An experiment with mathematical statistics

In this chapter we present a model for using case studies for the core of an undergraduate theoretical statistics course. These case studies have more depth than most examples found in typical mathematical statistics texts. Each case study provides a real-world question to be addressed as well as background information on the question, data to address the problem, and suggestions for investigating the problem. An important goal is to encourage and develop statistical thinking. This chapter provides both a sample case study on quality control and analysis of variance and a directed project on experimental design that includes a discussion of how we have changed the class to fit these activities into the course. In essence, the case studies become the centerpiece of the course, and as a result, the curriculum, lectures, and assignments are significantly different from a traditional mathematical statistics course.

The purpose of this research is to analyze the difficulty of students in solving problems related to mathematical creative thinking ability comprehensively as a result of the implementation of PACE (Project, Activity, Cooperative learning, Exercise) model and conventional learning in Mathematical Statistics Course. This research was a qualitative research. The subjects in this research were students of Mathematics Education Program who took Mathematical Statistics Course at one of the private universities in East Jakarta. This research used purposive sampling and various instruments. They were test of mathematical prior knowledge (MPK), test of mathematical creative thinking ability (MCTA), observation sheet, interview sheet, and researcher. The data of this research was collected using triangulation techniques. The result of this research show that students are still experiencing difficulties based on MPK level and overall in solving problems related to mathematical creative thinking ability in both learning (PACE model and conventional). However, the difficulty of students who have obtained PACE Model learning is lower than students who have obtained conventional learning. The most of difficulty of students in both learning (PACE model and conventional) is in indicator of ‘originality’. Keywords: mathematical creative thinking ability, Mathematical Statistics, PACE model

Considerable research suggests that traditional assessments in higher education are outdated, and newer, more authentic assessment strategies should be pursued. One such form of authentic assessment is a portfolio of learning evidence, and another is the oral interview or oral assessment. However, the portfolio and interview are not only methods of assessment, but also primary drivers of learning and creative thinking. In statistics and data science education, the portfolio of learning evidence and interview assessment have not been implemented widely, although some authors have explored the implications of assessments. This mixed-methods study examines the interplay between the use of portfolios and interview assessments in a mathematical statistics course and Fink’s framework for significant learning. The study finds that there are indeed numerous advantages that can be gained from using these novel methods instead of traditional learning and teaching methods.

Probability theory and mathematical statistics is an important basic course of science and technology, economics and management, and has close contact with industry, agriculture, military and science and technology, so it is particularly important for students to study probability theory and mathematical statistics. The learning effect of students largely depends on the teaching methods of teachers. Under the background of the Internet, the emerging development of Internet education, in order to conform to the development of The Times and improve the quality of teaching, the innovative teaching methods of probability theory and mathematical statistics should also be studied. Based on the analysis of the characteristics and advantages of Internet education, combined with the characteristics of probability theory and mathematical statistics and the shortcomings of current teaching, this paper designs an innovative teaching method based on the application of flipped classroom teaching mode under the Internet. Taking the rain classroom as an example, the teaching method is designed from three links: preview before class, teaching in class and feedback after class. Under the background of Internet education, it is of great significance to improve the teaching quality of probability theory and mathematical statistics courses.

Statistical power is an important topic taught in most graduate-level and undergraduate-level mathematical statistics courses, but it is often difficult to understand conceptually. Visualizing the power curve, sampling distributions, and how they interact can help students more easily conceptualize power, but the creation of such visuals can be difficult and time-consuming. Interactive web applications provide a way for students to dynamically visualize power, and many web applications for understanding power exist. However, most of these applications assume samples are drawn from a Normal population, concern only the sample mean, and/or were created for introductory classes. In this paper, we describe a web application suitable for undergraduate-level and graduate-level mathematical statistics courses that was created to allow users to visualize the complex relationships underlying power for multiple different statistics and population distributions (available at https://powerapp.shinyapps.io/powerapp/); source code is provided for instructors who wish to modify the application. Our experience implementing this application across two different semesters is also discussed, and example activities are provided in the appendix.

