Visualizing and Understanding Regression and Correlation Using Dynamic Software

A "Reverse Order" Elementary Statistics Course

Students in elementary statistics traditionally see experiments and data as words and numbers in a text. They receive little exposure to the important statistical activities of sample selection, data collection, experimental design, development of statistical models, the need for randomization, selection of factors, etc. They often leave the first course without a firm understanding of the role of applied statistics or of the statistician in scientific investigations. In an attempt to improve elementary statistics education, we have developed a statistics laboratory similar to those of other elementary science courses. We will discuss our experiences in teaching a laboratory component with the traditional elementary statistics course. In each lab session, students, working in teams, discuss the design of an experiment, carry out the experiment, and analyze their data using Minitab on a Macintosh or MS-DOS based computer. The students then individually either answer a series of short answer questions or write a formal scientific report. The labs are designed to be relatively inexpensive and portable. They do not require a prior background in science, statistics or computing.

Hand calculators have come into common use for performing calculations in high school and undergraduate courses in elementary statistics. Computers have also been used to do standard statistical computations and tests. However, computers can be used in elementary statistics courses not only to do standard statistical tests, but to perform simulations which clarify concepts and theorems of statistics. These simulations allow the nonmathematically oriented student in elementary statistics to have inductive experiences with statistical concepts in a time-efficient manner. A basic concept in statistics is that of a sample distribution. The sampling distribution of a sample statistic is defined to be the theoretical probability distribution of values for that sample statistic obtained from all possible samples of a certain fixed size from the population. Typically elementary texts illustrate this concept by using a very small finite population which has a small number of possible samples of a given size. Although large or infinite populations are of interest since they have a large or infinite number of samples of a certain size it is difficult to develop a sample distribution. Computer simulation, however, makes this easy to explore. The computer can simulate taking many samples of a certain size from a population. By taking many samples the law of large numbers guarantees that the observed frequency distribution of the sample statistic is close to the theoretical sample distribution. The sample distribution of the mean is the most important sample distribution. The computer simulations in Figure 1 explore a sample distribution of the mean. These simulations employ an elementary statistical package called MINITAB developed at Pennsylvania State University. The MINITAB command URANDOM simulates random observations from a population where numbers between 0 and 1 all have an equal probability of occurring. The first simulation consists of drawing 200 samples of size 2 and computing the means of the samples. A histogram of the results is given. Continuing with samples of size 2, drawing 750 samples is simulated and then 1500 samples. As the number of samples increases the student can see the isosceles triangle-shaped theoretical sample distribution emerging. Note that, because of the small sample size and the nonnormal population, the sample distribution should not be expected to be normal.

One of the main themes of statistics courses is to teach about variability, as well as location. This is especially important for non-statistics students, who often overlook variability. We consider particularly the problem of comparing variability among k samples (k > 2) that are not necessarily drawn from Gaussian populations. This can also be viewed as testing for homoskedasticity of samples. We examine tools for this problem from the perspective of their suitability for inclusion in elementary statistics courses for students of non-mathematical subjects. The ideas are illustrated by an example that arose in a student project.Supplemental data for this article can be accessed on the publisher's website.

The Statistics Anxiety Rating Scale has 51 items, each scored on a 5-point rating scale to measure statistics anxiety with six subscales, Worth of Statistics, Interpretation Anxiety, Test and Class Anxiety, Computational Self-concept, Fear of Asking for Help, and Fear of Statistics Teachers. Psychometric properties included analyses of construct and concurrent validities an internal consistency and test-retest reliability. 221 college students (74% women; M age=28 yr.) in elementary statistics courses at several southwestern state universities participated. The findings are consistent with previous reports and indicate adequate concurrent validity, internal consistency, and split-half reliability, but for construct validity confirmatory factor analysis yielded marginal support.

This month, we bring you some “Technology Tips” that are related to probability and statistics. We begin with a contribution by Bishnu Naraine and Peiyi Zhao: an exploration that uses The Geometer's Sketchpad to discuss the broken-stick problem, which is also known as the spaghetti problem. This totally geometric solution nicely complements the analytic solution of the same problem, given in “The Spaghetti Problem Problem,” by Dennis Ippolito, which appeared on pages 422–26 of the May 2000 issue of the Mathematics Teacher. Next is a feature by Allen Holmes that discusses obtaining an explicit definition for a recursively defined sequence by using the built-in statistics regression ability of graphing calculators to model the equation of the sequence. Then a short program by John Jose for the TI-83 and TI-83 Plus graphing calculator helps investigate the law of large numbers by simulating the results of a simple two-dice experiment. Finally our “Surfing note” features a Web site that is devoted to statistics. It contains many resources for an elementary statistics course.

A collaborative, statistics education research project (Lovett, 2001) is discussed. Some results of the project were applied in the computer lab sessions of my elementary statistics course. I detail the process of applying these research results, as well as the use of knowledge surveys. Furthermore, I give general suggestions to teachers who want to put educational research results into effective use in their own classes.

This study qualitatively investigates the nature of the challenges that a student who is highly successful in mathematics faces in learning college-level elementary statistics. The study draws on the constructs of eagerness, flexibility and willingness to characterize the necessary disposition for critical thinking that is essential in learning statistics. The case study is based on data collected through a survey assessment and a follow-up interview with a mathematics major enrolled in an elementary college statistics course at the time of the study. The qualitative analysis relies on the student's verbal descriptions of the challenges he was experiencing in the course. The findings suggest that while his strong inclination towards inquisitive learning and strong understanding of mathematical concepts supported this student's mathematics learning, the same characteristics might have been causing him difficulties in learning college-level elementary statistics.

