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### An Introduction to Statistics

This textbook covers the basic principles of statistics required for students studying a first-year undergraduate business degree but it is also suitable for most undergraduate programmes. The course material assumes little prior subject knowledge taking the reader through the fundamental concepts and calculations of descriptive statistics and classical probability, before delving into discrete and continuous probability distributions and inferential statistics. The book contains numerous examples and worked solutions in addition to some advanced sections such as the derivation of the poisson distribution from the binomial and a least-squares derivation of the regression line formulae.

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Email the Author(s) Michael Hughes

Michael Hughes has over thirty years experience in research and higher education teaching. Since 2012 he has taught an `Introduction to Statistics` module to international students at INTO University of Exeter. This book is a consequence of collating his teaching notes into a single source for his students.

## Contents

1 Introduction

•                  1.1 A very brief introduction
•                  1.2 Why study statistics?
•                  1.3 What is statistics? What are statistics?
•                  1.4 Sample and population
•                  1.5 Data types and Levels of Measurement 14
•                  1.6 The Levels of Measurement 15
•                  1.7 Chapter 1 questions

2 Presenting data

•                  2.1 Organising data
•                  2.2 Building a frequency distribution
•                  2.3 Adjusting classes to close gaps
•                  2.4 Plotting the histogram
•                  2.5 Plotting a cumulative frequency graph
•                  2.6 The Box and Whisker plot
•                  2.7 Stem & Leaf diagram
•                  2.8 Chapter 2 questions
•                  2.9 Chapter 2 tutorial questions
• 3 Measures of location and spread
•                  3.1 Measures of data centre
•                  3.2.1 Variance & Standard Deviation
•                  3.2.2 Inter Quartile Range (IQR)
•                  3.3 Standard Deviation in context
•                  3.3.1 The Empirical Law
•                  3.3.2 Chebychev’s Inequality
•                  3.3.3 Z scores
•                  3.3.4 Detecting outliers
•                  3.4 Measures of skew
•                  3.4.1 Pearson’s coefficient of skew
•                  3.4.2 Software coefficient of skew
•                  3.5 Chapter 3 questions
•                  3.6 Chapter 3 tutorial questions

4 Classical probability

•                  4.1 Introduction
•                  4.2 Ways to assign probability
•                  4.2.1 Classical probabilities
•                  4.2.2 Empirical probability
•                  4.2.3 Subjective probability
•                  4.3 Rules of addition: OR
•                  4.3.1 Mutual exclusivity: The special rule of addition
•                  4.3.2 Compliment rule
•                  4.3.3 Not mutually exclusive: The general rule of addition
•                  4.4 Rules of multiplication: AND
•                  4.4.1 Independence: The special rule of multiplication
•                  4.4.2 Not independent: The general rule of multiplication
•                  4.5 Conditional probability
•                  4.5.1 General rule for conditional probability
•                  4.6 Tree diagrams
•                  4.7 Chapter 4 questions
•                  4.8 Chapter 4 tutorial questions

5 DRV: Discrete random variables

•                  5.1 General DRV
•                  5.2 Specific DRV’s namely: Binomial & Poisson
•                  5.2.1 The binomially distributed DRV
•                  5.2.2 The Poisson distributed DRV
•                  5.2.3 The Poisson probability formula
•                  5.3 Chapter 5 questions
•                  5.4 Chapter 5 tutorial questions

6 The Normal Distribution

•                  6.1 What is the Normal Distribution?
•                  6.2 The Normal Probability Density Function
•                  6.3 Calculating normal probabilities
•                  6.3.1 Calculating probabilities for given data values
•                  6.3.2 Reverse-Lookup: Finding a data value for a given probability
•                  6.4 The Central Limit Theorem (CLT)
•                  6.4.1 What is a sampling distribution?
•                  6.5 Calculating probability for sample means and proportions
•                  6.6 Chapter 6 questions
•                  6.7 Chapter 6 tutorial questions

7 In Brief: Sampling distributions and sampling methods

•                  7.1 Sampling distribution
•                  7.1.1 Student’s t-distribution
•                  7.1.2 Sampling distributions and hypothesis testing
•                  7.2 Sampling methods
•                  7.2.1 Probabilistic sampling methods
•                  7.2.3 Sampling errors
•                  7.3 Chapter 7: CLT probability questions
•                  7.4 Chapter 7: Basic sampling method questions

8 Parameter estimation: Confidence intervals for means and proportions

•                  8.1 Diagrammatic explanation of confidence intervals
•                  8.1.1 Using z or t distributions for estimating the mean
•                  8.1.2 Degrees of freedom
•                  8.2 Formulae for confidence intervals to estimate mean
•                  8.3 Formulae for confidence intervals to estimate proportion
•                  8.4 Finding the sample size to control the width of the CI
•                  8.5 Chapter 8 questions
•                  8.6 Chapter 8 tutorial questions

9 Hypothesis testing: Single sample tests for means and proportions

•                  9.1 The types of hypotheses
•                  9.2 Explaining hypothesis testing for the mean
•                  9.3 Hypothesis testing steps
•                  9.4 Hypotheses tests for means using the t-distribution
•                  9.5 Hypothesis tests for proportions
•                  9.6 Type I and II errors
•                  9.6.1 Type I and II examples
•                  9.7 Chapter 9 questions
•                  9.8 Chapter 9 tutorial questions

10 Two sample hypothesis tests for means and proportions

•                  10.1 Testing a difference between two means from independent samples
•                  10.1.1 Conditions for using the t-distribution
•                  10.2 Testing for differences in proportions from independent samples
•                  10.3 Paired t-test for dependent samples
•                  10.4 Chapter 10 questions
•                  10.5 Chapter 10 tutorial questions

11 Chi-Squared tests

•                  11.1 Goodness of fit tests
•                  11.2 Testing for independence
•                  11.3 Comments on Chi-squared tests
•                  11.4 Chapter 11 questions
•                  11.5 Chapter 11 tutorial questions

12 F-Tests & ANOVA: Single Factor Analysis of variance

•                  12.1 F-Test for equality of variance
•                  12.2 ANOVA: Single factor Analysis of variance
•                  12.3 The ANOVA calculations
•                  12.3.1 Putting it all together: The hypothesis test procedure
•                  12.3.2 Which group means are significantly different?
•                  12.4 The assumptions behind ANOVA
•                  12.5 Chapter 12 questions
•                  12.6 Chapter 12 tutorial questions

13 Correlation and linear regression

•                  13.1 Simple linear regression & correlation
•                  13.2 Product Moment Correlation Coefficient (PMCC)
•                  13.3 Observing correlation
•                  13.4 Spearman’s Rank Correlation Coefficient SRCC
•                  13.5 Testing the significance of the correlation coefficients
•                  13.6 The Least Squares Regression line
•                  13.7 Deriving the formulae for slope and intercept
•                  13.8 Chapter 13 questions
•                  13.9 Chapter 13 tutorial questions

14 Appendix A: Tables

•                  14.1 Normal distribution tables
•                  14.2 Students’ t-distribution
•                  14.3 Fishers F-distribution
•                  14.4 Chi-Squared percentage points
•                  14.5 Random number tables
•                  14.6 Critical values for correlation

15 Appendix B: Stats mode on the Casio fx83-85 calculators

•                  15.1 A Brief guide to Stats Mode on Casio fx83GT and fx85GT
•                  15.2 A Brief guide to Stats Mode on Casio fx83GTX and fx85GTX

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