An analytical model for assessing the knowledge of statistical procedures amongst postgraduate students in a higher educational environment

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AN ANALYTICAL MODEL FOR ASSESSING

THE KNOWLEDGE OF STATISTICAL PROCEDURES

AMONGST POSTGRADUATE STUDENTS IN A

HIGHER EDUCATIONAL ENVIRONMENT

Germaine Kamleu

Student No: 3008010

Thesis submitted in fulfilment of the requirements

for the degree of Doctor of Philosophy in Applied Mathematics in the Department of Mathematics and Applied Mathematics,

Faculty of Natural Sciences, University of the Western Cape

Supervisors: Prof. Lorna Holtman Prof. Bingwen Yan

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ABSTRACT

Over the past decades, the use and application of statistical concepts for university students have been a big challenge learned from their previous courses. Aftermath of democracy, South African higher education focused on redressing issues of reparation and social imbalances inherited from Apartheid with the commitment to reconstruct a comprehensive educational quality framework. Growing activities lead to new models emphasised to support students and universities in their attempts to demonstrate evidence of enthusiastic statistics learning, with an acceptable degree of accuracy. This study combines quantitative and qualitative research approaches to assess the knowledge of postgraduate students in applying suitable statistical procedures in higher education (HE). The quantitative data were randomly collected from the postgraduate students (n1=307) while the qualitative data were collected

through semi-structured interviews (n2=19) from two institutions (University of Cape Town

[UCT] and University of the Western Cape [UWC]) in the Western Cape, South Africa. The SPSS V24 statistical package was used for quantitative data analysis and the explorative design was selected as a theoretical framework to guide the investigation, analysis and interpretation of the qualitative findings. UCT model achieved for all combined categories 73% high prediction accuracy. The UWC model revealed similar results, with ask for help, worth of statistics, fear of statistics monitors, affect, cognitive competence, support from significant others, marital status, ethnic groups and type of study as significant predictors with a high prediction accuracy of 75.49%. Additionally, the ethnic groups, marital status, postgraduate programmes, experiences in statistics and effort were significant contributed factors of SELS beliefs while findings of the combined data of UCT and UWC significantly explained the variation observed in SELS beliefs with only 60% model accuracy.

Nevertheless, the qualitative data outcomes indicated that the comments of the participants provided a rich understanding of the perceived failure to choose a relevant statistical test. The results further indicated that confusion and frustration characterised the attitude of students during the selection of a suitable statistical test. The original value of this current study is bridging the inequity gap, in terms of statistics learning, and building a substantial input to the achievement of the objectives of UNESCO, the World Education Forum and the White Paper 3, while ultimately, contributing to the sustainable development of learning statistics at universities in the Western Cape, South Africa. By logical extrapolation, this current study proffers significant insights to the rest of the universities in Africa, and beyond.

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KEYWORDS

Academic environment Postgraduate research Self-efficacy Social support Statistical procedures

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LIST OF ACRONYMS AND ABBREVIATIONS

ABE Adult Basic Education and Training

ANOVA Analysis of Variances

ANC African National Congress

ATS Attitudes towards Statistics

BUs Black Universities

CC Complete Confidence

CCA Canonical Correlation Analysis

CFA Confirmatory Factor Analysis

CFI Comparative Fit Index

CI Confidence Interval

COSATU Congress of South African Trade Unions

Cog Comp Cognitive Competence

CUMSA Curriculum Model for South Africa

DDA Descriptive Discriminant Analysis

DF Degree of freedom

ET Effective in service Training

ExpRmeth Experiences in Research Methodology ExpStats Experiences in Statistics

FET Further Education and Training

GET General Education and Training

GPA Grade Point Average

HEQC Higher Educational Quality Council

HES Higher Education System

HSRC Human Sciences Resource Council

MANOVA Multivariate Analysis of Variances

MBE Ministry of Basic Education

MHET Ministry of Higher Education and Training

MSPSS Multidimensional Scale of Perceived Social Support NCDC National Curriculum Development Committee NCHE National Commission on Higher Education

NGOs Non-Governmental Organisations

NLRD National Learners‟ Records Database

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OBE Outcome-based Education

OBET Outcome – Based Education and Training

OECD Organisation for Economic Co-operation and Development

P P-value

PCA Principal Component factor Analysis PNI Preference for Numerical Information

RC Regression Coefficient

RMSEA Root Mean Square Error of Approximation

RPL Recognition of Prior Learning

RSA Republic of South Africa

SA Strongly Agree

SAQA South African Qualifications Authority SATS Survey of Attitudes Towards Statistics SDQIII Self-Descriptive Questionnaire III SelfEffAbsMeanOrd Self-Efficacy Absolute Mean Ordinal SELS Self-Efficacy to Learn Statistics

SITSTATS Situations that are commonly associated with statistical anxiety SPSS Statistical Package for Social Sciences

STARS Statistics Anxiety Rating Scale STASTATS Statements related to statistics

UCT University of Cape Town

UN United Nations

UNDP United Nations Development Program

UNEP United Nations Environment Program

UNESCO United Nations Educational, Scientific and Cultural Organisations UNICEF United Nations Children‟s Fund

UNISA University of South Africa

UWC University of the Western Cape

WCEFA World Conference on Education for All

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DEDICATION

I would like to express my heartfelt gratitude to everyone who played a role, and assisted me in the course of this dissertation, not forgetting those who constantly prayed for me. I dedicate this work you all.

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ACKNOWLEDGEMENTS

In the name of God, the beneficent, the most merciful. I am grateful to God for all the blessings.

I would like to extend my outmost gratitude to my supervisors, Professor Lorna Holtman and Professor Bingwen Yan, for your enormous support, guidance, understanding and constant encouragement until the completion of this dissertation. Without your assistance, it might not have been possible for me to get to this level. Thank you Profs.

Today we are able to achieve our dream because of the decisive support of the National Research Foundation (NRF) scholarship that covered this PhD research. To Prof. Richard Fray, at the Department of Mathematics and Applied Mathematics, thank you so much for your support.

My heartfelt gratitude also extends to Mr Robert Brown for his valuable time, constant support and encouragement. Thank you for assisting me whenever I needed help, but most importantly, in attending several international conferences for oral and poster presentations in USA, your help is much appreciated.

I also thank all my colleagues, with whom I enjoyed every day of the 3 years, as well as the Francophone Catholic Community in Cape Town.