Distributed fiber-optic sensors employ the conventional fiber as the sensing medium and allow all locations along the optical fiber to be measured simultaneously. Thus the sensing scheme provides an efficient way of simultaneously achieving multi-points sensing with a convenient and low-cost configuration. One of the most popular technologies that enables distributed sensing is based on Brillouin scattering. Characteristics of Brillouin scattering are determined mostly by the material properties of the optical fiber and the Brillouin Frequency Shift (BFS) has linear dependence on both the temperature and strain of the fiber under test (FUT). Based on this principle, intensive research efforts have been made in distributed Brillouin fiber sensors during the past two decades. Among them, Brillouin Optical Time-Domain Analysis (BOTDA) has attracted intensive research interest due to its promising properties for temperature and strain monitoring. In BOTDA the local Brillouin Gain Spectrum (BGS) is reconstructed by scanning the frequency offset of a continuous wave and an optical pulse around BFS when they are counter-propagating inside FUT, and the local BFS and hence the temperature and/ or strain information are retrieved accordingly. Usually curve fitting techniques are employed in determining the BFSs of the measured BGSs along the FUT, in which the measured BGS is fitted using an ideal curve and the BFS is found to be the frequency with the peak gain on the fitted curve. Fitting techniques using different lineshape for ideal curves, e.g. Lorentzian curve fitting (LCF), Gaussian curve fitting (GCF), pseudo-Voigt curve fitting (pVCF), and quadratic curve fitting (QCF) have been used to find the BFS distribution along FUT. However, the curve fitting techniques require careful initialization of model parameters and as many data points as possible collected on the measured BGS to ensure the fitting accuracy, otherwise the accuracy in BFS determination will degrade significantly. In addition, long processing time is needed for the iteration of fitting algorithms to find the BFS, especially when the sensing distance is long with a large number of BGSs collected and processed. Recently we have utilized the nonlinear mapping capability of Artificial Neural Network (ANN) in a BOTDA system and has successfully extracted temperature distribution from the measured BGSs along the FUT. Before temperature extraction ANN is trained to acquire the knowledge about the BGS patterns under different temperatures, thus it allows better accuracy even if the data points on the collected BGS become fewer when large frequency scanning step is adopted during the acquisition of BGSs. Moreover, the processing speed of ANN for temperature extraction is much faster than that of curving fitting techniques. In this presentation, we will review our work on using ANN for temperature extraction in a BOTDA system. Different idea spectrum profiles are used for training ANN, and the corresponding performances are compared with those using conventional curve fitting techniques. Some potential applications using ANN in BOTDA are also discussed. We believe that ANN can be an attractive tool for direct temperature or strain extraction in BOTDA system at high accuracy and fast speed.

In view of the disadvantages of traditional phase calibration methods for single channel monopulse tracking systems, this paper presents a new phase calibration method based on the curve fitting technique. The operational principle of a single channel monopulse tracking (SCMT) system is first illustrated in detail. Based on the curve fitting technique, a new phase calibration algorithm is presented by using the error voltage data from satellite tracking. Finally, the effectiveness of the proposed method is evaluated by the simulated simulation. The results show that the proposed method can be applied to the tracking systems with single channel monopulse mechanism and can perform the phase calibration only based on the error voltage's convergence trajectory data but independent of the signal source such as calibration towers, counteracting the shortcoming of the traditional phase calibration methods.

A computational study of entropy measures and topological indices for Magnesium Nitride using curve fitting techniques.

Wireless multimedia sensor network (WMSN) comprising of miniature sensor nodes is capable of processing multimedia data traffic such as still images and video from the environment. There is a wide range of applications which get benefited from such network. Unprocessed multimedia transmission is always expensive in terms of processing power, storage, and bandwidth. So, data processing is a challenge in WMSN. Exploring low-overhead data compression technique is a solution towards this problem. In this work we propose an energy saving image compression technique for WMSN using curve fitting technique considering the application of post-disaster situation analysis through image capturing of the affected area. Upon employing the method on the macroblocks of sensory image, curve fitting coefficients are generated and transmitted towards the sink thereby saves energy by transmitting reduced volume of data. Finally the design feasibility along with simulation results including statistical analysis is presented to evaluate efficacy of the scheme in terms of two conflicting parameters viz. energy consumption and peak signal to noise ratio. The comparative results confirm our scheme’s supremacy in WMSN application domain over existing methods.

The potential of computer technology to enhance the student’ s learning experience is being increasingly recognized and utilized. Many, many interactive, dynamic software packages have been developed which allow students to explore complex statistics concepts in new and exciting ways, with the goal of enhancing understanding. There are, however, other roles for the technology which allow us to enrich the students experience in ways which are not only educationally defensible, but also efficient and economical. This paper describes an internet based assessment tool which has been developed and successfully utilized in an Introductory Statistics course taught at tertiary level to a group of about 800 undergraduate students each year.

In this paper, a novel method for perturbed trajectory prediction of post-fault rotor angle is presented based on PMU (phasor measurement unit) measurements and trigonometric function curve fitting technique. A mathematical model about the relationship of post-fault rotor angle and time is derived from a speed-governing system model, which reveals that the perturbed trajectory of the post-fault rotor angle can be expressed by trigonometric functions. By curve fitting technique, a least square method to predict the perturbed trajectory using trigonometric function is proposed. Numerical simulations are conducted on IEEE 9-bus system to test the effectiveness of this method. The proposed method can give more reasonable and more promising prediction results as compared with polynomial curve fitting technique

A T-wave alternans assessment method based on least squares curve fitting technique