Students' Understanding Of Statistical Inference .
Students’ Understanding of StatisticalInference: Implications for TeachingbyRobyn Reaburn, B.App.Sci.(Medical Technology),BA, Dip.Teach.,Grad.Dip.Sci.,MSc.Submitted in fulfilment of therequirements for the Degree of Doctor ofPhilosophyUniversity of Tasmania, October, 2011.
This thesis contains no material which has been accepted for a degree or diplomaby the University or any other institution, except by way of background information and duly acknowledged in the thesis, and to the best of my knowledge andbelief no material previously published or written by another person except wheredue acknowledgement is made in the text of the thesis, nor does the thesis containany material that infringes copyright.ii
This thesis may be made available for loan and limited copying in accordancewith the Copyright Act of 1968.iii
The research associated with this thesis abides by the international and Australiancodes on human and animal experimentation, the guidelines by the AustralianGovernment‟s Office of the Gene Technology Regulator and the rulings of theSafety, Ethics and Institutional Biosafety Committees of the University.iv
AbstractIt was of concern to the researcher that students were successfully completing introductory tertiary statistics units (if success is measured by grades received),without having the ability to explain the principles behind statistical inference. Inother words, students were applying procedural knowledge (surface learning)without concurrent conceptual knowledge.This study had the aim of investigating if alternative teaching strategies could assist students in gaining the ability to explain the principles behind two tools ofstatistical inference: P-values and confidence intervals for the population mean.Computer simulations were used to introduce students to statistical concepts. Students were also introduced to alternative representations of hypothesis tests, andwere encouraged to give written explanations of their reasoning. Time for reflection, writing and discussion was also introduced into the lectures.It was the contention of the researcher that students are unfamiliar with the hypothetical, probabilistic reasoning that statistical inference requires. Therefore students were introduced to this form of reasoning gradually throughout the teachingsemester, starting with simple examples that the students could understand. It washoped that by the use of these examples students could make connections thatwould form the basis of further understanding.It was found that in general, students‟ understanding of P-values, as demonstratedby the reasoning used in their written explanations, did improve over the four semesters of the study. Students‟ understanding of confidence intervals also improved over the time of the study. However for confidence intervals, where simv
ple examples were more difficult to find, student understanding did not improveto the extent that it did for P-values.It is recommended that statistics instructors need to appreciate that tertiary students, even those with pre-tertiary mathematics, may not have a good appreciationof probabilistic processes. Students will also be unfamiliar with hypothetical,probabilistic reasoning, and will find this difficult. Statistics instructors, therefore,need to find connections that students can make to more familiar contexts, usealternative representations of statistical processes, and give students time to reflect and write on their work.vi