To my family members, I am very thankful for their patience, encouragement and love. To my late parents and eldest sister, for their prayers and blessings, which always directed me ‒ may their souls rests in peace.

To all my brothers and sisters, with whom I shared a wonderful childhood.

Finaly, to my husband, Ndouma Bernard, and my children Belinga Ndouma, Ndouma Teudjeu, Ndouma Kemajou and Ndouma Ndengue, for their understanding, compassion, unfailing encouragement and all the happiness they bring to my life.

To everyone who supported me during this study, Thank you. GOD BLESS YOU ALL.

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6.2.1.1.3. Marital status ... 181 6.2.1.1.4. Ethnic groups ... 181 6.2.1.1.5. Postgraduate programmes ... 182 6.2.1.1.6. Student status ... 183 6.2.1.1.7. Type of study ... 183 6.2.1.2. Experiences ... 184 6.2.1.3. STARS ... 184 6.2.1.4. SATS ... 185 6.2.1.5. Social support ... 186 6.2.1.6. Self-efficacy ... 187 6.2.1.6.1. Current self-efficacy ... 187 6.2.1.6.2. Self-efficacy to learn ... 188

6.2.1.6.3. Assessment scores for self-efficacy to learn statistics ... 189

6.2.2. Reliability of the Scales for UWC ... 190

6.2.3. Impact of Independent Variables on SELS ... 191

6.2.3.1. Impact of background information on SELS ... 191

6.2.3.2. Impact of STARS on SELS ... 195

6.2.3.3. Experiences differences ... 197

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6.2.3.5. Social support differences ... 202

6.2.4. Multivariate Analysis UWC ... 204

6.2.4.1. Predictive value of the model ... 205

6.2.4.4. Classification table ... 207

6.2.4.6. Principle of parsimony ... 212

6.2.4.7. Predictive value of the model with Logit link ... 212

6.2.4.11. Evaluation of the accuracy of the classification response categories ... 216

6.2.4.12. Correlation estimated classification probability between predicted and actual category ... 217

6.3. Qualitative findings ... 218

6.3.1. Description of the Study Participants ... 218

6.3.2. Participants‟ Responses ... 218

CHAPTER 7: COMPARATIVE QUANTITATIVE RESULTS OF BOTH UNIVERSITIES ... 228

7.2. Descriptive analysis ... 228

7.2.1. Background Information ... 228

7.2.2. Experience components, mean scores and standard deviation ... 230

7.2.3. STARS Components, Mean Scores and Standard Deviation ... 230

7.2.4. SATS Components, Mean Scores and Standard Deviation ... 231

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7.2.6. Self-Efficacy to Learn ... 233

7.2.6.1. Current self-efficacy (to solve) ... 233

7.2.6.2. Self-efficacy to learn (expectation) ... 234

7.2.6.3. Dependent variable: SELS ... 234

7.3. Reliability of the scales for both universities ... 235

7.4. Comparison analysis ... 237

7.4.1. Comparison of each independent variable across Universities ... 237

7.4.1.1. Background information on both universities ... 237

7.4.1.2. Comparison of experience components across universities ... 239

7.4.1.3. Comparison of STARS components across universities ... 241

7.4.1.4. Comparison of SATS components across universities ... 242

7.4.1.5. Comparison of social support across universities ... 244

7.4.1.6 Self-efficacy to learn ... 246

7.4.2. Comparison effect of independent variables on the SELS across universities ... 247

7.4.2.1. Impact of background on the SELS across universities ... 247

7.4.2.2. STARS differences ... 250

7.4.2.3. Experience differences ... 253

7.4.2.4. SATS differences ... 255

7.4.2.5. Social support differences ... 257

7.5. Multivariate analysis: Ordinal Regression Model ... 260

7.5.1. Predictive value of the model ... 261

7.5.2. Tests for goodness-of-fit ... 262

7.5.3. Pseudo R-squared measures ... 262

7.5.4. Test of parallel lines ... 263

7.5.5. Parameter estimates ... 264

7.5.6. Predictive value of the model ... 266

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7.5.8. Test of parallel lines ... 267

7.5.9. Pseudo R-Squared Measures... 267

7.5.10. Interpreting the Coefficients ... 268

7.5.11. Principle of Parsimony ... 270

7.5.12. Examination of the accuracy of the classification results ... 270

7.5.13. Correlation Estimated Classification Probability... 271

CHAPTER 8: DISCUSSION OF THE FINDINGS... 273

8.2. Main procedures followed in the research design... 273

8.3. Discussion of the findings of each academic institution ... 276

8.3.1. Assessment of the findings at UCT ... 276

8.3.1.1. Assessment of the quantitative findings at UCT ... 277

8.3.1.2. Assessment of the qualitative findings at UCT ... 285

8.3.2. Assessment of the findings at UWC ... 296

8.3.2.1. Assessment of the quantitative findings at UWC ... 296

8.3.2.2. Assessment of the qualitative findings at UWC ... 306

8.4. Discussion of the findings‟ combined data ... 322

8.4.1. Assessment of quantitative findings‟ combined data... 322

8.4.1.1. Assessment comparison of each variable across universities ... 322

8.4.1.2. Assessment of the impact of the predictors on SELS ... 328

8.4.1.3. Assessment findings multivariate combined data ... 332

8.4.2. Summary of the evaluation of the qualitative findings at UCT and UWC ... 334

8.5. Limitations of ordinal regression and Likert-type scales ... 337

8.5.1. Strengths and limitations of ordinal regression in this current study ... 337

8.5.2. Limitations of Likert-Type Scales on using and learning statistics ... 338

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CHAPTER 9: CONCLUSIONS, RECOMMENDATIONS, CONTRIBUTIONS AND

LIMITATIONS OF THE STUDY... 341

9.2. Conclusions ... 341

9.2.1. UCT... 342

9.2.2. UWC ... 345

9.2.3. Combined data from both universities ... 348

9.2.4. Summary of the qualitative findings from UCT and UWC ... 349

9.3. Integrating the qualitative and quantitative results ... 350

9.3.1. UCT... 350

9.3.2. UWC ... 351

9.3.3. Both UCT and UWC ... 354

9.4. Policy recommendations ... 355

9.4.1. A benchmark as a guide for future research ... 355

9.4.2. Consideration of a multi-disciplinary character of statistics learning in related programmes and projects ... 356

9.4.3. Knowledge transfer improvement (statistics monitors, supervisors & peers) ... 356

9.4.4. Strengthen learning programmes with an important statistics education aspect .. 357

9.4.5. Promote student empowerment for self-regulated learning ... 357

9.4.6. Particular attention to young postgraduate students ... 357

9.4.7. Consideration of specific population groups with important impact, such as females and ethnic groups ... 358

9.4.8. Consideration of a university approach to knowledge failure in statistics learning, through related programmes and projects design ... 358