Probability theory and mathematical statistics are fundamental courses for various majors in science and engineering. In response to the current teaching situation, we should integrate theory with practice, implement teaching reform, and carry out teaching innovation. The article carries out blended teaching with deep integration of online and offline modes and within and outside of class, constructing innovative measures of “four integrations and four reshaping.” The article conducts diversified evaluations to stimulate learning motivation and help achieve talent cultivation goals. Through the close integration of probability theory and mathematical statistics course teaching with professional education and practical application, the “three-in-one” teaching goal of value shaping, ability cultivation, and knowledge exploration is achieved. The fundamental task of “cultivating morality and talents” is implemented.

In this department The American Statistician publishes articles, reviews, and notes of interest to teachers of the first mathematical statistics course and of applied statistics courses. The department includes the Accent on Teaching Materials section; suitable contents for the section are described under the section heading. Articles and notes for the department, but not intended specifically for the section, should be useful to a substantial number of teachers of the indicated types of courses or should have the potential for fundamentally affecting the way in which a course is taught. This article examines the usefulness of multimedia technology for teaching statistics, with attention to both promises and pitfalls. We suggest some principles for the design and use of multimedia, and we offer opinions on the role of human teachers in a multimedia educational environment.

In this department The American Statistician publishes articles, reviews, and notes of interest to teachers of the first mathematical statistics course and of applied statistics courses. The department includes the Accent on Teaching Materials section; suitable contents for the section are described under the section heading. Articles and notes for the department, but not intended specifically for the section, should be useful to a substantial number of teachers of the indicated types of courses or should have the potential for fundamentally affecting the way in which a course is taught. When the residuals from a generalized linear model are to be plotted against the fitted values, the plot can be made to resemble that for a normal error model by transforming the fitted values to the constant-information scale.

Many undergraduate students are introduced to frequentist or classical methods of parameter estimation such as maximum likelihood estimation, uniformly minimum variance unbiased estimation, and minimum mean square error estimation in a reliability, probability, or mathematical statistics course. Rossman, Short, and Parks (1998) present some thought provoking insights on the relationship between Bayesian and classical estimation using the continuous uniform distribution. Our aim is to explore these relationships using the exponential distribution. We show how the classical estimators can be obtained from various choices made within a Bayesian framework.

