Distance Learning and Examining Systems

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1 Lodz University of Technology Distance Learning and Examining Systems - Theory and Applications edited by Sławomir Wiak Konrad Szumigaj HUMAN CAPITAL - THE BEST INVESTMENT The project is part-financed by the European Social Fund.

2 Edited by: prof. dr hab. inż. Sławomir Wiak mgr inż. Konrad szumigaj Distance Learning and Examining Systems strategy, logic of the system, architecture, evaluation. (IT platform e matura) Reviewers: prof. dr hab. inż. Maria Dems prof. dr hab. inż. Andrzej Krawczyk The book prepared under the E matura project, part financed by the European Social Fund in the frame of Human Capital Operational Program National cohesion strategy, Prioryty III High quality of the educational system, Action 3.3 Improving the quality of eduaction, Sub action Modernization of the content and methods of eduaction competition projects.

3 Book is distributed free of charge Reviewers: prof. dr hab. inż. Maria Dems prof. dr hab. inż. Andrzej Krawczyk Edited by: prof. dr hab. inż. Sławomir Wiak mgr inż. Konrad szumigaj graphic design Niceday The book prepared under the E matura project, part financed by the European Social Fund in the frame of Human Capital Operational Program National cohesion strategy, Prioryty III High quality of the educational system, Action 3.3 Improving the quality of eduaction, Sub action Modernization of the content and methods of eduaction competition projects.

4 Dr. Adam Depta, Department of Management, University of Lodz. Dr Joanna Kucner, Centre for Teaching Mathematics and Physics, Technical University of Lodz. Dr. Christopher Kisiel, Centre for Teaching Mathematics and Physics, Technical University of Lodz. Michael Kucner, student, Department of Mechanical Engineering, Technical University of Lodz. GeoGebra possibilities in solving statistical tasks used by students 7.1. Introduction Participants of the process of teaching and learning have more and more programs at their disposal to facilitate and support this process. There are both free and commercial programs. Mathematical programs can greatly facilitate calculation and sometimes also allow visualization of the problem. It is known that it is easier to understand and hence solve the task when we see a problem not only imagine. For solving school and matriculation tasks sudents can successfully use GeoGebra [1]. This program can also be used by both pupils and students to explore new areas of mathematics not covered in the curriculum [2]. In this work we will focus on the use of GeoGebra to solve problems in statistics both at the level of matriculation and at the level of studies. The following picture shows the possibilities which GeoGebra gives in this field of mathematics. Screen 1. Statistics in GeoGebra. 123

5 Theoretical information is not as useful as its application in practice. Below the reader will learn about using GeoGebra to solve specific tasks in statistics. Task 1 [3]. The median of a non-decreasing set of six numbers 1, 2, 3, a, 5, 8 equals 4. Then: A. B. a = 2 and a = 3 C = 4 and D = 5 Using the possibilities of the slider, you can easily determine the value of the searched number a. GeoGebra will help us answer the question from the task, however, in this particular task the student should use it as a tool which supports and verifies the correctness of the calculations made in the traditional way - on paper. Screen 2. Solving matriculation statistics tasks using GeoGebra. 124

6 Task 2 [4]. The following table presents some statistical data: Table 1. Contractual statistics. value number a) Calculate the arithmetic mean of the data. b) Determine the variance of the data. c) Calculate the standard deviation of the data with an accuracy of Screen 3. Solving a task in statistics using GeoGebra. It should be noted that GeoGebra gives all the calculations with an accuracy to two decimal places. Thus, in this task the student received directly the standard deviation with the desired accuracy. 125

7 When examining the population in terms of two features at the same time, which are in mutual relationship, one deals with a two-dimensional analysis. With more characteristics we deal with a multidimensional analysis. The study of the relationship between several traits is carried out by correlation and regression analysis. Correlation analysis measures the strength and direction of the relationship, whereas regression analysis describes the shape of the relationship [5]. GeoGebra has a module that allows to carry out regression analysis. The following model shows contractual values of the two features: X and Y. Screen 4. A linear regression model, statistical parameters and a scatter plot in GeoGebra. In this program it is possible to predict [6] or forecast, look at the middle bottom part of the screen "Calculate the expression" 126

