STA1501 Descriptive Statistics and Probability Study Guide(with summary notes) and A+ assured EXAM PACK

Department of Statistics STA1501 Descriptive Statistics and Probability Study Guide i STA1501/1 ii iii STA1501/1 CONTENTS ORIENTATION vi STUDY UNIT 1 1.1 Introduction 1 What is Statistics? 1.2 Types of Data and Information 4 1.3 Self-correcting Exercises for Unit 1 8 1.4 Solutions to Self-correcting Exercises for Unit 1 10 1.5 Learning Outcomes 12 1.6 Study Unit 1: Summary 13 STUDY UNIT 2 2.1 Introduction 14 2.2 Graphical and Tabular Techniques to describe Nominal Data 14 2.3 Graphical Techniques to Describe Interval data 18 2.4 Describing the Relationship between Two Variables and Describing Time Series Data 24 2.5 Self-correcting Exercises for Unit 2 27 2.6 Solutions to Self-correcting Exercises for Unit 2 29 2.7 Learning Outcomes 32 2.8 Study Unit 2: Summary 33 STUDY UNIT 3 3.1 Introduction 34 3.2 Graphical Excellence and Graphical Deception 34 3.3 Presenting Statistics: Written Reports and Oral Representations 40 3.4 Measures of Central Location 40 3.5 Measures of Variablity 45 3.6 Self-correcting Exercises for Unit 3 50 3.7 Solutions to Self-correcting Exercises for Unit 3 51 3.8 Learning Outcomes 55 3.9 Study Unit 3: Summary 56 STUDY UNIT 4 4.1 Introduction 58 4.2 Measures of Relative Standing and Box Plots 59 4.3 Measures of Linear Relationship 62 4.4 Comparing Graphical and Numerical Techniques 64 4.5 General Guidelines for Exploring Data 64 4.6 Self-correcting Exercises for Unit 4 64 iv 4.7 Solutions to Self-correcting Exercises for Unit 4 65 4.8 Learning Outcomes 67 4.9 Study Unit 4: Summary 68 STUDY UNIT 5 5.1 Introduction 70 5.2 Methods of Collecting Data and Sampling 70 5.3 Sampling Plans 72 5.4 Sampling and Nonsampling Errors 81 5.5 Self-correcting Exercises for Unit 5 84 5.6 Solutions to Self-correcting Exercises for Unit 5 85 5.7 Learning Outcomes 89 5.8 Study Unit 5: Summary 90 STUDY UNIT 6 6.1 Introduction 91 6.2 A basis for probability 92 6.3 Sophisticated methods and rules in probability theory 96 6.4 The rule of Bayes 108 6.5 Learning Outcomes 113 6.6 Study Unit 6: Summary 114 STUDY UNIT 7 7.1 Introduction 116 7.2 Discrete probability distributions 117 7.3 Bivariate distributions 122 7.4 Binomial distribution 127 7.5 Poisson distribution 131 7.6 Learning Outcomes 135 7.7 Study Unit 7: Summary 136 STUDY UNIT 8 8.1 Introduction 137 8.2 Continuous probability distributions 138 8.3 Continuous probability distributions: Normal distribution 140 8.4 Other Continuous probability distributions 155 8.5 Learning Outcomes 159 8.6 Study Unit 8: Summary 160 v STA1501/1 STUDY UNIT 9 9.1 Introduction 161 9.2 Sampling distribution of the mean 166 9.3 Sampling distribution of a proportion 172 9.4 Sampling distribution of the difference between two means 179 9.5 Self-Correcting Exercises for Unit 9 183 9.6 Solutions to Self-Correcting Exercises for Unit 9 184 9.7 Learning Outcomes 186 9.8 Study Unit 9: Summary 187 STUDY UNIT 10 10.1 Introduction 189 10.2 Concepts of Estimation 190 10.3 Estimating the Population Mean when the Population Standard Deviation is Known 192 10.4 Selecting the Sample Size 198 10.5 Self-correcting Exercises for Unit 10 200 10.6 Solutions to Self-correcting Exercises for Unit 10 203 10.7 Learning Outcomes 206 10.8 Study Unit 10: Summary 208 STUDY UNIT 11 11.1 Introduction 209 11.2 Concepts of Hypothesis Testing 210 11.3 Testing the Population Mean when the Population Standard Deviation is Known 213 11.4 Calculating the Probability of a Type II error 221 11.5 The Road Ahead 224 11.6 Self-Correcting Exercises for Unit 11 224 11.7 Solutions to Self-Correcting Exercises for Unit 11 225 11.8 Learning Outcomes 226 11.9 Study Unit 11: Summary 228 STUDY UNIT 12 12.1 Introduction 229 12.2 Inference about a Population Mean when the standard deviation is unknown 229 12.3 Inference about the mean: What else need you keep in mind? 242 12.4 Inference about a Population Variance 244 12.5 Inference about a Population Proportion 247 12.6 Self-correcting Exercises for Unit 12 252 12.7 Solutions to Self-correcting Exercises for Unit 12 253 12.8 Learning Outcomes 255 12.9 Study Unit 12: Summary 256 vi ORIENTATION Introduction Welcome to STA1501. This module consits of the first half of the first-year statistics course for students in the College of Economic and Management Sciences. The two modules form an integrated whole and are focused on