COURSE AIMS AND OBJECTIVES: To adopt the basic terms and classical methods of statistical data analysis.
COURSE DESCRIPTION AND SYLLABUS:
1. Introduction. Examples of statistical problems. Statistical data. Classification of statistical variables. Frequency distributions of discrete variables. Tables and graphs. Continuous variables. Grouping data. Histogram. Point diagram. Line diagram. Stem and leaf diagram.
2. Measures of central tendencies. Mean (arithmetic, geometric, harmonic). Median. Mod. Measures of location (quartiles, deciles, percentiles, quantiles). Measures of variability. Range. Interquartile range. Standard deviation. Box and whisker diagram. Geometric interpretations of arithmetic mean and median. Chebyshev's inequality and its interpretation. Moments. Standardization of data. Measures of shape (skewness and kurtosis).
3. Frequency distributions of two-dimensional vectors (contingency tables). Marginal and conditional frequency distributions. Regression function. Statistical dependencies/independencies. Measures of statistical dependence in contingency tables. Covariance and correlation coefficient. Correlation coefficient as measure of linear dependence.
4. Scatter diagram. Regression line. The method of least squares. Decomposition of variance (for regression line). Examples. The Projection Theorem in Rn. Geometric interpretation of variance decomposition.
5. Population and sample. Population parameters and statistics. Simple random sample (with and without repetition, finite and infinite populations). Sample distribution. Example: estimation of proportion parameter of finite population based on simple random sampling with and without repetition, and of infinite population. Definition of random sample.
6. Empirical distribution function. Glivenko - Cantelli's theorem. Binomial and multinomial models for statistical data. Normal model.
7. Standard normal random vector. Chi-square-distribution. Cochran's theorem. Sample distribution and independency of and S2. t-distribution. F-distribution.
8. Point parameter estimation. The moment method. Estimation of mean and variance. Unbiased estimators. Mean squared error. Consistency (application of law of large numbers). Standard error. Asymptotic distribution of and S2 (application of central limit theorem). Maximum likelihood estimation. Asymptotic distribution of MLE. Examples.
9. Interval estimation. Confidence intervals. Pivotal method. Examples. Approximate confidence intervals. Examples. Confidence intervals for proportion parameters.
10. Testing statistical hypotheses. Statistical tests. Errors of 1. and 2. types. Classical testing. Neyman - Pearson's lemma. Example (normal model, simple hypotheses). Confidence level. p-value.
11. Testing hypotheses about normal model parameters (t-test, Chi-square-test). Comparison of two normal populations (t-test, F-test). Large sample tests. Comparison of proportions.
12. One-way analysis of variance. The model. Parameter estimation. ANOVA table. F-test. Bivariate normal model. Testing correlations.
13. Linear regression model. Parameter estimation. Gauss - Markov's theorem. Sample distribution of estimators. ANOVA table. Prediction.
14. Chi-square-goodness of fit test. Kolmogorov - Smirnov test. Chi-square-test of distribution homogeneity and independency in contingency tables.
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