Statistical Problem Solving Research Articles

Designing a web-based collaborative-learning module for statistical problem solving

Designing a web-based collaborative-learning module for statistical problem solving

Design for Six Sigma: caveat emptor

PurposeThe purpose of this paper is to describe the historical approach to concurrent engineering (CE) which has resulted in product line management (PLM) and then evaluates the theoretical models that have been proposed for design for Six Sigma (DFSS) in order to determine which model is able to provide the most consistent approach with historical development of PLM.Design/methodology/approachThe approach begins with an overview of the approach taken by the Union of Japanese Scientists and Engineers (JUSE) in the development of a coherent quality methodology for structured analysis and problem solving – the Deming Wheel of plan‐do‐check‐act (PDCA) which has become the standard model in Japanese total quality management to define a logical decomposition in process management. In Japan, PDCA is the single logical model which has been broadly accepted as the construct for understanding how to develop both strategic and operational quality methods. The second step in the approach is to examine a similar American development of the model for statistical problem solving that is applied in the Six Sigma method for statistical problem solving: define‐measure‐analyze‐improve‐control (DMAIC). Next, the paper examines the historical sequence in the way the product development process has developed over the past forty years, with emphasis on its military origins (especially CE) and which resulted in the generic model for PLM. The final part of this paper examines the models that have been proposed to implement DFSS over the past ten years and evaluate their logical congruence with the engineering community's design process.FindingsProblems in alignment with the engineering design process were identified with all of the DFSS models and with the non‐structured or “heuristic” approach to developing a coherent body of knowledge related to DFSS.Originality/valueThis paper provides a challenge to the quality community as well as to the academic community. The paper points out the need for rigorous examination of logical models that are proposed for guiding the thinking of practitioners in the use of quality methods for both the engineering of products and business systems. An expose of lack of rationality in the way an approach to DFSS has been investigated calls for more responsibility in the management of the development of this body of knowledge.

Mean, Median and Mode from a Decision Perspective

The classroom activity described here allows mathematically mature students to explore the role of mean, median and mode in a decision-making environment. While students discover the importance of choosing a measure of central tendency, their understanding of probability distributions, maximization, and prediction is reinforced through active learning. The lesson incorporates the GAISE recommendations by actively engaging students in the process of statistical problem-solving in a realistic situation.

Response surface optimization of enzymatic hydrolysis of Cistus ladanifer and Cytisus striatus for bioethanol production

Current ethanol production processes using crops such as sugar cane and corn are well established; however, utilization of a cheaper substrate such as lignocellulose could make bioethanol more competitive with fossil fuel, without the ethical concerns associated with the use of potential food resources. The sequential configuration employed to obtain cellulosic ethanol implies that the solid fraction of pretreated lignocellulosic material undergoes hydrolysis. In this work, the enzymatic hydrolysis of pretreated Cistus ladanifer and Cytisus striatus was studied following an experimental design as a statistical problem solving approach. Plackett–Burman design was used in order to select the most important variables from the simultaneous study on influence of operating and reactional conditions, and central composite design to optimize the process of enzymatic hydrolysis. The optimization of enzymatic hydrolysis using the response surface methodology allowed a study on the influence of the variables (pH, temperature, cellulases concentration, polymer (PEG) concentration and incubation time) and variability due to the type of substrate ( C. ladanifer and C. striatus) used. From the obtained results it can be concluded that the enzymatic hydrolysis was clearly enhanced by temperature, cellulase concentration, and incubation time. Model validation showed a good agreement between experimental results and the predicted responses.

Teaching, Learning and Assessing Statistical Problem Solving

In this paper we report the results from a major UK government-funded project, started in 2005, to review statistics and handling data within the school mathematics curriculum for students up to age 16. As a result of a survey of teachers we developed new teaching materials that explicitly use a problem-solving approach for the teaching and learning of statistics through real contexts. We also report the development of a corresponding assessment regime and how this works in the classroom.Controversially, in September 2006 the UK government announced that coursework was to be dropped for mathematics exams sat by 16-year-olds. A consequence of this decision is that areas of the curriculum previously only assessed via this method will no longer be assessed. These include the stages of design, collection of data, analysis and reporting which are essential components of a statistical investigation. The mechanism outlined here could provide some new and useful ways of coupling new teaching methods with learning and doing assessment — in short, they could go some way towards making up for the educational loss of not doing coursework. Also, our findings have implications for teaching, learning and assessing statistics for students of the subject at all ages.

