Essentials of Statistics
Publisher: Bookboon.com
Published: 2013
ISBN-13: 9788776814083
Table of contents: title: Content; title: 1. Preface; title: 2. Basic concepts of probability theory; title: 2.1. Probability space, probability function, sample space, event; title: 2.2. Conditional probability; title: 2.3. Independent events; title: 2.4. The Inclusion-Exclusion Formula; title: 2.5. Binomial coefficients; title: 2.6. Multinomial coefficients; title: 3. Random variables; title: 3.1. Random variables, definition; title: 3.2. The distribution function; title: 3.3. Discrete random variables, point probabilities; title: 3.4. Continuous random variables, density function; title: 3.5. Continuous random variables, distribution function; title: 3.6. Independent random variables; title: 3.7. Random vector, simultaneous density and distribution function; title: 4. Expected value and variance; title: 4.1. Expected value of random variables; title: 4.2. Variance and standard deviation of random variables; title: 4.3. Example (computation of expected value, variance and standard deviation); title: 4.4. Estimation of expected value µ and standard deviation s by eye; title: 4.5. Addition and multiplication formulas for expected value and variance; title: 4.6. Covariance and correlation coefficient; title: 5. The Law of Large Numbers; title: 5.1. Chebyshev’s Inequality; title: 5.2. The Law of Large Numbers; title: 5.3. The Central Limit Theorem; title: 5.4. Example (distribution functions converge to F); title: 6. Descriptive statistics; title: 6.1. Median and quartiles; title: 6.2. Mean value; title: 6.3. Empirical variance and empirical standard deviation; title: 6.4. Empirical covariance and empirical correlation coefficient; title: 7. Statistical hypothesis testing; title: 7.1. Null hypothesis and alternative hypothesis; title: 7.2. Significance probability and significance level; title: 7.3. Errors of type I and II; title: 7.4. Example; title: 8. The binomial distribution Bin(n, p); title: 8.1. Parameters; title: 8.2. Description; title: 8.3. Point probabilities; title: 8.4. Expected value and variance; title: 8.5. Significance probabilities for tests in the binomial distribution; title: 8.6. The normal approximation to the binomial distribution; title: 8.7. Estimators; title: 8.8. Confidence intervals; title: 9. The Poisson distribution Pois(?); title: 9.1. Parameters; title: 9.2. Description; title: 9.3. Point probabilities; title: 9.4. Expected value and variance; title: 9.5. Addition formula; title: 9.6. Significance probabilities for tests in the Poisson distribution; title: 9.7. Example (significant increase in sale of Skodas); title: 9.8. The binomial approximation to the Poisson distribution; title: 9.9. The normal approximation to the Poisson distribution; title: 9.10. Example (significant decrease in number of complaints); title: 9.11. Estimators; title: 9.12. Confidence intervals; title: 10. The geometrical distribution Geo(p); title: 10.1. Parameters; title: 10.2. Description; title: 10.3. Point probabilities and tail probabilities; title: 10.4. Expected value and variance; title: 11. The hypergeometrical distribution HG(n, r,N); title: 11.1. Parameters; title: 11.2. Description; title: 11.3. Point probabilities and tail probabilities; title: 11.4. Expected value and variance; title: 11.5. The binomial approximation to the hypergeometrical distribution; title: 11.6. The normal approximation to the hypergeometrical distribution; title: 12. The multinomial distribution Mult(n, p1, . . . , pr); title: 12.1. Parameters; title: 12.2. Description; title: 12.3. Point probabilities; title: 12.4. Estimatorer; title: 13. The negative binomial distribution NB(n, p); title: 13.1. Parameters; title: 13.2. Description; title: 13.3. Point probabilities; title: 13.4. Expected value and