AcknowledgementsI would like to thank my supervisors, Professor Jane Watson and Associate Professor Kim Beswick, and my research supervisor, Associate Professor RosemaryCallingham, for their help and support over the time of my candidature. Theyhave been consistently helpful, cheerful, and encouraging. It has been my goodfortune to work with them. Thanks also to Dr. Des Fitzgerald, coordinator of thestatistics unit described in this study, for his advice.I wish to thank my parents, Kenneth and Phyllis Duck. They did all they could sothat their daughters could have the education that they could never have. My father died during the time this thesis was written; he would have been thrilled toknow that it has been finished.Thanks to my daughters, who have managed well with a mother who has beentired and occasionally very absent minded. Most of all thanks must go to myhusband Nicholas, for his consistent love and support. Much has happened in ourlives over the past year – this thesis would never have been finished without him.vii
.1. INTRODUCTION . 151.1 WHY DO THIS RESEARCH? . 151.2 THE RESEARCH QUESTIONS . 171.3 A NOTE ON THE TERMINOLOGY . 182. LITERATURE REVIEW – PART I: STATISTICAL REASONING . 202.1 WHAT IS STATISTICS? . 202.2 STATISTICAL REASONING . 212.3 HYPOTHESIS TESTING AND CONFIDENCE INTERVALS . 222.3.1 Hypothesis testing .222.3.2 Confidence intervals .262.4 MISCONCEPTIONS WITH PROBABILISTIC REASONING . 282.4.1 Introduction .282.4.2 The contribution of Tversky and Kahneman .292.4.3 Other misconceptions about probability .322.4.4 Misconceptions about conditional probability . 332.5 OTHER MISCONCEPTIONS ABOUT STATISTICAL REASONING . 342.5.1 Misconceptions about randomness .342.5.2 Misconceptions about sampling .352.5.3 Misconceptions about measures of central tendency .362.5.4 Misconceptions about statistical inference .382.6 THE PERSISTENCE OF PRECONCEIVED VIEWS . 472.6.1 Introduction .472.6.2 How students change their previous conceptions . 472.7 IMPLICATIONS OF THE LITERATURE FOR TEACHING STATISTICS . 483. LITERATURE REVIEW PART II: THE NATURE OF LEARNING . 503.1 INTRODUCTION - WHAT IS LEARNING? . 503.2 HOW LEARNING OCCURS . 513.2.1 Introduction .513.2.2 Information processing theory .528
3.2.3 Constructivist theories of learning . 533.2.4 Implications of the cognitive models for teaching . 553.3 AFFECTIVE FACTORS . 563.4 THE USE OF THE SOLO TAXONOMY IN ASSESSING LEARNING . 584. LITERATURE REVIEW PART III: MEASUREMENT IN THE SOCIAL SCIENCES . 604.1 SCALES USED IN MEASUREMENT . 604.2 MEASUREMENT THEORY . 624.3 SHOULD THE SOCIAL SCIENCES USE MEASUREMENT?. 664.4 ITEM RESPONSE THEORY . 674.5 THE MATHEMATICS OF THE RASCH MODEL. 734.5.1 The Dichotomous model . 734.5.2 The Partial Credit Model . 754.5.3 How Rasch analysis was used in this study . 785. THE USE OF COMPUTER TECHNOLOGY IN STATISTICS. 83EDUCATION . 835.1 Introduction . 835.2 Discovery Learning and Simulation . 835.3 Simulation in statistics . 855.4 How computers were used in this study . 886. THE STUDY DESIGN . 896.1 INTRODUCTION . 896.2 RESEARCH DESIGNS IN EDUCATION . 906.2.1 Scientific Research in Education. 916.2.2 How “Scientific” Does Knowledge Have To Be? . 946.2.3 The Action Research Method of educational research . 956.2.4 How this study fitted the research paradigms . 976.3 THE STUDY – AIMS, PARTICIPANTS, TASKS, INTERVENTIONS . 976.3.1 The aims of the study . 976.3.2 The participants in the study . 986.3.3 The sources and analysis of the questionnaires and the test items . 989