9.4.9. Need for a specific national survey on statistics learning ... 359

9.5. Contributions and limits of the study ... 359

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APPENDICES... 400

Appendix 1: Additional tables ... 400

Appendix 2: Application to do research Form DSA 100 – UCT ... 405

Appendix 3: UWC Ethics clearance letter ... 406

Appendix 4: Request for participation in Quantitative phase of research study letter ... 407

Appendix 5: Information sheet for participants ... 408

Appendix 6: Quantitative Instrument ... 409

Appendix 7: Qualitative Questionnaire ... 416

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LIST OF TABLES

Table 4.1: Cronbach Alpha coefficient results of the preliminary instruments ... 110

Table 5.1: Characteristics of respondents (UCT) ... 135

Table 5.2: Distribution of student‟s experiences ... 141

Table 5.3: Distribution of students‟ STARS ... 142

Table 5.4: Distribution of student‟s SATS ... 143

Table 5.5: Distribution of student‟s social support ... 144

Table 5.6: Distribution self-efficacy to solve ... 145

Table 5.7: Distribution self-efficacy to learn ... 146

Table 5.8: Distribution of current responses of SELS ... 147

Table 5.9a: Group statistics... 149

Table 5.9b: Independent Samples Test ... 150

Table 5.9c: Test of homogeneity of variances ... 151

Table 5.9d: ANOVA ... 152

Table 5.10b: Independent samples test ... 153

Table 5.11a: Test of homogeneity of variances ... 154

Table 5.11b: ANOVA experiences ... 155

Table 5.11c: Multiple comparisons... 156

Table 5.12b: Robust tests of equality of means ... 157

Table 5.12c: SATS ANOVA ... 158

Table 5.12d: Post Hoc Tests ... 159

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Table 5.14 Model-fitting information ... 163

Table 5.15 Test of parallel linesa ... 164

Table 5.16: Pseudo R-squared measures ... 164

Table 5.17: Confusion matrix for the initial model ... 165

Table 5.18: Parameter estimates ... 166

Table 5.19: Model-fitting information ... 168

Table 5.20: Pseudo R2 ... 168

Table 5.21: Test of Parallel Linesa ... 169

Table 5.22: Correlation between predicted and actual category ... 170

Table 6.1 Background information (UWC) ... 183

Table 6.2: Experience components ... 188

Table 6.3: STARS components... 189

Table 6.4: SATS components ... 190

Table 6.5: Social support components ... 191

Table 6.6: Self-efficacy to solve ... 192

Table 6.7: Self-efficacy to learn... 192

Table 6.8: Distribution current responses of SELS ... 193

Table 6.9b: Independent samples Test ... 197

Table 6.9c: Test of homogeneity of variances ... 198

Table 6.9d ANOVA ... 198

Table 6.10b Independent samples test ... 200

Table 6.11a: Test homogeneity of variances ... 201

Table 6.11c: ANOVA experiences ... 202

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Table 6.12a: Test of homogeneity of variances ... 204 Table 6.12b: SATS ANOVA ... 205 Table 6.13a: Test of homogeneity of variances ... 206 Table 6.13b: Social support ANOVA ... 207 Table 6.14: Model-fitting information ... 209 Table 6.15: Test of parallel linesa ... 210 Table 6.16: Pseudo R-squared measures ... 211 Table 6.17: Confusion matrix for the initial model ... 211 Table 6.18: Parameter estimates ... 214 Table 6.19: Model-fitting information ... 216 Table 6.20: Test of parallel linesa ... 217 Table 6.21: Pseudo R-squared measures ... 218 Table 6.22: Parameter estimates ... 219 Table 6.23: Cross-tabulation ... 220 Table 6.24: Correlation between predicted and actual category ... 221 Table 7.1: Description and Chi-square individual characteristics for both universities ... 233 Table 7.2: Experience components ... 234 Table 7.3: STARS components... 235 Table 7.4: SATS components ... 236 Table 7.5: Social support ... 236 Table 7.6: Self-efficacy to solve ... 237 Table 7.7: Expected self-efficacy (to learn) ... 238 Table 7.8: Distribution current response category SELS ... 239 Table 7.9: Results of Cronbach‟s Alpha of test instruments ... 240 Table 7.10: Description and Chi-square of individual characteristics at both universities ... 243 Table 7.11: Chi-square test for independence of experiences components ... 244 Table 7.12: Chi-square test for independence of STARS components across universities ... 245

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Table 7.13: Chi-square test for Independence of SATS components across universities ... 247 Table 7.14: Chi-square test for independence of social support components ... 249 Table 7.15a: Group statistics... 250 Table 7.15b: Independent samples test ... 250 Table 7.16a: Group statistics... 251 Table 7.16b: Independent samples test ... 252 Table 7.16c Test of homogeneity of variances ... 253 Table 7.16d ANOVA ... 254 Table 7.17a: Group statistics for STARS ... 255 Table 7.17b: Independent samples test STARS ... 256 Table 7.18a: Test of homogeneity of variances ... 257 Table 7.18b: ANOVA experiences ... 258 Table 7.18c: Multiple comparisons of experiences ... 258 Table 7.19a: Test of homogeneity of variances ... 259 Table 7.19b: SATS ANOVA ... 260 Table 7.20a: Test of homogeneity of variances ... 261 Table 7.20b: Robust tests of equality of means ... 262 Table 7.20c: ANOVA social support ... 263 Table 7.20d: Post hoc comparisons social support ... 264 Table 7.21: Model-fitting information ... 266 Table 7.22: Goodness of fit with scale model... 266 Table 7.23: Pseudo R-square ... 267 Table 7.24: Test of parallel lines... 267 Table 7.25: Parameter estimates ... 269 Table 7.26: Model-fitting information ... 270 Table 7.27: Goodness of fit with scale model... 271 Table 7.28: Test of parallel linesa ... 271

