Estimation in Real Life

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  Stochastics

  Introduction to Probability and Statistics

  • Hans-Otto Georgii, Marcel Ortgiese, Ellen Baake, Hans-Otto Georgii, Marcel Ortgiese, Ellen Baake, Hans-Otto Georgii(Authors)
  • 2008(Publication Date)
  • De Gruyter

    (Publisher)

  7 Estimation Statistics is the art of fi nding rational answers to the following key question: Given a speci fi c phenomenon of chance, how can one uncover its underlying probability law from a number of random observations? The laws that appear to be possible are described by a family of suitable probability measures, and one intends to identify the true probability measure that governs the observations. In this chapter, we will fi rst give an overview over the basic methods and then discuss the most elementary one, the method of estimation, where one simply tries to guess the true probability measure. The problem is to fi nd reliable ways of guessing. 7.1 The Approach of Statistics Suppose you are faced with a situation governed by chance, and you make a series of observations. What can you then say about the type and the features of the underlying random mechanism? Let us look at an example. (7.1) Example. Quality control. An importer of oranges receives a delivery of N = 10 000 oranges. Quite naturally, he would like to know how many of these have gone bad. To fi nd out, he takes a sample of n = 50 oranges. A random number x of these is rotten. What can the importer then conclude about the true number r of rotten oranges? The following three procedures suggest themselves, and each of them corresponds to a fundamental statistical method. Approach 1: Naive estimation. As a rule of thumb, one would suspect that the proportion of bad oranges in the sample is close to the proportion of bad oranges in the total delivery, in other words that x / n ≈ r / N . Therefore, the importer would guess that approximately R ( x ) := N x / n oranges are rotten. That is, the number R ( x ) = N x / n (or more precisely, the nearest integer) is a natural estimate of r resulting from the observed value x . We thus come up with a mapping R that assigns to the observed value x an estimate R ( x ) of the unknown quantity.

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  The Pragmatics of Mathematics Education

  Vagueness and Mathematical Discourse

  • Tim Rowland(Author)
  • 2003(Publication Date)
  • Routledge

    (Publisher)

  estimation of the number of objects in a set. This choice of focus is partly for the sake of addressing what is perhaps the most obvious aspect of mathematical activity in which one would expect vague language to play a part. Moreover, it is possible in a short (5–10 minute) interview to present appropriate estimation tasks to children in a meaningful way, to obtain responses, and to follow these up from a restricted menu of probes. It is therefore convenient, in designing an age-related study, to use estimation rather than generalization tasks to elicit vague language when dealing with a pupil sample numbered in hundreds rather than tens.

  Estimation

  Clayton (1992, p. 11) classifies the diffuse notion of estimation into three broad categories.

  Computational estimation involves the determination of approximate (typically, mental) answers to arithmetic calculations, e.g. 97π is roughly 100×3. 1 or 310. Such competence is commended by the National Curriculum for Mathematics in England and Wales (DFE, 1995, p. 25) for the purpose of checking answers to precise calculations for their ‘reasonableness’; pupils, however, seem to regard such checks as trivial or pointless (Clayton, 1992, p. 163).

  Quantitative estimation indicates the magnitude of some continuous physical measure such as the weight of a book, the length of a stick.

  Numerical estimation entails a judgement of ‘numerosity’—the number of objects in a collection. In principle, such a set could be precisely quantified by counting. In practice such precise enumeration may be impracticable or simply judged to be unnecessary, excess to pragmatic requirements.

  Ellis (1968, p. 159) observes that counting may be considered to be a measuring procedure, unique in the non-arbitrariness of the unit of measure. In fact, Clayton merges numerical estimation and quantitative estimation into one analytical category. Sowder (1992) notes that ‘there simply is not a rich research base in estimation’ and that most such research has been on computational estimation. Moreover, ‘Numerosity estimation has received the least research attention, and […] the only two studies located combine it with measurement estimation’ (p. 372). One those two studies was reported in a short article—by Clayton himself—in Mathematics Teaching

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  Helping Children Learn Mathematics

  • Robert Reys, Mary Lindquist, Diana V. Lambdin, Nancy L. Smith(Authors)
  • 2014(Publication Date)
  • Wiley

    (Publisher)