6.4 THE DESIGN FOR THIS STUDY . 1076.4.1 Introduction .1076.4.2 The pre-intervention semester .1086.4.3 The first cycle of the intervention .1106.4.4 The second cycle of the intervention .1146.4.5 The third cycle of the intervention .1176.5 A SUMMARY OF THE STUDY DESIGN . 1206.6 CONSTRAINTS ON THE RESEARCH. 1227. RESULTS OF THE QUANTITATIVE AND QUALITATIVE ANALYSIS OF THEFIRST QUESTIONNAIRE . 1247.1 INTRODUCTION . 1247.2 RASCH ANALYSIS OF THE FIRST QUESTIONNAIRE . 1267.2.1 Introduction .1267.2.2 Items in the First Questionnaire .1267.2.3 The Rasch Analysis of the Items (Partial Credit Model) .1287.2.4 Rasch analysis of persons . 1377.3 QUALITATIVE ANALYSIS OF THE FIRST QUESTIONNAIRE . 1407.3.1 Questions requiring interpretation of verbal probabilistic statements . 1407.3.2 Questions requiring an understanding of statistical independence . 1447.3.3 Students‟ Awareness of Variation in Stochastic Processes .1517.3.4 Questions requiring judgements of differences between groups .1617.3.5 Conditional probability questions .1717.4 SUMMARY AND DISCUSSION . 1768. RESULTS OF THE QUANTITATIVE AND QUALITATIVE ANALYSIS OF THESECOND QUESTIONNAIRE . 1808.1 INTRODUCTION . 1808.2. RASCH ANALYSIS OF THE SECOND QUESTIONNAIRE . 1818.2.1 Introduction .1818.2.2 Items in the Second Questionnaire .1818.2.3 The Rasch Analysis of the Items (Partial Credit Model).1838.2.4 Rasch analysis of persons . 19110
8.3 QUALITATIVE ANALYSIS OF THE SECOND QUESTIONNAIRE . 1958.3.1 The circuit breaker questions . 1958.3.2 Explaining the meaning of “significant difference” . 1988.3.3 Judgement as to the likelihood of sample means, given a population mean . 2018.3.4 The Use of Informal Inference . 2078.3.5 Questions that deal with randomness – what is random, and why randomise? . 2098.3.6 Repeated questions from the first questionnaire . 2158.4 RELATIONSHIPS AMONG ABILITY MEASURES AND SCORES FROM FORMAL ASSESSMENTS . 2228.5 SUMMARY AND DISCUSSION . 2269. AN ANALYSIS OF STUDENTS’ UNDERSTANDING OF P-VALUES. . 2289.1. INTRODUCTION . 2289.2 RESULTS OF THE PRE-INTERVENTION SEMESTER (SEMESTER 2 – 2007) . 2309.2.1 Teaching strategies . 2309.2.2 Student answers to the P-value items in the test . 2319.3 RESULTS OF THE FIRST CYCLE OF THE INTERVENTION (SEMESTER 1 – 2008) . 2329.3.1 Teaching strategies . 2329.3.2 Student answers to the P-value items in the test . 2329.4 RESULTS OF THE SECOND CYCLE OF THE INTERVENTION (SEMESTER 2 – 2008) . 2349.4.1 Teaching strategies . 2349.4.2 Student answers . 2369.5 RESULTS OF THE THIRD CYCLE OF THE INTERVENTION (SEMESTER 1 – 2009) . 2379.5.1 Teaching strategies . 2379.5.2 Student answers to the P-value items in the test . 2409.6 A DESCRIPTION OF THE TEACHING STRATEGIES USED IN THE TEACHING OF P-VALUES INSECOND CYCLE OF THE INTERVENTION . 2429.6.1 Stage 1 – Introduction to probabilistic reasoning . 2439.6.2 Stage 2 – Consolidation – hypothetical probabilistic reasoning in another context . 2469.6.3 Stage 3 – Simulation of P-values . 2479.6.4 Stage 4 – Introduction to the formal hypothesis testing procedure . 25011
9.6.5 Stage 5 – P-values in other contexts – chi-squared tests for independence, the analysisof variance, and linear regression . 2559.6.6 Stage 6 - revision .2609.7 SUMMARY OF THE MISCONCEPTIONS IDENTIFIED DURING THE STUDY IN COMPARISON TO THELITERATURE . 2619.8 HOW STUDENTS‟ UNDERSTANDING CHANGED OVER THE INTERVENTION . 2629.9 IMPLICATIONS FOR TEACHING . 26410. AN ANALYSIS OF STUDENTS’ UNDERSTANDING OF CONFIDENCE INTERVALS . 26710.1 INTRODUCTION . 26710.2 RESULTS OF THE PRE-INTERVENTION SEMES
sist students in gaining the ability to explain the principles behind two tools of statistical inference: P-values and confidence intervals for the population mean. Computer simulations were used to introduce students to statistical concepts. Stu-dents were also introduced to alternative representations of hypothesis tests, and
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