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Table 7.29: Pseudo R-squared measures ... 272 Table 7.30: Parameter estimates ... 273 Table 7.31: Current responses against predicted response categories ... 275 Table 7.32: Correlation between predicted and actual category ... 275 Table 8.1: Description of the qualitative data ... 279

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LIST OF FIGURES

Figure 2.1: Conceptual Framework Diagram ... 26 Figure 5.1: Percentage distribution of gender ... 136 Figure 5.2: Percentage distribution of age groups of the respondents ... 136 Figure 5.3: Percentage distribution of marital of the status respondents ... 137 Figure 5.4: Percentage distribution of ethnic groups of the respondents ... 138 Figure 5.5: Percentage distribution of Postgraduate programmes ... 139 Figure 5.6: Percentage distribution of student status of the respondents ... 140 Figure 5.7: Percentage distribution of type of study ... 140 Figure 5.8 Distribution of students‟ overall self-efficacy to learn statistics at UCT ... 163 Figure 6.1: Percentage distribution of respondents‟ gender ... 184 Figure 6.2: Percentage distribution of respondents‟ age groups ... 184 Figure 6.3: Percentage distribution of respondents‟ marital status ... 185 Figure 6.4: Percentage distribution of respondents‟ ethnic groups ... 186 Figure 6.5: Percentage of respondents‟ distribution across postgraduate programmes ... 186 Figure 6.6: Percentage distribution of student status ... 187 Figure 6.7 Percentage distribution of type of study ... 187 Figure 6.8: Distribution of the overall students‟ self-efficacy to learn statistics at UWC ... 209 Figure 7.1: Distribution of students‟ self-efficacy to learn statistics in both universities ... 265

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CHAPTER ONE

SCOPE OF THE RESEARCH

1.1. Introduction

According to Mda and Mothata (2000), the education system of a country reflects what the country values and holds dear. Since the 1994 general elections, the South African government has focused on the development of new educational policies. These policies have not only transformed the education system, which was highly fragmented and chaotic during the apartheid era, but also provided direction and support to students and academic institutions (Barth, Godemann, Rieckmann & Stoltenberg, 2007; Reddy, Juan & Meyiwa, 2013). This current research, therefore, focusses on lifelong learning (postgraduate programmes), from formal, through non-formal to informal education (Laal & Salamati, 2012).

Batanero and Díaz (2012) acknowledge that statistics is a section of the mathematics programmes for secondary and primary school in various nations. Various scholars highlight the utility of likelihood and statistics for regular life, its active role in other subjects, the requirement for a simple theoretical awareness in numerous occupations, and the vital role of statistics in emergent critical reasoning (Ottaviani & Batanero, 1999; Gal, 2002; Franklin et

al., 2005). Students from secondary and primary school are likely to develop statistical

reasoning by designing research studies; formulating research questions; collecting data, applying comments, investigations, then tests; define and associate data sets; use and appreciate diagrams and events; suggest and validate decisions, as well as estimates that are built on data (Garfield & Ben-Zvi, 2008; Batanero, Burrill & Reading, 2011).

However, statistical cognitive is diverse from mathematical perceptive. Although together, these ways are vital to daily life, supplementing each other support the mathematics curriculum in general for students (Batanero et al., 2011; Batanero & Díaz, 2012). As mentioned earlier, the teaching of statistics often connects statistics with the general process of inquiry, and involves doing calculations, or showing mathematical formulas, with limited break to design tests and examine statistics (Batanero & Díaz, 2010). Consequently, students complete their academic education with slight thoughtful of the elementary values underlying

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statistics examination, which explains many of the problems they meet with the practice of statistics in everyday and expert lives, or in statistics courses at university level (Batanero & Díaz, 2010; Cohen, Manion & Morrison, 2013).

The relationship between statistics learning quality and statistics learning productivity is crucial to the development of a high-quality educational system (Van der Wende, 2007). Within this relationship, initial learning statistics serve as the substance to re-define the self-development of statistics learners, facing the present challenges of a globalised society (Engeström, 2014). Pajares (1996) acknowledges that the self-efficacy of students has been the focus of research for decades, yet “there is no agreement on what factors enhance learner quality”. Since quality learning of statistics is the critical factor, there is an urgent need to re-conceptualise how to prepare a generation of students, equipped to meet the demands of the 21st-century student (Tishkovskaya & Lancaster, 2012).

1.2. Study Problem

Although various studies on statistics education have been conducted worldwide (Shaffer, Hall & Van der Bilt, 1999; Ben-Zvi & Garfield, 2004), many aspects of statistics learning in middle developing countries, such as South Africa, has not been clarified, or explored (Bray & Thomas 1995). This indicates that knowledge failure in statistics is a serious educational learning issue of students all over the world. The impact of the problem is more accentuated in developing countries, particularly in Africa (Sawyerr, 2004). The consequences of knowledge failures in learning statistics have a dramatic effect on social, economic and innovation (Argote, 2012). Statistics form an integral part of many academy programmes (Nasser, 2004). Because of the consequences that knowledge failure in learning statistics could have on the sustainable development of a country, actions are constantly taken at universities, nationally and internationally, to overcome the problem (Delors, 1998).

Recommendations have been made at many international meetings regarding the reduction of knowledge failure in education to address the transformation of the HES (United Nations Educational, Scientific and Cultural Organisations [UNESCO], 2014; World Bank, 2011; Republic of South Africa [RSA], National Department of Education [NDoE], 1997b). South Africa ratified all the international decisions related to knowledge failure in HES (Allais, 2007). National policy and programmes also incorporated the issue of knowledge failure in

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statistics education, in higher education (Subotzky, 1999). All these strategies in learning statistics have been implemented in the South Africa, but the level of knowledge failure in statistics education is still rated among the highest in the world (Onwuegbuzie & Wilson 2003; Pan & Tan, 2004). Unfortunately, there is a significant lack of data, while the existent data are of poor quality (Judge, Puckett & Cabuk, 2004; Cherkassky & Mulier, 2007; Janssen, Charalabidis & Zuiderwijk, 2012). In 2004, South Africa consolidated 36 universities into 21 institutions that could promote social cohesion among all South African students (Cloete, 2006). However, policy and programmes are still developed at national level and implemented at university level (Caldwell & Mays, 2012). This situation is partly motivated by the lack of experts at the university level and the real lack of data and study at that level (Tremblay, Lalancette & Roseveare, 2012). Very few numbers of scientific studies have been undertaken on the subject in Africa (Mji & Makgato, 2006).