the following objective: To collect, organise, analyse and interpret data for the purpose of making better decisions. STA1501 STA1502 STA1503 This is where you are First−year Statistics The first part of the module covers the “Descriptive Statistics” part, which is earthly and real and the focus is on the presentation of data. The first step is to carefully think about the type of variable that each measurement represents. This is extremely important as the type dictates what you can or can’t do in the rest of your data analysis. Then we will also consider the collection of data (which most often, for the social sciences and in business applications, involve administering questionnaires and/or survey data, and sampling plays an important role in this regard). Between the collection of data and the ultimate goal of analysis of data lies the very important step of organising and summarising the data. So, in this module we discuss how we organise and summarise the gathered information intelligibly and efficiently. The second part of the module covers the “Probability and Probability Distributions” part where we leave the practical familiarity of data and turn to the less familiar abstract concept of probability. This is almost like a shift in gears! A proper understanding of the laws of probability is essential to ensure a proper understanding of the mechanisms underlying statistical data analysis. Probability theory is the tool that makes statistical inference possible. The third part of the module covers the applications of the probability distributions. You have learned that the shape of the normal distribution is determined by the value of the mean µ and the variance σ2, whilst the shape of the binomial distribution is determined by the sample size n and the probability of vii STA1501/1 a success p. These critical values are called parameters. We most often don’t know what the values of the parameters are and thus we cannot “utilise” these distributions (i.e. use the mathematical formula to draw a probability density graph or compute specific probabilities) unless we somehow estimate these unknown parameters. It makes perfect logical sense that to estimate the value of an unknown population parameter, we compute a corresponding or comparable characteristic of the sample. The objective is to draw inference about a population (a complete set of data) based on the limited information contained in a sample. In dictionary terms, inference is the act or process of inferring; to infer means to conclude or judge from premises or evidence; meaning to derive by reasoning. In general, the term implies a conclusion based on experience or knowledge. More specifically in statistics, we have as evidence the limited information contained in the outcome of a sample and we want to conclude something about the unknown population from which the sample was drawn. The set of principles, procedures and methods that we use to study populations by making use of information obtained from samples is called statistical inference. viii Learning objectives There are very specific outcomes for this module which we list below. Throughout your study of this module you must come back to this page, sit back and reflect upon these outcomes, think them through, digest them into your system and feel confident in the end that you have mastered them. • Analyse data considering different types of data and how they relate to relevant graphical and tabular presentations e.g. pie charts, bar charts, histograms, stem-and-leaf displays, line charts, scatter diagrams and box-and-whisker plots • Analyse data by calculating accurate numerical measures of central location, variability, relative standing and linear relationship. • Differentiate between simple random sampling, stratified random sampling and cluster sampling and implement a sampling plan for a given research problem with an awareness for the effect of sampling errors. • Describe the different concepts and laws of probability and apply definitions of joint, marginal and conditional probability. • Apply the complement, multiplication and addition rules and probability trees for calculation of more complex events and calculate complicated events from the probabilities of related events. • Understand the role of probability in decision making and the application in basic statistical inference. • Describe random variables and the probabilities