear and also predicts continuous variables which are different than what was asked of participants in the study, (Dowling). Furthermore, linear regression analysis does not express the association between qualitative data, hence does not forecast categorical data.

** **

**Lagrange Interpolation Formulae **

A Lagrange interpolation formula is a method of obtaining a polynomial equation of degree n from a given set of points in data. A formula for obtaining polynomials of degree n (the Lagrange interpolation polynomial) that interpolates a given function f(x) at nodes , (Hazewinkel). First, definition of basic terms used in Lagrange interpolation formulae will broaden the understanding in this method. A polynomial is a mathematical expression that contains more than one variable and coefficients. Interpolate is construction of new data points within a subset of known discrete data. Generally, Lagrange interpolation formulae is

(x – x_{1}) . . . (x – x_{n}) |
(x – x_{0}) . . . (x – x_{n-1}) |
|||

f(x) = | f_{0 }+ . . . + |
f_{n} |
||

(x_{0 }– x_{1}) . . . (x_{0 }– x_{n}) |
(x_{n }– x_{0}) . . . (x_{n} – x_{n-1}) |

** **

n | ( | n |
)f_{i} |
|||

S | | | | x – x_{j} |
||||

j = 0 | (x_{i} – x_{j}) |
|||||

i = 0 | j ¹ 1 |

Obtaining a polynomial equation will be used to forecast the growth of beef cattle, Angus. Below is a line graph showing the weight of Angus Against period in months.

One, from the line graph above, obtain the dataset at any point in the line, for example

X –axis | 0 | 6 | 12 | 18 |

Y –axis | 40 | 150 | 280 | 430 |

Two, using Lagrange interpolation formulae,

(430) =

The solving polynomial obtained above can predict any month’s weight a beef cattle will be having. Solving the weight of the cow in the 27^{th} month, substitute the value of x in the polynomial obtained above.

= 692.671 kilograms

An advantage of using Lagrange interpolation formulae is that the method does not require evenly spaced variables of x. Also, the method can be used to obtain high-order degree polynomials.

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**Conclusion **

Forecasting of beef cattle weight is a type of estimation; it gives a rough idea of the weight of the beef in the given future. Since it is an estimation, this explains the difference in weight in the 27^{th} month of Angus beef cattle. However, though the figures are different, they form in the same range of not less six hundred kilograms and not more than seven hundred kilograms.

In conclusion, determining the weight of the Angus beef cattle will enable the beef cattle rearing firms to plan when to give the beef cattle more pasture to add more weight. Also by forecasting the Angus weight in a given month, will help the beef cattle rearing firms identify a problem in the weight of the cattle since they will be having a rough idea of the expected weight. Forecasting is an essential tool that a beef rearing firm cannot omit in its operations.

Works Cited

Dowling, Richard G.* Multimodal level of service analysis for urban streets*__.__ Washington, D.C: Transportation Research Board, 2008.

Hazewinkel, M. *Encyclopaedia of Mathematics Volume 3: Heaps and Semi-Heaps — Moments, Method of (in Probability Theory)*. Boston: Springer, 2013.

Jekel, James F.* Epidemiology, biostatistics, and preventive medicine*__.__ Philadelphia: Saunders/Elsevier, 2007.

Montgomery, Douglas C. *Introduction to linear regression analysis*__.__ Oxford: Wiley-Blackwell, 2011.

Yan, Xin and Xiaogang Su. *Linear regression analysis: theory and computing*__.__ Singapore; Hackensack, N.J: World Scientific Pub. Co., 2009.