Among the following authors cited in this paragraph, one category focuses on undergraduate students and another category concentrates on post-graduate studies. The former category is separated into two categories; the principal category assesses the statistical reasoning of students (Garfield & Ahlgren, 1988; Gal & Ginsburg, 1994; Mvududu, 2003), while the second group does course evaluation (Latief, 2005; Makapela, 2009) in South Africa. The latter category concentrates on post-graduate studies (Gardner & Hudson, 1999; Perepiczka, Chandler & Becerra, 2011), with an exception that Gardner and Hudson (1999) mix undergraduate and post-graduate students in their studies. However, the small sample size and the diverse background of their students/participants make it difficult to generalise accurately to a wider population. Even though Perepiczka et al. (2011) focus on post-graduate students in USA, they do not consider the marital status, ethnic groups, postgraduate programmes, student status, and type of study, which could explain some influences in the learning status of the students. In addition, most of these research projects are only implemented in developed countries, such as Australia and USA. Little empirical research exists for similar initiatives in developing countries (Fuller, 1986). These facts, therefore, highlight the need for a university approach to the problem of knowledge failure in statistics education.

This situation raised many interrogations, which motivated the initiation of this current research. The principal research question is, therefore, “How can statistics learning oriented policies in South Africa be improved efficiently?” Many subsidiary questions are also raised; “What are the explanatory factors for the postgraduate students‟ low level of statistics ability

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at UCT, UWC and the combined data of both universities?” “What are the attempts made, or actions already implemented, to reduce statistics learning failure?” and “What are the current and future consequences of statistics learning failure, if more attention is not paid to the problem?” The intention of this current research is to examine the knowledge level of postgraduate students, in applying appropriate statistical procedure in their studies.

Due to the progress made in statistical techniques over the years, involving new statistics tasks, as well as the differences in students‟ academic backgrounds, the Self-efficacy to Learn Statistics (SELS) beliefs of students might also be different (Zieffler et al., 2008; Artino, 2012). The lack of statistical knowledge and practical experiences leads to anxiety among postgraduate students (in terms of attitude toward statistics, individual characteristics, social support, etc.) in their academic research (Perepiczka et al., 2011; Zhang et al., 2012). Students still encounter difficulties in identifying the extent to which statistical procedure is appropriate to answer research questions, or to evaluate a variety of misconceptions, when applying real world problems (Garfield, 1995; National Research Council [NRC], 2004). There is a lack of an effective approach for students to follow, in order for them to gain a better understanding and interpretation (self-regulation) of statistical procedures (Clark, 2012). Therefore, the poor performance of post-graduate students is differentially distributed among ethnic groups, and according to student status (Froehlich, 2015). Urgent action is needed to address this specific problem for solutions, as the number of students with inadequate knowledge in statistics could decrease dramatically, if more effort is applied to reducing the grave differences between universities (Gurian, 2010).

This phenomenon is not only the concern of statisticians, as many researchers are trying to find the suitable methodology, with the smallest possible bias, for estimating statistics learning (Ioannidis, 2008). Finding a lasting solution to this problem requires more knowledge about learning statistics, as well as the provision of reliable indicators that reflect the reality of different geographical areas (Ioannidis, 2008). Many methods have been applied, such as the multiple regression and structural modelling equation, to build models; however, they still have critical limitations and need to be improved (Mac Nally, 2000; Hox & Bechger, 2007). As with most social issues, the reliability of methods varies from one university to another, depending on the specific socio-cultural and economic environment (Fougeyrollas et al., 1998; Nunnally & Durham, 1975).

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In South Africa, as in many African countries, the problem of statistics learning is a major challenge for postgraduate students because of its complexity, lack of data and poor quality of existing data (Cherkassky & Mulier, 2007; Janssen et al., 2012). Another concern for researchers and policy makers is the explanation of learning statistics failure, which involves a statistical model that is difficult to implement because of the weaknesses of existent data (Majone, 1989). In addition to these difficulties, an important insufficiency of scientific work, related to this issue in South Africa, has been identified (Banks & Banks, 2004). However, estimating the level of SELS beliefs at the university, and the combined data of both UCT and UWC, is an important scientific issue; therefore, finding research factors to explain the problem is a real challenge (Pajares, 1997). The development of a practical approach for post-graduate students to have a better understanding of applying statistical procedures effectively, adds to the significance of this current study.

To contribute to the body of knowledge on statistics learning issues, the researcher, in this current study, has set as the main objective, the provision of statistical information on the causes of statistics learning failure at the university level, for a better orientation of related programmes. The aims are, specifically, to:

 Determine explanatory factors of SELS beliefs;

 Assess the estimates of SELS beliefs levels at UCT, UWC and both scales;

 Compare the levels of SELS beliefs across universities;

 Predict the levels and trends of SELS beliefs across universities and for combined data of both universities; and

 Investigate a set of empirical results from multiple situations, as well as develop possible theories that explain and predict how to choose a relevant statistical test. 1.3. The status of students’ knowledge of using statistical procedures

1.3.1. Context of Developed Countries

A broader responsibility, beyond national interests for developed countries, exists in the interdependent world to address universal issues of higher education. The system of higher education in the USA includes community colleges, four-year colleges and research universities. There are state, religious affiliated and private universities. One

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can hardly call it a system. Thullen Heyl and Brownell (2002) mention that no central body at national level controls, or coordinates higher education in the USA, even with respect to international programmes and activities. Concerning research and graduate education, the United States is reputed to have one of the greatest systems of advanced education in the world (Kezar, Chambers & Burkhardt, 2015; Alexander, 2000).

However, the weakening of public asset in education and academic research tends to reduce the ability of America‟s academic research to yield new acquaintance for revolution (Mohrman, Ma & Baker, 2008); although, the specific heterogeneous character of the American HES keeps them dominant in the world (De Wit, 2009; Clark, 1986). Many countries set the striving objective of providing their people with prevalent, permanent learning. This creates factual learning environment. However, the constant expansion of education and social capital are driving economic prosperity, national security, and social welfare (Duderstadt, 2009). In addition, HES in the world currently tends to solve significant problems linked to the general dynamic conditions of students, in which their particular interest is based on solutions; therefore, institutions are compelled to produce the real changes in social interactions (Fullan, 2007).

