Learning Objectives
5 objectives- Understand fundamental concepts and key terminologies in mathematical statistics.
- Apply descriptive statistics and probability distributions to analyze data sets.
- Analyze sampling methods and estimate population parameters using estimation theory.
- Conduct hypothesis testing, regression analysis, and analysis of variance (ANOVA).
- Explore nonparametric and Bayesian statistical methods and their applications.
Content Outline
PreviewUnit 1197 - Mathematical Statistics
1. Introduction to Mathematical Statistics
- Role of statistics in data analysis
- Types of data: qualitative vs. quantitative, discrete vs. continuous
- Key terminologies: population, sample, parameter, statistic, variable
2. Descriptive Statistics
- Measures of central tendency: mean, median, mode
- Measures of dispersion: variance, standard deviation, range, interquartile range
- Graphical representations:
- Histograms
- Box plots
- Stem-and-leaf plots
- Frequency distributions
3. Probability Distributions
- Overview of probability concepts
- Discrete distributions:
- Binomial distribution: definition, properties, applications
- Poisson distribution: definition, properties, applications
- Continuous distributions:
- Normal distribution: properties, standard normal distribution, empirical rule
- Uniform distribution: definition and applications
4. Sampling Techniques
- Importance of sampling in statistics
- Sampling methods:
- Simple random sampling
- Stratified sampling
- Cluster sampling
- Systematic sampling
- Sampling biases and errors
5. Estimation Theory
- Point estimation:
- Definition and properties (unbiasedness, consistency, efficiency)
- Common estimators for mean and variance
- Interval estimation:
- Confidence intervals for population mean and proportion
- Interpretation of confidence levels
- Methods of estimation: Method of moments, Maximum likelihood estimation (overview)
6. Hypothesis Testing
- Formulating hypotheses: null and alternative
- Significance levels and p-values
- Types of errors: Type I and Type II
- Test statistics and critical values
- Common hypothesis tests:
- Z-test
- t-test (one-sample, two-sample)
- Chi-square test for independence and goodness-of-fit
7. Regression Analysis
- Introduction to regression and correlation
- Simple linear regression:
- Model formulation
- Least squares method
- Interpretation of regression coefficients
- Multiple regression overview
- Assessing model fit:
- Coefficient of determination (R²)
- Residual analysis
8. Analysis of Variance (ANOVA)
- Purpose and assumptions of ANOVA
- Sources of variation: between-group and within-group
- One-way ANOVA:
- Calculations of sums of squares
- F-test statistic
- Interpreting ANOVA results
- Post-hoc tests (brief overview)
9. Nonparametric Statistics
- When to use nonparametric methods
- Wilcoxon rank-sum test (Mann-Whitney U test)
- Kruskal-Wallis test
- Spearman's rank correlation coefficient
10. Bayesian Statistics
- Introduction to Bayesian reasoning
- Bayes' theorem and its interpretation
- Prior, likelihood, and posterior probabilities
- Bayesian inference concepts
- Applications and advantages of Bayesian methods
Each section includes theory, examples, and practical exercises to reinforce learning.
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