Foundations of Statistics

D.G. Rees

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.