1.3.2. Context of Developing Countries

During the course of the last decades, much of Sub-Saharan Africa has suffered from deep poverty, corruption, disease, natural disasters and international conflict (Ake, 1991; Ikejiaku, 2009). More remains to be done; the capacity to use statistical data is still limited; the challenges are to fit in indicators into policymaking, in order to endorse exposed right to use data, to improve their use, and to increase statistics systems (World Bank, 2011; De Francesco, 2016). Data are useful and relevant for refining people‟s life everywhere. Higher education in Africa is vitally important to its development. African universities have the ability to transform themselves and promote national development (Altbach, 2009). Unfortunately, its universities suffered from an enormous decline in government resources for education. Evaluating contemporary education systems in Africa is difficult, since it is a challenge to generalise (Sawyerr, 2004).

Regarding sub-Saharan countries, there is a great variety in the traditional cultural institutions and practices of ethnic groups. In many cultures, authority is centralised, leadership tends to be inherited, and advancement is limited to particular lineages and clans (Omolewa & Kellaghan, 2003). Position of prestige, authority and leadership are

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achieved based on the individual‟s ability, knowledge and skills. However, the meritocratic assumptions in Western systems of education and evaluation would differ from the African traditions. African societies are different because some regions were under Western cultural influences. Such cultural contact was marked by the willingness and openness of traditional cultures to change, and their acceptance of a new ideology (Omolewa & Kellaghan, 2003).

An attempt to address these limitations of knowledge and space (variation in educational activity on the continent and the complexity of that variation) is inadequately presented. Learning had standard criteria, by which those, who engaged in learning, were certified as having been successful (Omolewa & Kellaghan, 2003). This creates an avenue to improve the quality of trans-border higher education, in the logic of ensuring its occasions, enhanced by innovative actions in higher education. The potential is summed up in these two aspects, adding the internationalisation of higher education, as a root for the expansion and exhibition of scientific knowledge at various levels. Since international donors (partners such as World Bank [2008] and United Nations Educational, Scientific and Cultural Organisations [UNESCO] 2009]) supported statistical capacity and the application of statistical procedures, as well as development in developing countries in 2008, much progress has been made concerning these issues. Recently, the 2009 Dakar Declaration on the development of statistics, reiterated that intensive effort is required to use statistical data more effectively, to support programmes, as well as strengthen and sustain the capacity of statistical systems, especially in developing countries (UNESCO, 2009).

Empirical studies, conducted at various universities, are genuine evidence that confirm the interactions between changes in academic context, the students‟ influence on teaching, as well as their learning outcomes (Laurillard, 2013; Biggs, 2011). An attempt to restructure learning cultures to respond to frameworks, may unfortunately discourage innovation and enhance passive, as well as instrumental attitudes to learning. Regarding statistical learning, considerable literature is available on self-efficacy to learn statistics among students in developed countries; however, comparatively little scholarly attention was given to Africa (Biggs, 2011; Knight & Yorke, 2004).

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Concerning the application of statistical procedures, universities are expected to increase graduates‟ total understanding of statistics, as well as assist them with special and broad skills, regarding their capacity to innovate the ability to learn (Schunk, 2008). The real impact of the learning programme focusses on the interactions between certain key changes, such as academic aspects, teaching and the curricula, students‟ abilities and their feedback (Biggs, 2011). Education is likely to be transformed as a pedagogical exchange, in which the lecturer is considered the producer of knowledge and the student, the consumer of knowledge (Molesworth, Nixon & Scullion, 2009). Ultimately, rather than merely stipulating new procedures to enhance higher education, mechanisms may be perceived as a way to transform academic values and relationships, to comply with market frameworks (Naidoo, Shankar & Veer, 2011).

1.3.3. Context of South Africa

Given that little is known about statistics problems among university students in South Africa, the present study was conducted, in order to evaluate whether the postgraduate students had any differences about self-efficacy in learning statistics at Western Cape universities. The apartheid system promulgated that some ethnic groups had limited ability, and, therefore, were only allowed to study in specific programmes and courses (Swartz & Foley, 1996; Higher Education Quality Committee [HEQC], 2006). For a variety of reasons, black students entered university into fields such as social work, or divinity (Badat, 2016).

Therefore, South African advanced education reform was challenged with collective, administrative and financial problems, arising from global environments, after the democratic elections of 1994 (Scott, Yeld & Hendry, 2007). Higher education in South Africa involves the interaction of policies to engage problems of reparation, and admission to higher education remains one mechanism for the achievement thereof in South Africa (Horsthemke, 2004). An introduction of a quality assurance framework enables admission, by improving student success, as opposed to simply ensuring their participation (Akoojee & Nkomo, 2007). A comprehensive quality assurance agenda, with fixed pledge to admission, would probably reply suitably to national development prerogatives of higher education access (Akoojee & Nkomo, 2007). South African universities have engaged various initiatives to substitute, as well as respond to the changes in society, while preparing students for the realities of increasing globalization.

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The proper expansion of expertise could help to achieve academic performance, enabling students to encounter the requirements of industry. University staff members are progressively redirecting students to address the social imbalances inherited from apartheid, as well as respond to social needs (Nicol & Macfarlane-Dick, 2006).

Real world difficulties are multifaceted; therefore, it is vital to mix knowledge and information from various academic disciplines, including education, engineering, art & humanity, applied, natural and social sciences (Griffiths*, 2004). Based on what is learnt from diverse experiences, the aforementioned challenges are addressed simultaneously, in order to fulfil the reality of learning statistics (Fink, 2013). Shared learning, through teamwork, includes the organisation of academic programmes, as well as the pathway for careers (Brundiers, Wiek & Redman, 2010). This current study, however, promotes a better understanding of the interaction between graduate students and their environment, and collaborates on common environmental issues. Understanding how the notion of statistics has been combined into academic research, is indispensable to contextualising practices and behaviours for quality assurance (Brundiers, Wiek & Redman, 2010). This current study opts for a large contextual background to assess the postgraduate students‟ ability to apply statistics skills and deal with social and behavioural sciences at the relevant universities. An important aspect to consider is that these students have learned statistics courses, in some cases, for one or two semesters, in other cases, for three years, as part of their programme. As mentioned previously, the statistics course is a tool for the understanding of research projects. It also contributes to the accomplishment of a specific research situation, instead of being considered a pure academic discipline (National Research Council [NRC], 2012). However, the research methodology and statistics courses are intended to expand students‟ computational skills, with or without the conceptual understanding of theory, or the use of computer packages (Chance, Ben-Zvi, Garfield & Medina, 2007).

