Topics in the Statistical inference course ----------------------------------------- Statistical inference vs probability theory Statistical models Population distribution Population mean and standard deviation, population proportion IID sampling (with replacement) Simple random sample (without replacement), negative dependence Point estimate, sampling distribution Mean square error, systematic and random errors Sample mean, sample variance, sample standard deviation, sample proportion Finite population correction Standard error of the sample mean and sample proportion Approximate confidence interval for the mean Stratified random sampling Optimal allocation of observations, proportional allocation Parametric models, population parameters Method of moment for point estimation Binomial, geometric, Poisson, discrete uniform models Maximum likelihood estimate (MLE) Likelihood function Normal approximation for the sampling distribution of MLE Continuous uniform, normal, exponential, gamma models Sufficient statistics for population parameters Chi-square and t-distributions Exact confidence intervals for the mean and variance Statistical hypotheses, simple and composite, null and alternative Two types of error Significance level, test power Rejection region P-value of the test, one-sided and two-sided p-values Large-sample test for the proportion Small-sample test for the proportion Large-sample test for the mean One-sample t-test Nested hypotheses, generalized likelihood ratio test Pearson's chi-square test, its approximate nature Multinomial distribution Empirical cumulative distribution function Survival function and hazard function Empirical survival function Kernel density estimate Steam-and-leaf plot Population quantiles Ordered sample and empirical quantiles Q-Q plots, normal probability plot Light tails and heavy tails of probability distributions Coefficient of skewness and kurtosis Leptokurtic and platykurtic distributions Population mean, mode, and median Sample median, outliers Sign test and non-parametric confidence interval for the median Trimmed means Parametric and non-parametric bootstraps Bootstrap confidence interval Sample range, quartiles, IQR and MAD Boxplots Two independent versus paired samples Approximate CI and large sample test for the mean difference Two-sample t-test, pooled sample variance Exact CI for the mean difference Transformation of variables Wilcoxon rank sum test, ranks vs exact measurements Wilcoxon signed rank test Double-blind randomized controlled experiments Confounding factor, Simpson's paradox One-way ANOVA, sums of squares and mean squares Normal theory model, F-test F-distribution Normal probability plots for residuals Multple comparison or multiple testing problem Simultaneous CI, Bonferroni's method and Tukey's method Kruskal-Wallis test Two-way ANOVA, main effects and iteraction Three F-tests Additive model Randomized block design Friedman's test Categorical data, Fisher's exact test Chi-square test of homogeneity Chi-square test of independence Prospective and retrospective studies Matched-pairs design, McNemar's test Odds ratio Simple linear regression model Three model parameters Least square estimates Normal equations Sample correlation coefficient, sample covariance Corrected MLE of the noise variance Coefficient of determination CI and hypotheses testing for the intercept and slope Model utilty test Prediction interval for a new observation Standardized residuals Multiple regression Coefficient of multiple determination and adjusted coefficient of multiple determination Collinearity problem Bayes formulas for probabilities and densities Prior and posterior distributions Loss function, posterior risk Conjugate priors Normal-normal model Beta and Dirichlet distributions Beta-binomial model and Dirichlet-multinomial model Bayesian estimation, MAP (0-1 loss function) and PME (squared error loss) Credibility interval Posterior odds ratio test