Foundations of Statistics

by D.G. Rees

Overview

This text provides a through, straightforward first course on basics statistics. Emphasizing the application of theory, it contains 200 fully worked examples and supplies exercises in each chapter-complete with hints and answers.

Product Details

ISBN-13:	9781138469723
Publisher:	CRC Press
Publication date:	12/18/2020
Pages:	560
Product dimensions:	6.12(w) x 9.19(h) x (d)

Table of Contents

1 Diagrams and tables — 1.1 Introduction — 1.2 Data and an example of a data set — 1.3 Tables and diagrams for continuous variables — 1.4 Tables and diagrams for discrete variables — 1.5 Tables and diagrams for categorical variables — 1.6 Summary — Worksheet 1 — 2 Measures of location — 2.1 Introduction — 2.2 Mean of ungrouped data — 2.3 Mean of grouped data — 2.4 Median of ungrouped data — 2.5 Median of grouped data — 2.6 Mode of ungrouped data — 2.7 Mode of grouped data — 2.8 When to use the mean, median and mode — 2.9 Geometric mean, weighted mean and index numbers — 2.10 Summary — Worksheet 2 — 3 Measures of dispersion and skewness — 3.1 Introduction — 3.2 Standard deviation and variance of ungrouped data — 3.3 Standard deviation and variance of grouped data — 3.4 Inter-quartile range, percentiles and deciles of grouped data — 3.5 Which measure of dispersion to use? — 3.6 Range — 3.7 Measures of skewness — 3.8 Summary Appendix to Chapter 3 — Worksheet 3 — 4 Basic ideas of probability — 4.1 Introduction — 4.2 Some terminology — 4.3 The definition of probability for the case of equally likely outcomes — 4.4 The relative frequency definition of probability — 4.5 Probability, proportion, percentage and odds — 4.6 Probabilities of the intersection of events; the multiplication law — 4.7 Probabilities of the union of events; the addition law — 4.8 Complementary events, a mutually exclusive and exhaustive — set of events, and the probability of ‘at least one’ — 4.9 Using both laws of probability, tree diagrams — 4.10 Permutations and combinations — 4.11 The law of total probability and Bayes’ formula — 4.12 Summary — Worksheet 4 — 5 Random variables and their probability distributions — 5.1 Introduction — 5.2 Discrete random variables, probability function — 5.3 Expectation, mean and variance of a discrete random variable — 5.4 Probability generating function for a discrete random variable — 5.5 Continuous random variables, probability density function — 5.6 Expectation, mean and variance of a continuous random variable — 5.7 Distribution function for a continuous random variable — 5.8 Median of a continuous random variable — 5.9 Moment generating function for a continuous random variable — 5.10 Mean and variance of a linear function of a random variable — 5.11 The probability distribution for a function of a continuous random variable — 5.12 Summary — Appendix to Chapter 5 — Worksheet 5 — 6 Some standard discrete and continuous probability distributions — 6.1 Introduction — 6.2 Binomial distribution — 6.3 Poisson distribution — 6.4 Geometric distribution — 6.5 Rectangular (uniform) distribution — 6.6 Normal distribution — 6.7 Exponential distribution — 6.8 Summary Worksheet 6 — 7 Approximations to the binomial and Poisson distributions — 7.1 Introduction — 7.2 Poisson approximation to the binomial distribution — 7.3 Normal approximation to the binomial distribution — 7.4 Normal approximation to the Poisson distribution — 7.5 Summary — Worksheet 7 — 8 Linear functions of random variables, and joint distributions — 8.1 Introduction — 8.2 The mean and variance of aX + bY — 8.3 The distribution of a linear function of independent normally distributed variables — 8.4 The distribution of the sum of independent Poisson variables — 8.5 The distribution of the sum of independent and identically distributed geometric variables — 8.6 Joint, conditional and marginal distributions — 8.7 Summary — Worksheet 8 — 9 Samples, populations and point estimation — 9.1 Introduction — 9.2 Samples and populations — 9.3 Random sampling — 9.4 Properties of point estimators — 9.5 Sampling distribution of the sample mean — 9.6 Point estimation of the mean of a normal distribution — 9.7 Point estimation of the variance of a normal distribution — 9.8 Point estimation of the binomial parameter, p — 9.9 Point estimation of the common variance of two normal distributions, data from two samples — 9.10 Point estimation of the binomial parameter, p, data from two binomial experiments — 9.11 Summary — Worksheet 9 — 10 Interval estimation — 10.1 Introduction — 10.2 Confidence interval for the mean of a normal distribution with known variance — 10.3 The t distribution and degrees of freedom — 10.4 Confidence interval for the mean of a normal distribution with unknown variance — 10.5 The sample size required to estimate the mean of a normal distribution — 10.6 Confidence interval for the difference between the means of two normal distributions (unpaired samples data) — 10.7 Confidence interval for the mean of a normal distribution of differences (paired samples data) — 10.8 The x2 distribution — 10.9 Confidence interval for the variance of a normal distribution — 10.10 Confidence interval for a binomial parameter, p — 10.11 The sample size required to estimate a binomial parameter, p — 10.12 Confidence interval for the difference between two binomial parameters — 10.13 Confidence intervals based on the central limit theorem — 10.14 Summary — Worksheet 10 — 11 Hypothesis tests for the mean and variance of normal distributions — 11.1 Introduction — 11.2 The null and alternative hypotheses — 11.3 Hypothesis test for the mean of a normal distribution with known variance — 11.4 Hypothesis test for the mean of a normal distribution with unknown variance — 11.5 Hypothesis test for the difference between the means of two normal distributions (unpaired samples data) — 11.6 Hypothesis test for the mean of a normal distribution of differences (paired samples data) — 11.7 Hypothesis test for the variance of a normal distribution — 11.8 Hypothesis test for the equality of the variances of two normal distributions — 11.9 How a confidence interval can be used to test hypotheses — 11.10 Type I and II errors, and the power of a test — 11.11 Note on assumptions made in hypothesis tests — 11.12 Summary — Worksheet 11 — 12 Hypothesis tests for the binomial parameter, p — 12.1 Introduction — 12.2 An exact test for a binomial parameter — 12.3 An approximate test for a binomial parameter — 12.4 An approximate test for the difference between two binomial parameters — 12.5 Summary — Worksheet 12 — 13 Hypothesis tests for independence and goodness-of-fit — 13.1 Introduction — 13.2 x2 test for independence, contingency table data — 13.3 2x2 contingency table, x2 test — 13.4 x2 goodness-of-fit test for a simple proportion distribution — 13.5 x2 goodness-of-fit test for a binomial distribution — 13.6 x2 goodness-of-fit test for a Poisson distribution — 13.7 Graphical method of testing for a Poisson distribution 13 8 x2 goodness-of-fit test for a normal distribution — 13.9 Graphical methods of testing for a normal distribution — 13.10 Summary — Worksheet 13 — 14 Non-parametric hypothesis tests — 14.1 Introduction — 14.2 Sign test — 14.3 Wilcoxon signed rank test — 14.4 Mann-Whitney U test — 14.5 Summary Worksheet 14 — 15 Correlation — 15.1 Introduction — 15.2 The correlation coefficient between two variables — 15.3 Calculation and interpretation of Pearson’s correlation coefficient, r — 15.4 The coding method of calculating Pearson’s r — 15.5 Hypothesis test for non-zero values of p (Fisher’s transformation) — 15.6 Confidence interval for p — 15.7 Hypothesis test for the difference between two correlation coefficients, p i and p2 — 15.8 Spearman’s coefficient of rank correlation, rs — 15.9 Kendall’s tau (x) — 15.10 Correlation coefficients between linear functions of two variables — 15.11 Summary — Appendix to Chapter 15 — Worksheet 15 — Regression — 16.1 Introduction — 16.2 Method of least squares — 16.3 The equation of the regression line of Y on X: an example — 16.4 A linear statistical model for regression — 16.5 Inferences about the slope, /?, of the regression line, a2 known — 16.6 Inferences about /?, g2 unknown — 16.7 Inferences about a — 16.8 Inferences about predicted mean values of Y — 16.9 Inferences about the difference between two predicted mean values of Y — 16.10 Regression when both variables are random — 16.11 Transformations to produce linearity — 16.12 Summary — Worksheet 16 — Elements of experimental design and analysis — 17.1 Introduction — 17.2 A completely randomized design with two treatments: an example — 17.3 Analysis of variance for a completely randomized design with two treatments — 17.4 One-way analysis of variance for a completely randomized design with more than two treatments — 17.5 Further analysis following the analysis of variance for a completely randomized design — 17.6 Two-way analysis of variance for a randomized block design — 17.7 Further analysis following the analysis of variance for a randomized block design — 17.8 Summary — Appendix to Chapter 17 — Worksheet 17 — 18 Quality control charts and acceptance sampling — 18.1 Introduction — 18.2 Control charts for the mean and range of a continuous variable — 18.3 Control charts for fraction defective — 18.4 Acceptance sampling, a single sampling plan — 18.5 Acceptance sampling, a double sampling plan — 18.6 Single versus double sampling plans — 18.7 Summary Worksheet 18 — Information on projects in statistics at A-level — Appendix A Answers to worksheets — Appendix B Glossary of notation — Appendix C Statistical tables — Index.