8 Screen 5. A regression model (a form of a polynomial of degree 2), statistical parameters and a scatter plot in GeoGebra. Screen 6. A regression model (polynomial of degree 4), statistical parameters and a scatter plot in GeoGebra. 127

9 The Module "Statistics of one variable" allows the calculation of the basic characteristics of descriptive statistics and provides data in the form of a histogram (where the number of columns depends on the number of classes (categories) adopted for the analysis. Numerical characteristics called descriptive parameters are used for the quantitative comparison of statistical properties. Discriptive parameters of a statistical population are numbers which synthetically define the characteristics of the surveyed population. The main task of these parameters is to determine the properties of the distributions of the tested population and compare several communities in terms of the same feature and the same population in terms of a number of different features [7]. Screen 7. Basic characteristics of descriptive statistics and a histogram (6 classes) in GeoGebra. 128

10 Screen 8. Basic characteristics of descriptive statistics and a histogram (9 classes) in GeoGebra. In addition, there is a possibility in GeoGebra to draw conclusions about the entire population based on a statistical sample, that is its portion. Statistical inference or mathematical statistics consists in: estimation, ie, estimating distribution parameters of the studied characteristic in general population, verification (testing) of statistical hypotheses concerning the distribution of the characteristic in general population, the relationship between the measured traits or the compliance of the schedules of the studied traits [8]. 129

11 Screen 9. The confidence interval for the mean - normal distribution Z and a histogram GeoGebra. Screen 10. A test for the mean - normal distribution Z and a histogram GeoGebra. 130

12 Screen 11. A test for the mean (small sample) and a histogram in GeoGebra. Screen 12. Box plots of two variables in GeoGebra. 131

13 Screen 13. A test for the mean (small sample), a histogram and a frequency table in GeoGebra. Screen 14 A t-student's test - the case of connected (dependent) variables, and box plots of variables in GeoGebra 132

14 Screen 15. Estimation - the case of connected (dependent) variables, and box plots of variables in GeoGebra. Screen 16 A sample distribution of students' height with the expected value of 172 cm and a standard deviation of 10 cm and P (160 X 182) in GeoGebra 133

15 Screen 17. A sample distribution of students' height with the expected value of 172 cm and a standard deviation of 10 cm and P (X 160) in GeoGebra. Screen 18 A sample distribution of students' height with the expected value of 172 cm and a standard deviation of 10 cm and P (190 X) in GeoGebra 134

16 Poisson's distribution Poisson's distribution was applied, among other things, in nuclear physics, for example, to record particles by the Geiger-Muller counter, the dissociaton of radioactive elements but also the timing of incoming telephone calls, etc. Task [9]. There was a study of the distribution of flying bombs that fell on London during the Second World War. One area was divided into to ¼ km each. The following table shows the number of k bombs: N 576 parts of an area equal N k of these parts, hit by exactly k and more N p(k;0,9323) N k Compliance with the model is striking. Exponential distribution Task [10]. Random variable X has an exponential distribution and Write the density of probability, calculate the quantile of order It is easy to see that 3 and so the probability density is: 1 VarX

17 Screen 19. A chart of probability density for the variable of exponential distribution with the use of GeoGebra. Screen 20 Determination of the quantile of order 3/4 using GeoGebra 136