South African scholars face serious challenges to sustain research and intellectual activities around socio-cultural diversity, which is still under pressure. The legacies of Apartheid remain a barrier, when developing a critical theory of diversity for the recognition of difference that advances the cause of social equality, through students who retard it, or undermine it (Jansen, 2004; Higham, 2012). The South African government faces very hard queries about how best to reform the HES. The significant

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skills allow students to think more critically and learn independently, thereby reducing dropout rates, achieving full potential, and utilising better knowledge in industry. In addition, good statistics are essential for developing policies that effectively improve the welfare and productivity of students (Altbach, Reisberg & Rumbley, 2010).

1.4. The Higher Education System Reconstruction in South Africa

The South African Qualifications Authority [SAQA] Act (Republic of South Africa, Act 58 of 1995) offers the background, which accommodates South African education and training, in the democratic era. South Africa has inhabitants closely 50 million, of which more than 41 per cent stayed in the worse countryside areas, and near to 54 per cent aged less or equal to 24 years (Blom, Parker & Keevy, 2007). South Africa is experiencing fast urban relocation, particularly males aged between 20 and 34 years. As a result, a high proportion of households in rural areas are headed by young women (Blom et al., 2007). The inheritance of Apartheid remains an important task in South Africa, with various formal and informal initiatives underway to advance the level of education of black people (Fiske & Ladd, 2004; Leibbrandt, Woolard, McEwen & Koep, 2010). Although the levels of discrimination persist great (Gini coefficient = 0.35), they are no longer exclusively founded on ethnic separations (Blom et al., 2007). The joblessness occurs mainly in the less than 30 age group (49 per cent in 2002). Unemployment is advanced among females compared to males. Most black employees remain in elementary occupations (Blom et al., 2007).

All countrywide known diplomas are chronicled in the National Learners‟ Records Database (NLRD), which include both unit based qualifications, and non-unit standards-based qualifications (South African Qualifications Authority [SAQA], 2013). After a lengthy review process, the new National Qualifications Framework [NQF] Act (Republic of South Africa [RSA], Act No. 67 of 2008) was passed in 2008, with effect from 1 June 2009. A single Ministry of Education was divided into two: a Ministry of Higher Education and Training (MHET), and a Ministry of Basic Education (MBE). This meant that the MHET would be accountable for universities, universities of technology, training colleges (for example, nursing, agricultural, etc.), colleges of further learning and practice, mature elementary instruction and the entire working out sector.

Diplomas and unit values are described as education results and evaluation standards. One recognition is equal to ten theoretical hours of knowledge. If non-formal and relaxed learning

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leads to recorded criterions in the NQF, and the providers are accredited, the learning can be formally recognised (RSA, SAQA Act No. 58 of 1995). RPL rule permits applicants to attain credentials, in share, or in filled, done the recognition of prior learning. SAQA arrived into a research corporation with the University of the Western Cape, to research why recognition of prior learning (RPL) has failed to take off as envisaged. The research is being conducted at a variety of different sites. Extensive regulation is approved, determined mainly through the developmental program, again leading in this country (SAQA, 2013; Singh & Duvekot, 2013).

Higher education shows an important role in the community, traditional, and financial growth of the country (National Commission on Higher Education [NCHE], 1996). South Africa‟s change from Apartheid to Democracy needs that all current rehearses in education, institutions and values to be revised in terms of their appropriateness for the new era (Harvey & Knight, 1996; Henrard, 2002). The development of learning stimulates intellectual dynamisms of all the people, towards achieving the challenge of renovation and expansion of higher education in South Africa (Republic of South Africa [RSA], National Department of Education [NDoE], 1997b). However, the challenges in South Africa are to re-dress past disparities, alter the HES, help a novel communal instruction to light countrywide requirements, as well as opportunities, and respond to new realities (RSA, NDoE, 1997b). Studies on statistics learning in South Africa, conducted over several decades, reveal that progress has been made in research methods leading to greater observed effectiveness in learning. This progress also emphasizes the reform in the learning process (Sammons, Hillman & Mortimore, 1995; Nomlomo, 2007).

The Education White Paper 3 entitled, “A programme for the transformation of higher education”, emphasizes the framework for change that enables the eradication of fragmentation, inequality, inefficiency, and creates an education environment, which releases the original and intelligent dynamisms of students, to achieving the objectives of change of HES (RSA, NDoE, 1997b). The reconstruction of the advanced learning system contributes to satisfying the learning needs of students, through the development of abilities (Boud, 2000; Biggs, 2011). Singh (2001) and Singh (2011) argue that the clarity of social justice issues in higher education responsiveness is most likely difficult to achieve. This change prepares students to brand the greatest usage of their aptitudes, and the options accessible at the university, for self-fulfilment. Cook-Gumperz (1986) emphasises that bringing about social

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change and stability, provides a set of guidelines about the use of knowledge. In case the outcomes are not as expected, the problems are seen as attributable to educational failure.

This transformation attempts to achieve equity in the distribution of options among students. It addresses the development needs of learning in a driven and knowledge-dependent society (Singh, 2001; Badat, 2010). In addition, it provides the ever-changing labour market with high levels of abilities and skill, which are essential for the progress and wealth of a student (Brennan & Naidoo, 2008). Therefore, Fiske and Ladd (2004) argue that the evaluation of transformation strategies have to apply equally to educational opportunity and competence.

Regarding the socialisation aspect, the reconstruction of higher education enables students to become responsible and to perform critical constructs (students express their knowledge, using critical thinking, which allows diverse values to surface, be questioned and more deeply understood). Transformation inspires the expansion of a deep size and a readiness to review, and restart, main ideas, strategies and practices, based on the shared moral (Pityana, 2003; Maharajh, Motala & Scerri, 2011). In its present state, the HES is unable to meet the moral, political, social and economic demands of the new South Africa (Maharajh et al., 2011). However, despite the deficiencies of the apartheid legacy, an academic expertise of higher education remains harmful to the countrywide attention. (Tate, 1997; Ross, 2009; Arnold, 2014). Nevertheless, the reconstruction of higher education has to address the discriminations, inequities and misrepresentations that resulting from the apartheid legacy. Higher education has to be renovated to see the contests of a new democratic culture, dedicated to equity and justice, in order develop a better environment for students (Viljoen, 2005).