Iconic Representations and Statistical Problem Solving: The Case of Greek Pupils

Iconic Representations and Statistical Problem Solving: The Case of Greek Pupils

Selection and use of propositional knowledge in statistical problem solving

Central to this study is the question of why subjects who possess the necessary factual or propositional knowledge needed to solve a particular statistical problem, often fail to find the solution to that problem. Ten undergraduate psychology students were trained so as to possess all the relevant knowledge needed to solve five multiple choice problems on descriptive regression analysis. They were asked to think aloud while attempting to solve the problems. Analysis of the think-aloud protocols showed that a failure to select the relevant information in the text, together with a failure to retrieve relevant propositional knowledge from memory and a difficulty with logical reasoning combined to produce incorrect responses. Factual knowledge was less likely to be successfully retrieved when it was acquired only recently or when it concerned relationships of a highly abstract nature. Furthermore, the existence of misconceptions appeared to inhibit the use of correct factual knowledge.

Statistical Thinking in Empirical Enquiry

SummaryThis paper discusses the thought processes involved in statistical problem solving in the broad sense from problem formulation to conclusions. It draws on the literature and in‐depth interviews with statistics students and practising statisticians aimed at uncovering their statistical reasoning processes. From these interviews, a four‐dimensional framework has been identified for statistical thinking in empirical enquiry. It includes an investigative cycle, an interrogative cycle, types of thinking and dispositions. We have begun to characterise these processes through models that can be used as a basis for thinking tools or frameworks for the enhancement of problem‐solving. Tools of this form would complement the mathematical models used in analysis and address areas of the process of statistical investigation that the mathematical models do not, particularly areas requiring the synthesis of problem‐contextual and statistical understanding. The central element of published definitions of statistical thinking is “variation”. We further discuss the role of variation in the statistical conception of real‐world problems, including the search for causes.

Pennies from Heaven—Nickels from Where?

Major efforts to increase the attention that statistics receives in school mathematics have been under way since the appearance of NCTM's Curriculum and Evaluation Standards for School Mathematics (1989). The issue is what and how statistics should be integrated into the mathematics curriculum. The Standards promote the notion that students should be involved in all phases of statistical investigations and problem solving rather than focus on calculation and technique. Figure 1 proposes a model (Kader and Perry 1994) that displays the five components of the statistical problem-solving process.

Statistical Problem Solving in Quality Engineering

Statistical Problem Solving in Quality Engineering

Statistical Problem Solving in Quality Engineering

Introduction to statistical problem solving pareto analysis and brainstorming techniques introduction to variation and statistics measurement system and analysis introduction to Shewart control charts for variables capability studies more variables control charting, pre-control, multi-vary charting attribute control charts check sheets scatter plots design of experiment.

Problem Solving: A Statistician's Guide

"Problem Solving: A Statistician's Guide." Journal of the Operational Research Society, 47(6), pp. 835–836

Mixture models in haematology: a series of case studies.

Two specific applications of finite mixture distributions in haematology include (1) the analysis of the distribution of red blood cell volumes to characterize and quantify alterations in erythrocyte subpopulations in anaemia and (2) the analysis of the distribution of transferrin saturation to estimate subpopulation parameters based on individual genotype for haemochromatosis. For these applications, we describe the comprehensive approach that was taken to distribution modeling. The choice of distributions and the number of mixture components to be fit were based upon theoretical and physiological considerations and a variety of additional statistical problems were considered. These included fitting mixtures of doubly-truncated data and evaluating model fit in the presence of extremely large sample sizes. As a result of the comprehensive approach taken to statistical problem solving in haematology, methods developed for analysis of red blood cell volume distributions have now been adopted by the International Council for Standardization in Haematology. Additionally, analysis of population transferrin saturation distributions from the white population of the USA has led to an independent estimate of the prevalence of homozygotes for haemochromatosis, confirming that the gene for haemochromatosis is common.