variance; title: 13.5. Estimatorer; title: 14. The exponential distribution Exp(?); title: 14.1. Parameters; title: 14.2. Description; title: 14.3. Density and distribution function; title: 14.4. Expected value and variance; title: 15. The normal distribution; title: 15.1. Parameters; title: 15.2. Description; title: 15.3. Density and distribution function; title: 15.4. The standard normal distribution; title: 15.5. Properties of F; title: 15.6. Estimation of the expected value µ; title: 15.7. Estimation of the variance s2; title: 15.8. Confidence intervals for the expected value µ; title: 15.9. Confidence intervals for the variance s2 and the standard deviation s; title: 15.10. Addition formula; title: 16. Distributions connected with the normal distribution; title: 16.1. The ?2-distribution; title: 16.2. Student’s t-distribution; title: 16.3. Fisher’s F-distribution; title: 17. Tests in the normal distribution; title: 17.1. One sample, known variance, H0 : µ = µ0; title: 17.2. One sample, unknown variance, H0 : µ = µ0 (Student’s t-test); title: 17.3. One sample, unknown expected value, H0 : s2 = s2 0; title: 17.4. Example; title: 17.5. Two samples, known variances, H0 : µ1 = µ2; title: 17.6. Two samples, unknown variances, H0 : µ1 = µ2 (Fisher-Behrens); title: 17.7. samples, unknown expected values, H0 : s2 1 = s2 2; title: 17.8. Two samples, unknown common variance, H0 : µ1 = µ2; title: 17.9. Example (comparison of two expected values); title: 18. Analysis of Variance (ANOVA); title: 18.1. Aim and motivation; title: 18.2. k samples, unknown common variance, H0 : µ1 = · · · = µk; title: 18.3. Two examples (comparison of mean values from 3 samples); title: 19. The chi-square test (or ?2-test); title: 19.1. ?2-test for equality of distribution; title: 19.2. The assumption of normal distribution; title: 19.3. Standardised residuals; title: 19.4. Example (women with 5 children); title: 19.5. Example (election); title: 19.6. Example (deaths in the Prussian cavalry); title: 20. Contingency tables; title: 20.1. Definition, method; title: 20.2. Standardised residuals; title: 20.3. Example (students’ political orientation); title: 20.4. ?2-test for 2 × 2 tables; title: 20.5. Fisher’s exact test for 2 × 2 tables; title: 20.6. Example (Fisher’s exact test); title: 21. Distribution free tests; title: 21.1. Wilcoxon’s test for one set of observations; title: 21.2. Example; title: 21.3. The normal approximation to Wilcoxon’s test for one set of observations; title: 21.4. Wilcoxon’s test for two sets of observations; title: 21.5. The normal approximation to Wilcoxon’s test for two sets of observations; title: 22. Linear regression; title: 22.1. The model; title: 22.2. Estimation of the parameters ß0 and ß1; title: 22.3. The distribution of the estimators; title: 22.4. Predicted values ˆyi and residuals ˆei; title: 22.5. Estimation of the variance s2; title: 22.6. Confidence intervals for the parameters ß0 and ß1; title: 22.7. The determination coefficient R2; title: 22.8. Predictions and prediction intervals; title: 22.9. Overview of formulas; title: 22.10. Example; title: 23. A Overview of discrete distributions; title: 24. B Tables; title: 24.1. B.1 How to read the tables; title: 24.2. B.2 The standard normal distribution; title: 24.3. B.3 The ?2-distribution (values x with F?2(x) = 0.500 etc.); title: 24.4. B.4 Student’s t-distribution (values x with FStudent(x) = 0.600 etc.); title: 24.5. B.5 Fisher’s F-distribution (values x with FFisher(x) = 0.90); title: 24.6. B.6 Fisher’s F-distribution (values x with FFisher(x) = 0.95); title: 24.7. B.7 Fisher’s F-distribution (values x with FFisher(x) = 0.99); title: 24.8. B.8 Wilcoxon’s test for one set of observations; title: 24.9. B.9 Wilcoxon’s test for two sets of observations, a = 5%; title: 25. C Explanation of symbols; title: 26. D Index