1.5. Research Questions

The primary research question of this study is:

 Which approach can be used effectively to provide post-graduate students with a better understanding and interpretation (self-regulation) of statistical procedures in academic institutions?

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1. (a) What is the graduate student‟s experience level in statistics, research methodology?

(b) What is the graduate student‟s statistics anxiety level? (c) What is the graduate student‟s attitude toward statistics?

(d) What is the graduate student‟s level of perceived social support, and (e) What is the graduate student self-efficacy level?

2. What are the effects of the individual‟s characteristics, emotion, behaviour and social support on the SELS beliefs per university?

3. What are the similarities and differences of the dependent variables across universities, and their impact on the SELS beliefs at both universities?

4. What are the factors that predict SELS beliefs, significantly, per academic institution? 5. How do the students choose the appropriate statistical test?

1.6. Justification/Rationale

The researcher‟s intention of conducting this current study is to contribute towards the learning ability development of postgraduate students, in the application of skills in statistics that would enhance their performance in the academic environment, particularly in the field of medical, social and behavioural science research. Considering that statistical effectiveness requires a skilled student, the aim of the researcher is to design a statistical model, to assist in assessing the learning level of students. The proposed model of this current study is a replicable statistical evaluation learning process that could be applied to measure the effectiveness of statistical skills, as well as answer questions related to real world problems. This framework could be used as a model for academic research in a higher educational environment that could continuously progress the excellence of learning statistics. The potential contribution of this current study is that this proposed model could evaluate the incoming students‟ profile and easily identify their level of understanding and interpretation of statistics. Therefore, it would equip universities with practical measures that could be used to assess learning problems. Based on the aforementioned rationale, a set of solutions will be generated through this research study‟s findings.

Students‟ attitudes affect their learning process (Mvududu, 2003; Perepiczka et al., 2011). If the attitude or perception of a student is negative, the learning process can be delayed. If the

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attitude of students is positive, on the other hand, the learning process can be enhanced. Barrell (1998) identifies that, in developing countries, high school graduates entering university do not appear to be sufficiently prepared in statistics. The influence of the type of study environment affects the performance of students at post-graduate level (Ali, Haider, Munir, Khan & Ahmed, 2013; Alos, Caranto & David, 2015). In a South African context, various authors (Altbach & Knight, 2007; Heath, Rothon & Kilpi, 2008; Altbach Reisberg & Rumbley, 2009; Kuh, 2009) emphasize that students from foreign countries tend to perform better in their statistical programme in higher education, as compared to the local students. In addition, Perna, (2005) explores the economic benefit of Higher Education that graduate students realize, in terms of differences in ethnic groups, gender and even in post-graduate programme achievement rates. Therefore, the researcher investigates the indicators leading to these gaps, focusing on postgraduate students of medical, social and behavioural sciences, according to their context.

Universities in South Africa are committed to providing the best quality of supervision to their students, through an adequately qualified and committed team. The researcher‟s attempt in this current study is of a similar nature. The researcher anticipates that, if the model, which this current study proposes, is adopted by South African universities, they will be able to produce skilled graduate students, and establish an effective learning environment. In addition, this current research aims to identify factors associated with learning statistics that might inform researchers, police-makers and other stakeholders to develop relevant preventions and interventions for research.

1.7. Limitations

The first limitation to consider is that this current study is only restricted to the students in the medical, social and behavioural sciences departments of their respective academic institutions. Therefore, to generalise the outcomes to all students at all universities in South Africa requires careful application. Secondly, it is important to consider that no method is faultless and will be successful with all students. Many studies in statistics education reveal that misconceptions are often compelling and resilient among students (Garfield, 1995; Garfield & Ben-Zvi, 2007; Huck, 2009). Students are reluctant to transformation, even when challenged with indication that their perceptions are inaccurate, which is only part of the problem. Additional limitation is whether students are sufficiently committed to engage with learning new concepts.

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1.8. Definition of Keywords

 Statistical procedure is a systematic way of doing a task, which implies a logical arrangement of steps (Burns & Burns, 2008). The appropriate statistical procedure is subjected to the study problem and the type of data collected (Kothari, 2004; Onwuegbuzie & Teddlie, 2003). Lehmann and Romano (2006) define statistical procedure as a way of providing a suitable summary of the data, or designating what evidence is accessible about the unidentified parameter or distribution. The relevant information is used as a guide in many matters; however, it cannot provide the only foundation for any specific choices, unless the emphasis is on the inference, rather than the decision aspect of the problems (National Research Council [NRC], 2002). Additionally, Akaike (1970) describes statistical procedure as the ideinvestigation of the predictor, when the

stochastic

process under comment is an autoregressive process made from inactive and independent inventions.

 Academic environment is a setting where instruction is provided to students (Zimmerman, 1989). It also defines as an atmosphere that greatest makes students for their upcoming expert life and contributes toward their individual expansion and social well-being (McLean & Gibbs, 2010). Various issues meaningfully affects the manner students observe and understanding their edification (Bandura, 1993).

 Postgraduate research represents a prescribed area of study that is recognized by a university, or institute of higher learning (Clark, 1995). It involves investigating and writing critical, accurate, valid, timely research, based on scientific inquiry (National Research Council [NRC], 2006). The scientific method is based on the statement that everything in the universe is linked by cause and effect (Ary, Jacobs, Irvine & Walker, 2013).

 Social support denotes to the several sources of support (for example, assistance/help that people receive from others) and generally classified into two (sometimes three) main categories: emotional, active [and sometimes informational] support (Jang, 2012). An example of supportive resources can be emotional [nurturance], tangible [financial assistance], intangible [personal advice], informational [advice], or companionship [sense of belonging] (Jang, 2012; Williams, 2014). Support can come from many sources, namely: family, friends, student peers, neighbours, co-workers, organizations, or government (Williams, 2014).

An analytical model for assessing the knowledge of statistical procedures amongst postgraduate students in a higher educational environment