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Lecture note 2

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Explain the law of diminishing return Solution Law of diminishing returns is an economic law which states that as one unit of input is increased (all other inputs being constant), the output will increase at an increasing rate intially but a point will eventually be reached when these additions of the input will yield diminishing increases in output. In other words, the output will increase but at a diminishing rate. Let us take an example of a chocolate factory and let the variable input be labor hours. As it can be seen from the above table, earlier as the labor hours increased, the chocolates produced increased at an increasing rate till 5 hours. From the 6th hour of additional labor, the marginal chocolates produced started decreasing which shows that the production increased at a diminishing rate. This is what is stated by the Law of Diminishing Returns.Labor HoursChocolates ProducedMarginal chocolates produced15521163198432135491766314774118828987510892.

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- Explain the logic behind the least-squares regression solution. What is the mean goal of analysis? How are errors expressed? Solution The method of least squares is a standard approach to the approximate solution of overdetermined systems, i.e., sets of equations in which there are more equations than unknowns. "Least squares" means that the overall solution minimizes the sum of the squares of the errors made in the results of every single equation. The most important application is in data fitting. The best fit in the least-squares sense minimizes the sum of squared residuals, a residual being the difference between an observed value and the fitted value provided by a model. When the problem has substantial uncertainties in the independent variable (the 'x' variable), then simple regression and least squares methods have problems; in such cases, the methodology required for fitting errors-in-variables models may be considered instead of that for least squares. The knowledge we have of the physical world is obtained by doing experiments and making measurements. It is important to understand how to express such data and how to analyze and draw meaningful conclusions from it. In doing this it is crucial to understand that all measurements of physical quantities are subject to uncertainties. It is never possible to measure anything exactly. It is good, of course, to make the error as small as possible but it is always there. And in order to draw valid conclusions the error must be indicated and dealt with properly. Take the measurement of a person's height as an example. Assuming that her height has been determined to be 5' 8", how accurate is our result? Well, the height of a person depends on how straight she stands, whether she just got up (most people are slightly taller when getting up from a long rest in horizontal position), whether she has her shoes on, and how long her hair is and how it is made up. These inaccuracies could all be called errors of definition. A quantity such as height is not exactly defined without specifying many other circumstances. Even if you could precisely specify the "circumstances," your result would still have an error associated with it. The scale you are using is of limited accuracy; when you read the scale, you may have to estimate a fraction between the marks on the scale, etc. If the result of a measurement is to have meaning it cannot consist of the measured value alone. An indication of how accurate the result is must be included also. Indeed, typically more effort is required to determine the error or uncertainty in a measurement than to perform the measurement itself. Thus, the result of any physical measurement has two essential components: (1) A numerical value (in a specified system of units) giving the best estimate possible of the quantity measured, and (2) the degree of uncertainty associated with this estimated value. For example, a measurement of the width of a table would yield a result such as 95.3 +/- 0.1 cm.

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