Summary STA1502-Statistical Inference I EXAM PACK 2022.

STA1502-Statistical Inference I EXAM PACK 2022. CONTENTS ORIENTATION iii STUDY UNIT 1 1.1 Introduction 1 1.2 Inference about the Difference Between Two Population Means: Independent Samples 1 1.3 Observational and Experimental Data 9 1.4 Inference about the Difference Between Two Population Means: Matched Pairs Experiment 9 1.5 Inference about the Ratio of Two Variances 19 1.6 Self-correcting Exercises for Unit 1 22 1.7 Solutions to Self-correcting Exercises for Unit 1 23 1.8 Learning Outcomes 27 STUDY UNIT 2 2.1 Introduction 28 2.2 Inference about the Difference Between Two Population Proportions 28 2.3 One-Way Analysis of Variance 34 2.4 Multiple Comparisons 43 2.5 Analysis of Variance experimental designs (read only) 47 2.6 Randomized Block(two-way) Analysis of Variance 47 2.7 Self-correcting Exercises for Unit 2 51 2.8 Solutions to Self-correcting Exercises for Unit 2 52 2.9 Learning Outcomes 55 STUDY UNIT 3 3.1 Chi–square test 57 3.2 Chi-squared goodness-of-fit test 58 3.3 Chi-squared test of a Contingency Table 62 3.4 Summary of test on nominal data 64 STUDY UNIT 4 4.1 Simple linear regression and correlation 70 4.2 Estimating the coefficients 70 4.3 Error variable: required conditions 75 4.4 Assessing the model 76 4.5 Using the regression equation 77 4.6 Regression diagnostics 77 Downloaded by Mandla Msibi () lOMoARcPSD| ii STUDY UNIT 5 5.1 Non parametric statistics 82 5.2 Wilcoxon Rank Sum Test 82 5.3 Sign test and Wilcoxon signed rank sum test 86 STUDY UNIT 6 6.1 Time series analysis and time series forecasting 96 6.2 Components of time series and smoothing possibilities 96 6.3 Smoothing techniques 97 6.4 Trend and seasonal effects 100 6.5 Introduction to forecasting 102 6.6 Forcasting models 102 Downloaded by Mandla Msibi () lOMoARcPSD| iii STA1502/1 ORIENTATION Welcome Welcome to STA1502. This module is the second one of the first-year statistics courses. STA1501 and STA1502 form the first year Statistics course for students from the College of Economic and Management Sciences. If you are a BSc student in the College of Science, Engineering and Technology, the three modules STA1501 and STA1502 and STA1503 form the first year in Statistics. In the preceding module STA1501, we treated probability and probability distributions, and unless one has a proper understanding of the laws of probability, the mechanisms underlying statistical data analysis will not be understood properly. Probability theory is the tool that makes statistical inference possible. In STA1502, we consider to the applications of the probability distributions. You have learned in STA1501 that the shape of the normal distribution is determined by the value of the mean µ and the variance σ 2 , whilst the shape of the binomial distribution is determined by the sample size n and the probability of 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 of this module is to focus on the issues related to prediction and inference in statistics and therefore it is called Statistical Inference and the "I" in the title indicates that it is a module at the first level. We 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. Learning outcomes There are very specific outcomes for this module, listed below. Throughout your study of this module you must come back to this page, sit back and reflect upon them, think them through, digest them into your system and feel confident in the end that you have mastered the following outcomes: Downloaded by Mandla Msibi () lOMoARcPSD| iv · Describing the behaviour of sample statistics in repeated sampling, focussing on sampling distributions of the sample mean and the sample proportion. · Evaluating the reliability of estimates of the population parameters with the use of the Central Limit Theorem and the sampling distributions of the corresponding sample statistics. · Considering point and interval estimators for single or compound population parameters. · Basic concepts of large-sample statistical estimation and hypothesis testing involving population means and proportions. · Small-sample tests and confidence intervals for population means and proportions · Employ three diferent non-parametric test to compare two populations of ordinal or interval data when normality cannot be accepted. · Applying the classical time series and its decomposition into trend, seasonal and random variation. · Measuring long-term trend using regression analysis and seasonal variation by computing seasonal indexes. · Describing four forecasting techniques, including the autoregressive model. The prescribed textbook For this module you have to study certain sections from six chapters of the prescribed textbook: Keller, G. (2009) International Student Edition (8 th edition) Managerial Statistics, South Western, a part of Cengage Learning. Chapter 13: INFERENCE ABOUT COMPARING TWO POPULATIONS Chapter 14: ANALYSIS OF VARIANCE (not 14.5and 14.6) Chapter 15: CHI-SQUARED TESTS Chapter 16: SIMPLE LINEAR REGRESSION AND CORRELATION Chapter 19: NONPARAMETRIC STATISTICS (only 19.1 and 19.2) Chapter 20: TIME SERIES ANALYSIS AND FORECASTING The study guide The study guide may be better describes as a textbook guide because it guides you through the textbook in a systematic way. It is no substitute for the textbook, where the different topics are explained in detail. You have to use the two together as the guide supplements with additional exercises and longer explanations, but is not repeating the basic theoretical knowledge.This study guide serves as an interactive workbook, where spaces are provided for your convenience. Should Downloaded by Mandla Msibi () lOMoARcPSD| v STA1502/1 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 6 chapters of the textbook into 5 study units or "sessions". Make very sure about the sections indicated in each study unit since some sections of the textbook are excluded and we do not want you frustrated by working through unnecessary work. 