BOOK REVIEWS

Book reviewed in this article: Robust Estimation and Testing. By R.G. Staudte and S.J. Sheather. Survey Sampling Principles. By E.K. Foreman Evaluation and Control of Measurements. By John Mandel. Statistical Problem Solving. By Wendell E. Carr. Computer‐Interactive Data Analysis. By A.D. Lunn and D.R. McNeil. Randomization and Monte Carlo Methods in Biology. By B.F.J. Manly. Probability—The Mathematics of Uncertainty. By Dorian Feldman and Martin Fox.

Simulation as an Aid in Practical Statistical Problem-Solving

Simulation as an Aid in Practical Statistical Problem-Solving

How to Think about Statistics

This edition helps readers to make sense of the numbers that they encounter: as consumers, voters, in business, and in school. Instead of focusing on mathematics and computations, the author explains the underlying logic of statistical analysis and problem solving, building one concept upon another to provide a solid framework for understanding how statistics are used, reported, and manipulated. Examples and demonstrations of real-life applications are included to enable the reader to think carefully about the statistics that confront us daily - in our newspapers, in the classroom, and in the claims of politicians and advertisers.

On the relation between justification of solution method and correctness of solution in statistical problem solving

The relation between the justification of a choice of solution method and the correctness of that choice in statistical problem solving was investigated. In the first of two studies 16 subjects were asked to think aloud while solving two statistical problems. The results showed that an incorrect choice of solution method was more common when subjects did not justify their choice of solution as compared to when they justified their choice with domain‐specific knowledge. Study 2 employed an experimental design. A group of 20 subjects were instructed to provide a justification for each choice of solution method, while another group of 20 subjects received no such instructions. The results showed no difference between the groups with respect to number of correct choices of solution method. A qualitative analysis of the justifications in the instructed group showed that the justifications for incorrect solution methods were more often incorrect than subjects’ justifications for correct solution methods. The results in Study 2 suggested that the association found between incorrect choice of solution method and lack of justification in Study 1 was not in the first place due to a strategical deficiency on the part of the subjects but due to a lack of domain‐specific knowledge.

Error detection processes in statistical problem solving

Error detection processes in statistical problem solving

Error Detection Processes in Statistical Problem Solving

Problem solvers' error detection processes were studied by instructing 16 subjects to think aloud when solving two statistical problems. The evaluative episodes occurring in subjects' protocols were analyzed into Affirmative evaluation, Direct error‐hypotheses, Error suspicion, and Standard check episodes, the last three of which are assumed to cover all main types of error detection processes. Most errors (78%) were found to have contributed to a solution part that triggered some evaluative episode. However, only one‐third of the undetected errors had contributed to such a solution part. The Standard check episodes, seen as centrally‐invoked, only led to the detection of few errors in proportion to the number of times they were performed. Evidence was found for two types of spontaneous error detections, one occurring abruptly and the other as a result of a more elaborated error detection process, initiated by the solver perceiving the solution as dissatisfying or strange. The perception of a symptom was a fairly reliable source of information about errors. However, subjects often did not manage to detect the error after having noticed a symptom. The closer a relevant Error suspicion episode followed an error, the greater was the probability of detecting the error. Good problem solvers detected a higher proportion of their errors compared to poor problem solvers, probably due to differences in the processes leading up to the triggering of an evaluative episode rather than to differences after the episode had been triggered.

Detection of errors in statistical problem solving

Ten subjects were asked to think aloud while solving two statistical problems. The subjects were instructed after each substep of his/her problem solving, to check in various ways the solution of the previous substep. The subjects detected 25 out of a total of 56 errors when they solved the problems. About half of the detected errors were computational errors. Nine errors were eliminated in response to the checking instructions. The think aloud data indicated that subjects' most common way of detecting their own errors was by noting that computations resulted in extreme values. Subjects also detected errors by (a) “spontaneous discovery”; (b) discontent with other aspects of a solution than the numerical value of the answer; (c) repeating a solution. The last mentioned type of error detection only occurred when subjects responded to the checking instructions. Finally it was found that subjects had a strong tendency to respond to the checking instructions either in a routinized or in a non‐elaborated way. It was discussed how the formulation of checking instructions can be improved in order to avoid this effect.