associated with them in the form of a table, formula or graph and also in terms of its parameters, usually the expected value and the variance. • Describe different probability distributions as either discrete or continuous and know the parameters of expected value and variance The prescribed textbook For this module you must study twelve chapters from the prescribed textbook: Keller,G. (2009, (8th edition)) Managerial Statistics, South–Western, Cengage Learning Chapter 1: WHAT IS STATISTICS? Chapter 2: GRAPHICAL AND TABULAR DESCRIPTIVE TECHNIQUES Chapter 3: ART AND SCIENCE OF GRAPHICAL PRESENTATIONS Chapter 4: NUMERICAL DESCRIPTIVE TECHNIQUES Chapter 5: DATA COLLECTION AND SAMPLING Chapter 6: PROBABILITY Chapter 7: RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTIONS ix STA1501/1 Chapter 8: CONTINUOUS PROBABILITY DISTRIBUTION Chapter 9 : SAMPLING DISTRIBUTIONS Chapter 10: INTRODUCTION TO ESTIMATION Chapter 11: INTRODUCTION TO HYPOTHESIS TESTING Chapter 12: INFERENCE ABOUT A POPULATION The study guide The study guide is exactly what its name implies: a guide through the textbook in a systematic way. The textbook will focus on the theoretical contents of the module and we have tried not to duplicate material from the textbook in the guide. For each separate study unit you should first study the work in the textbook and utilise the guide to assess your progress, test your knowledge and prepare for the examination. In other words, the study guide will provide you with an opportunity to apply your knowledge of the material that is covered in the textbook. This study guide serves as an interactive workbook, where spaces are provided for your convenience. Should you so prefer, you are welcome to write and reference your solutions in your own book or file, if the space we supply is insufficient or not to your liking. Study units and workload We realise that you might feel overwhelmed by the volumes and volumes of printed matter that you have to absorb as a student! How do you eat an elephant? Bite by bite! We have divided the twelve chapters of the textbook into 12 study units or “sessions”. Make very sure about the sections in each study unit since some sections of the textbook are not included and we do not want you frustrated by working through unnecessary work. The study units vary in length but you should try to spend on average 12 hours on each unit. Practically everybody should be able to do statistics. It depends on the amount of TIME you spend on the subject. Regular contact with statistics will ensure that your study becomes personally rewarding. Try to work through as many of the exercises as possible Doing exercises on your own will not only enhance your understanding of the work, but it will give you confidence as well. Feedback is given immediately after the activity to help you check whether you understand the specific concept. The activities are designed (i.e. specific exercises are selected) so that you can reflect on a concept discussed in the textbook. You can only obtain maximum benefit from this activity-feedback process if you discipline yourself not to peep at the solution before you have attempted it on your own! Final word: Attitude You are the master of your own destiny. Studying through distance education is neither easy nor quick. We know that many of you have some “math anxiety” to deal with, but we will do our best to make your statistics understandable and not too theoretic. Studying statistics is sometimes not “exciting” or “fun” and keep in mind that to master the content of a module can involve considerable x effort. However, we do claim that knowledge of statistics will enable you to make effective decisions in your business and to conduct quantitative research into the many larger and detailed data sources that are available. Statistical literacy will enable you to understand statistical reports you might encounter as a manager in your business. We are there to assist you in a process where you shift yourself from a supported school learner to an independent learner. There will be times when you feel frustrated and discouraged and then only your attitude will pull you through! In a paper by Sue Gordon1 (1995) from the University of Sydney, the following metaphor is given: “The learning of statistics is like building a road. It’s a wonderful road, it will take you to places you did not think you could reach. B