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 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" but keep in mind that the considerable effort to master the content of this module can be very rewarding. We 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. Studying through distance education is neither easy nor quick. There will be times when you feel frustrated and discouraged and then only your attitude will pull you through! You are the master of your own destiny. In a paper by Sue Gordon 1 (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. But when you have constructed one bit of road you cannot sit back and think ‘Oh, that’s a great piece of road!’ and stop at that. Each bit leads you on, shows the direction to go, opens the opportunity for more road to be built. And furthermore, the part of the road that 1 Gordon, Sue (1995) A theoretical Approach to Understanding Learners of Statistics. Journal of Statistics Education v. 3, n.3 University of Sydney. Downloaded by Mandla Msibi () lOMoARcPSD| vi you built a few weeks ago, that you thought you were finished with, is going to develop pot holes the instant you turn your back on it. This is not to be construed as failure on your part, this is not inadequacy. This is just part of road building. This is what learning statistics is about: go back and repair, go on and build, go back and repair." A few logistical problems (You can skip the following section if you have read through it when you did STA1501.) Decimal comma or point? We realise that in the South African schooling system commas are used to indicate the decimal digit values. You have been penalised at school for using a point. Now we sit between two fires: the school system and common practice in calculators and computers! Most computer packages use decimal points (ignoring the option to change it) and Keller (the author) also uses the decimal point in our textbook (Statistics for Management and Economics). Therefore we use the decimal point in our study guide, assignments and examination. Downloaded by Mandla Msibi () lOMoARcPSD| vii STA1502/1 Role of computers and statistical calculators: The emphasis in the textbook is well beyond the arithmetic of calculating statistics and the focus is on the identification of the correct technique, interpretation and decision making. This is achieved with a flexible design giving both manual calculations and computer steps. Every statistical technique that needs computation is illustrated in a three-step approach: Step 1 MANUALLY Step 2 EXCEL Step 3 MINITAB It is a good idea that you initially go through the laborious manual computations to enhance your understanding of the principles and mathematics but we strongly urge you to manage the Excel computations because using computers reflects the real world outside. The additional advantage of using a computer is that you can do calculations for larger and more realistic data sets. Whether you use a computer program or a statistical calculator as tool for your calculations is irrelevant to us. However, the emphasis in this module will always be on the interpretation and how to articulate the results in report writing. CD Appendixes and A Study Guide are provided on the CD-ROM (included in the textbook) in pdf format . The slide shot below is just to give you an idea of some of the topics covered. Although it will not be to your disadvantage if you do not use the CD, we encourage you to try your best to have at least a few sessions on a computer. Statistical Software makes Statistics exciting - so, play around on the computer should you have access! Downloaded by Mandla Msibi () lOMoARcPSD| viii Some Key Terms/Symbols Sampling distribution of the sample proportion Standard error of the proportion Sampling distribution of the difference between two sample means Standard error of the difference between two means Pooled variance estimator Matched pairs experiment Degrees of freedom Pooled proportion estimator Response variable Sum of squares for error Multinomial experiment Least squares method Distribution–free methods Random variation Trend analysis Downloaded by Mandla Msibi () lOMoARcPSD| 1 STA1502/1 STUDY UNIT 1 1.1 Introduction You should not attempt to do this module without knowledge of the contents of STA1501 as it is a continuation in the same textbook of the follow-up chapters. Chapters 2 and chapters 4 - 12 were covered in STA1501 and we now continue with Chapter 13. In chapter 12 you learnt about statistical inference for a single population and derived hypothesis tests and confidence intervals from the information contained in a single sample. You did this for • the population mean µ • the population variance σ 2 • the population proportion p In this study unit we will focus on statistical inference for two populations and derive hypothesis tests and confidence intervals from the information contained in two separate samples. Recall how a confidence interval is derived for (µ1 − µ 2 ) using the sampling distribution of (X 1 − X 2). Similar to the practical problems with inference for a single population mean,µ, you will understand that we again work with a t-distribution because of the more realistic set-up where we assume that both the population variances are unknown and we have to estimate them.