18 7.2. References 1. Bartos J., Dyczka W., Jakuszenkow H., Elementy rachunku prawdopodobieństwa i statystyki matematycznej dla włókienników, Przykład 7.7, Wydawnictwo Politechniki Łódzkiej, Łódź 1990, s Białek J., Depta A., Statystyka dla studentów z programem STAT_STUD 1.0, Wydawnictwo C.H. Beck, Warszawa Depta. A., Kisiel K., Kucner J., Lekcje na temat liczb zespolonych z użyciem programu GeoGebra dla klas ponadgimnazjalnych zachętą do studiowania na Politechnice Łódzkiej, w: System informatyczny zdalnego egzaminowania - strategia, logika systemu, architektura, ewaluacja (platforma informatyczna e-matura) (red.) Wiak S., Szumigaj K., Wydawnictwo Politechniki Łódzkiej, Łódź 2013, s Domański Cz., Testy statystyczne, PWE, Warszawa Karris S.T. (2007), Mathematics for Business, Science, and Technology, Third Edition, Orchard Publications. 6. Kucner J., Galewska E., Kisiel K., Stosowanie darmowego oprogramowania matematycznego do rozwiązywania zadań maturalnych, w: System informatyczny zdalnego egzaminowania - strategia, logika systemu, architektura, ewaluacja (platforma informatyczna e-matura) (red.) Wiak S., Szumigaj K., Wydawnictwo Politechniki Łódzkiej, Łódź 2013, s Egzamin maturalny z matematyki, poziom podstawowy, zad. 24, Maj 2013, s Orlińska M.: Matura 2013, Obowiązkowa matura z matematyki, Poziom podstawowy, Testy i arkusz, zad. 10, Operon 2012, s. 51, 9. Ross S.M. (2004), Introduction to Probability and Statistics for Engineers and Scientists, Third Edition, Elsevier Academic Press. 10. Traczyk T., Mączyński M., Matematyka stosowana w inżynierii chemicznej, Wydawnictwo Naukowo- Techniczne, Warszawa Footnotes 1. Kucner J., Galewska E., Kisiel K., Stosowanie darmowego oprogramowania matematycznego do rozwiązywania zadań maturalnych, w: System informatyczny zdalnego egzaminowania - strategia, logika systemu, architektura, ewaluacja (platforma informatyczna e-matura) (red.) Wiak S., Szumigaj K., Wydawnictwo Politechniki Łódzkiej, Łódź 2013, s Depta. A., Galewska E., Kucner J., Kisiel K., Lekcje na temat liczb zespolonych z użyciem programu GeoGebra dla klas ponadgimnazjalnych zachętą do studiowania na Politechnice Łódzkiej, w: System informatyczny zdalnego egzaminowania - strategia, logika systemu, architektura, ewaluacja (platforma informatyczna e-matura) (red.) Wiak S., Szumigaj K., Wydawnictwo Politechniki Łódzkiej, Łódź 2013, s Egzamin maturalny z matematyki, poziom podstawowy, zad. 24, str. 8, Maj Orlińska M.: Matura 2013, Obowiązkowa matura z matematyki, Poziom podstawowy, Testy i arkusz, zad. 10, str. 51, Operon Orlińska M.: Matura 2013, Obowiązkowa matura z matematyki, Poziom podstawowy, Testy i arkusz, zad. 10, str. 51, Operon Białek J., Depta A. (2010), Statystyka dla studentów z programem STAT_STUD 1.0, Wydawnictwo C.H. Beck, Warszawa, s Predykcji 8. Białek J., Depta A. (2010), Statystyka dla studentów z programem STAT_STUD 1.0, Wydawnictwo C.H. Beck, Warszawa, s Białek J., Depta A. (2010), Statystyka dla studentów z programem STAT_STUD 1.0, Wydawnictwo C.H. Beck, Warszawa, s Traczyk T., Mączyński M., Matematyka stosowana w inżynierii chemicznej, Przykład 1, str. 428, Wydawnictwo Naukowo-Techniczne, Warszawa Bartos J., Dyczka W., Jakuszenkow H.: Elementy rachunku prawdopodobieństwa i statystyki matematycznej dla włókienników, Zadanie 48, str. 259, Wydawnictwo Politechniki Łódzkiej, Łódź