<p><em>This qualitative descriptive study aims to describe the implementation of mathematics learning based on Lesson Study using the Reciprocal Teaching model in the Mathematical Statistics course at FKIP Untan and ISBI Singkawang. The research focuses on improving students’ ability to write definitions and theorems symbolically through collaborative interactions among lecturers, students, and classroom contexts. The research procedure followed the Lesson Study cycle (plan–do–see), with data collected through interviews, observations, field notes, and assessments. Data were analyzed using the Miles and Huberman model, involving data reduction, presentation, and conclusion drawing. The results show that implementing Lesson Study with Reciprocal Teaching fosters dialogic, collaborative, and continuous learning. In the plan stage, collaborative planning produced a learning design incorporating the strategies of predicting, questioning, clarifying, and summarizing, aligned with students’ characteristics. The do stage revealed active student engagement in group discussions, SPSS practice, and symbolic writing of concepts, although some misconceptions regarding probability, distribution, and variance were still observed. The see stage provided a space for collaborative reflection to identify strengths—such as increased participation, mathematical communication, and conceptual understanding and weaknesses, including uneven participation and limited time allocation. The study concludes that integrating Lesson Study with the Reciprocal Teaching model effectively enhances students’ ability to write definitions and theorems symbolically while strengthening their understanding of Mathematical Statistics concepts. This model is recommended as an innovative strategy for preparing prospective teachers capable of managing reflective, interactive, and meaningful learning.</em></p><p><strong>Pembelajaran Matematika Berbasis <em>Lesson Study</em> Dengan Menggunakan Model <em>Reciprocal Teaching</em> Pada Mata Kuliah Statistika Matematika</strong></p><p><strong>ABSTRAK<br /></strong>Penelitian deskriptif kualitatif ini bertujuan untuk mendeskripsikan implementasi pembelajaran matematika berbasis Lesson Study dengan menggunakan model Reciprocal Teaching pada mata kuliah Statistika Matematika di FKIP Untan dan ISBI Singkawang. Fokus penelitian diarahkan pada peningkatan kemampuan mahasiswa dalam menuliskan definisi dan teorema secara simbolik melalui interaksi kolaboratif antara dosen, mahasiswa, dan konteks kelas. Prosedur penelitian mengikuti siklus Lesson Study (plan–do–see) dengan teknik pengumpulan data berupa wawancara, observasi, catatan lapangan, dan asesmen. Analisis data dilakukan dengan model Miles dan Huberman melalui reduksi, penyajian, serta penarikan kesimpulan. Hasil penelitian menunjukkan bahwa implementasi Lesson Study dengan Reciprocal Teaching mampu menciptakan pembelajaran yang dialogis, kolaboratif, dan berkesinambungan. Pada tahap plan, perencanaan kolaboratif menghasilkan rancangan pembelajaran dengan strategi predicting, questioning, clarifying, dan summarizing yang sesuai dengan karakteristik mahasiswa. Tahap do memperlihatkan keterlibatan aktif mahasiswa dalam diskusi kelompok, praktik penggunaan SPSS, dan penulisan konsep secara simbolik, meskipun masih ditemukan miskonsepsi terkait peluang, distribusi, dan variansi. Tahap see memberikan ruang refleksi kolaboratif untuk mengidentifikasi kelebihan berupa peningkatan partisipasi, komunikasi matematis, dan pemahaman konseptual, serta kelemahan berupa distribusi partisipasi yang belum merata dan keterbatasan waktu. Kesimpulan penelitian menegaskan bahwa integrasi Lesson Study dengan Reciprocal Teaching efektif meningkatkan kemampuan menulis definisi dan teorema dalam bentuk simbolik, sekaligus memperkuat pemahaman konsep Statistika Matematika. Model ini direkomendasikan sebagai strategi inovatif untuk menyiapkan calon guru yang mampu mengelola pembelajaran reflektif, interaktif, dan bermakna.</p><p><strong>Kata Kunci</strong><strong> </strong>:<br />Lesson Study; Model Reciprocal Teaching; Statistika Matematika</p>

The aims of this study are to: a) develop Socrates-based mathematical statistics teaching materials; b) testing the quality of teaching materials in terms of validity, effectiveness, and practicality. The research was carried out in the even semester of 2021/2022 in the fourth semester of the mathematical statistics course for all classes. This study uses a development method using a 4-D model consisting of 4 main stages, namely define, design, develop, disseminate. The product developed is in the form of teaching materials that have been tested for feasibility with content validity, effectiveness tests by analyzing student learning outcomes after the teaching materials are used and practicality tests through student response analysis. The collection techniques and instruments are: 1) indirect communication techniques with the instruments being expert validation questionnaires and response questionnaires for instrument validation and student responses; 2) the measurement technique with the instrument is a test of learning outcomes for the effectiveness of the product; 3) direct communication techniques with interview sheets as instruments to find out the advantages and disadvantages of teaching materials when used for students. The data analysis techniques are: a) describe or describe each step of development with a 4-D model; b) expert validation analysis on the feasibility of teaching materials, namely qualitative descriptive, the results of the study are stated in the form of logical analysis; c) calculation of the validity of the assessment results and validation of the consistency of the experts using Aiken's formulation. The overall research results are: 1) the development of Socrates-based mathematical statistics teaching materials is carried out according to the main 4-D stages in sequence; 2) Socrates-based mathematical statistics teaching materials have content validity and reliability with very high criteria; 3) student learning outcomes after using Socrates-based mathematics teaching materials are classified as high criteria. So it can be concluded that the teaching materials developed are effective.