  Estimation produces an approximate rather than an exact answer. Estimation is useful before, during, and after exact computations, to get a sense of what kind of answer to expect, to check that the computation is moving in the right direction, and to check on the reasonableness of re- sults. Teachers must help children change their exact-answer mindset so they can see the power and usefulness of esti- mation. Estimation strategies include front-end estimation, using compatible numbers, flexible rounding, and cluster- ing. Regardless of the strategy used, children should learn to adjust their result to get a better estimate. Guidelines for helping children choose and use estimation strategies in- clude giving problems that encourage and reward estima- tion, making sure students are not computing exact answers and then rounding to produce estimates, getting students to talk about how they made their estimates, helping students overcome the one-right-answer syndrome, and encourag- ing students to think of real-world situations where estima- tion is needed. Studies across cultures show that students who excel at written computation and mental computation are not necessarily good at estimation. TABLE 10-1 • Performance of Taiwanese Sixth Graders on Written Computation and Parallel Estimation Items Written Computation Items Requiring an Exact Answer Percentage Correct by Sixth Graders 12/13 7/8 61 Estimation Items Requiring a Correct Choice Without calculating an exact answer, circle the best estimate for 12/13  7/8. A. 1 10 B. 2 a C. 19 D. 21 E. I don’t know 25 36 16 13 This multiplication has been carried out correctly except for placing the decimal point: 534.6  0.545  91357 Place the decimal point using estimation. A. 29.1357 B. 291.357 a C. Other answer 87 11 2 a Denotes correct answer choice. A GLANCE AT WHERE WE’VE BEEN T eachers must help students develop confidence and skill in choosing and using different methods of computa- tion.

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  Empowering Project Teams

  Using Project Followership to Improve Performance

  • Marco Sampietro, Tiziano Villa(Authors)
  • 2014(Publication Date)
  • Auerbach Publications

    (Publisher)

  7.3 PITFALLS IN THE ESTIMATION PROCESS Some points of the estimation process must be taken into careful consideration by the team member who provides the estimates during project planning. 7.3.1 Different Degrees of Tolerable Approximation A first point that very often creates difficulties, if not even embarrassment, is the fact that all projects are based on estimates and not accurate data, and that the approximation and uncertainty of these estimates may even vary within the same project. This aspect becomes more important the more the level of approxi-mation required in the estimation process moves away from the level of approximation (or certainty) that the team member is used to in perform-ing the nonproject work activities (Sampietro 2010). For instance, those working in fields where the margin of error or accepted approximation is very low (think of mathematics, accounting, measuring with high-precision tools) may find it very difficult to have to provide estimates with very high approximation margins and perhaps in a very short time, insofar as it requires a different mindset. On the contrary, those who are used to approximation (those who work on 148 • Empowering Project Teams advertising campaigns, creative types in general, teachers, strategists, etc.) may be uneasy working on projects that require highly detailed estimates. Moreover, even within the same project there may be times when a high level of detail is required (the planning of schedules in projects with penal-ties linked to significant delays), and others where approximation can be tolerated (the first budget simulation to understand whether the project makes economic sense). Below is an example highlighting the degree of approximation within projects.

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  Estimation Theory in Hydrology and Water Systems

  • K. Nacházel(Author)
  • 1993(Publication Date)
  • Elsevier Science

    (Publisher)

  In statistics [52], such an estimation is understood to be a rule (a decision function) with the help of which the value of an unknown parameter can be estimated on the basis of the probability properties of the sample, either as a number (point estimation) or as an interval within which the unknown parameter is most likely to lie (interval estimation). A detailed analysis of the properties of samples is thus fully justified by the need to estimate the parameters on which further progress in the solution of the problem is very often dependent”). In this respect, the solution of water-engineeringproblems can be regarded as a typical case of application of the theory of estimation, because the reliability of the solution (e. g. the reliability of the determination of the design parameters of a water-engineering project) depends fully upon the properties of the hydrolo- gical conditions estimated. 9 Instead of the older and more descriptiveterm “sample of the population with distribution qx)” use is now increasingly made in the more recently published literature of a shorter term, viz. “sample of F(x) distribution”. In some experiments one sample may be characterized by two, three, or more generally, p vectors of numbers. With each repetition we can thus observe a pdimensional random vector. We then speak of a random sample of a two- to p-dimensional distribution [65]. **I In their modern conception, the problems of decision-makinghave gained considerable impor- tance in the decision theory, which is gradually taking shape on the boundaries of the classical disciplines, such as probability theory, logic, psychology, the general theory of management, and cybernetics. These problems are of especial importance in systems disciplines [67], [94]; in water engineering they are particularly applicable to design of water resource systems [ 1 151. 16

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