THESE two books at once invite comparison with A Maurice Kendall's “Advanced Theory of Statistics” and A. C. Aitken's “Statistical Mathematics”. All four are mathematically flawless, and they all require from the reader some mathematical training and ability, but not to the same extent, nor are they all addressed to quite the same category of scientific workers. Kendall, in his preface, called his “a book on statistics, not on statistical mathematics” and declared his intention “to keep the mathematics to heel”. Cramer's book (based on his Stockholm lectures since 1930) is very definitely for the mathematician, and its avowed purpose is to join the two lines of development due on one hand to the British and American schools of statisticians, and on the other to the rigorous work of French and Russian mathematicians in developing the calculus of probability. The first twelve chapters are a purely mathematical introduction, dealing with the properties of sets, Lebesque integrals, Fourier integrals, matrices and quadratic forms, and the like, preparatory to the later chapters on random variables, probability distributions, and statistical inference. The author advises the reader to jump to Chapter 13 after reading the first three chapters and a few other paragraphs, and thereafter to use this earlier part only as needed. The advice is good, and for my part I would say jump to Chapter 13 at once. Many who ought to read the book would be highly discouraged by the apparent irrelevance of those first three chapters, yet might turn to them without dismay later after seeing their use and necessity. Mathematical Methods of Statistics By Prof. Harald Cramér. (Princeton Mathematical Series, 8.) Pp. xvi + 575. (Princeton, N.J.: Princeton University Press; London: Oxford University Press, 1946.) 33s. 6d. net. A First Course in Mathematical Statistics By Prof. C. E. Weatherburn. Pp. xv + 272, (Cambridge: At the University Press, 1946.) 15s. net.

We analyze learning data of an e-assessment platform for an introductory mathematical statistics course, more specifically the time of the day when students learn and the time they spend with exercises. We propose statistical models to predict students’ success and to describe their behavior with a special focus on the following aspects. First, we find that learning during daytime and not at nighttime is a relevant variable for predicting success in final exams. Second, we observe that good and very good students tend to learn in the afternoon, while some students who failed our course were more likely to study at night but not successfully so. Third, we discuss the average time spent on exercises. Regarding this, students who participated in an exam spent more time doing exercises than students who dropped the course before.

Proceedings of the London Mathematical SocietyVolume s3-48, Issue 3 p. 514-532 Articles Congruence Properties of Zariski-Dense Subgroups I C. R. Matthews, C. R. Matthews Department of Pure Mathematics and Mathematical Statistics, 16 Mill Lane, CambridgeSearch for more papers by this authorL. N. Vaserstein, L. N. Vaserstein Department of Pure Mathematics and Mathematical Statistics, 16 Mill Lane, CambridgeSearch for more papers by this authorB. Weisfeiler, B. Weisfeiler Department of Mathematics, Pennsylvania State University, University Park, PA, 16802 U.S.A.Search for more papers by this author C. R. Matthews, C. R. Matthews Department of Pure Mathematics and Mathematical Statistics, 16 Mill Lane, CambridgeSearch for more papers by this authorL. N. Vaserstein, L. N. Vaserstein Department of Pure Mathematics and Mathematical Statistics, 16 Mill Lane, CambridgeSearch for more papers by this authorB. Weisfeiler, B. Weisfeiler Department of Mathematics, Pennsylvania State University, University Park, PA, 16802 U.S.A.Search for more papers by this author First published: May 1984 https://doi.org/10.1112/plms/s3-48.3.514Citations: 46AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volumes3-48, Issue3May 1984Pages 514-532 RelatedInformation

1 Department of Mathematics and Mathematical Statistics, Umea University, 901 87 Umea, Sweden 2 Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel 3 Department of Mathematics and Statistics, University of Calgary, 2500 University Drive Northwest, Calgary AB, Canada T2N 1N4 4 Department of Mathematics, Faculty of Electrical Engineering and Computer Science, Brno University of Technology, 616 00 Brno, Czech Republic