PHD
  1. Algebra  1.1. Group Theory  1.1.1. Basic Properties of Groups  1.1.2. Subgroups  1.1.3. Cosets and Lagrange's Theorem  1.1.4. Normal Subgroups  1.1.5. Quotient Groups  1.1.6. Group Homomorphisms  1.1.7. Sylow Theorems  1.1.8. Free Groups  1.1.9. Group Actions  1.1.10. Simple and Solvable Groups  1.2. Ring Theory  1.2.1. Rings and Ideals  1.2.2. Ring Homomorphisms  1.2.3. Polynomial Rings  1.2.4. Principal Ideal Domains  1.2.5. Unique Factorization Domains  1.2.6. Noetherian Rings  1.2.7. Artinian Rings  1.3. Field Theory  1.3.1. Field Extensions  1.3.2. Splitting Fields  1.3.3. Galois Theory  1.3.4. Algebraic Closures  1.3.5. Finite Fields  1.3.6. Transcendental Extensions  1.4. Module Theory  1.4.1. Modules over Rings  1.4.2. Free Modules  1.4.3. Module Homomorphisms  1.4.4. Simple and Semi-Simple Modules  1.4.5. Projective Modules  1.4.6. Injective Modules  1.5. Linear Algebra  1.5.1. Vector Spaces  1.5.2. Linear Transformations  1.5.3. Eigenvalues and Eigenvectors  1.5.4. Canonical Forms  1.5.5. Inner Product Spaces  1.5.6. Spectral Theorem  1.5.7. Singular Value Decomposition
  2. Analysis  2.1. Real Analysis  2.1.1. Real Numbers and Completeness  2.1.2. Sequences and Series  2.1.3. Continuity  2.1.4. Differentiation  2.1.5. Riemann Integration  2.1.6. Metric Spaces  2.1.7. Lebesgue Measure  2.2. Complex Analysis  2.2.1. Complex Numbers  2.2.2. Analytic Functions  2.2.3. Contour Integration  2.2.4. Cauchy's Theorem  2.2.5. Residue Theorem  2.2.6. Conformal Mappings  2.3. Functional Analysis  2.3.1. Normed Spaces  2.3.2. Banach Spaces  2.3.3. Hilbert Spaces  2.3.4. Linear Operators  2.3.5. Spectral Theory  2.3.6. Compact Operators  2.4. Measure Theory  2.4.1. Sigma Algebras  2.4.2. Measures and Integrals  2.4.3. Lebesgue Integration  2.4.4. Convergence Theorems  2.4.5. Fubini's Theorem  2.5. Harmonic Analysis  2.5.1. Fourier Series  2.5.2. Fourier Transforms  2.5.3. Distribution Theory  2.5.4. Wavelets
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Continuity  3.1.3. Compactness  3.1.4. Connectedness  3.1.5. Metric Topology  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy Theory  3.2.3. Homology Theory  3.2.4. Cohomology  3.2.5. Exact Sequences  3.3. Differential Topology  3.3.1. Smooth Manifolds  3.3.2. Tangent Spaces  3.3.3. Vector Fields  3.3.4. Differential Forms
  4. Geometry  4.1. Differential Geometry  4.1.1. Curves and Surfaces  4.1.2. Riemannian Geometry  4.1.3. Geodesics  4.1.4. Curvature  4.2. Algebraic Geometry  4.2.1. Affine and Projective Varieties  4.2.2. Schemes  4.2.3. Divisors and Intersections  4.2.4. Cohomology in Algebraic Geometry  4.3. Computational Geometry  4.3.1. Convex Hulls  4.3.2. Triangulations  4.3.3. Voronoi Diagrams  4.3.4. Geometric Algorithms
  5. Number Theory  5.1. Elementary Number Theory  5.1.1. Divisibility  5.1.2. Congruences  5.1.3. Modular Arithmetic  5.2. Analytic Number Theory  5.2.1. Prime Number Theorem  5.2.2. Dirichlet Series  5.2.3. Zeta and L-functions  5.3. Algebraic Number Theory  5.3.1. Number Fields  5.3.2. Ideals  5.3.3. Class Groups  5.3.4. Galois Representations
  6. Combinatorics  6.1. Enumerative Combinatorics  6.1.1. Generating Functions  6.1.2. Recurrence Relations  6.1.3. Permutations and Combinations  6.2. Graph Theory  6.2.1. Graph Isomorphisms  6.2.2. Planarity  6.2.3. Graph Coloring  6.3. Extremal Combinatorics  6.3.1. Ramsey Theory  6.3.2. Probabilistic Methods in Extremal Combinatorics  6.4. Algebraic Combinatorics  6.4.1. Matrix Methods  6.4.2. Representation Theory
  7. Logic and Foundations  7.1. Set Theory  7.1.1. Zermelo-Fraenkel Axioms  7.1.2. Ordinals and Cardinals  7.2. Model Theory  7.2.1. Structures and Theories in Model Theory  7.2.2. Compactness Theorem  7.3. Proof Theory  7.3.1. Formal Systems  7.3.2. Consistency and Completeness
  8. Probability and Statistics  8.1. Probability Theory  8.1.1. Random Variables  8.1.2. Expectation and Variance  8.1.3. Central Limit Theorem  8.2. Stochastic Processes  8.2.1. Markov Chains  8.2.2. Brownian Motion  8.3. Statistical Inference  8.3.1. Hypothesis Testing  8.3.2. Regression Analysis
  9. Applied Mathematics  9.1. Numerical Analysis  9.1.1. Iterative Methods  9.1.2. Interpolation  9.1.3. Error Analysis  9.2. Partial Differential Equations  9.2.1. Elliptic Equations  9.2.2. Parabolic Equations  9.2.3. Hyperbolic Equations  9.3. Dynamical Systems  9.3.1. Chaos Theory  9.3.2. Stability Analysis in Dynamical Systems

PHD → Probability and Statistics ↓


Statistical Inference


Statistical inference is a branch of mathematics that deals with drawing conclusions about populations based on samples. It involves making predictions or generalizations about a larger group by examining a smaller, related group. Statistical inference includes a range of techniques and methodologies used to analyze data and draw meaningful conclusions. Its main purpose is to infer the properties of the underlying distribution by analyzing the data. This can be accomplished through estimating parameters, testing hypotheses, and making predictions.

Basic concepts

To understand statistical inference, we need to know some basic concepts:

  • Population: The collection of all the items we are interested in studying. This could be people, animals, events, or things.
  • Sample: A subset of a population that is selected to represent the entire population.
  • Parameter: A numerical characteristic of a population, such as the mean or variance.
  • Statistics: Numerical characteristics of a sample.

Statistical inference allows us to draw valid conclusions about a population based on sample data. For example, if we are interested in estimating the average height of all adults in a country, we can measure the height of a representative sample and use the estimate to generalize to the entire population.

Types of statistical inference

There are two primary types of statistical inference:

  • Estimation: It involves estimating the parameters of the population based on the sample data. There are two types of estimation:
    • Point estimation: Provides a single value as an estimate of a population parameter. For example, using the sample mean to estimate the population mean.
    • Interval estimation: Provides a range of values within which the parameter is expected to lie. A common example of this is a confidence interval.
  • Hypothesis testing: This involves making decisions about population parameters. It tests an assumption about a population parameter. The basic idea is to test a hypothesis by comparing it with sample data, then accept or reject it based on the evidence.

Point estimation

Point estimation involves using sample data to calculate a single value that serves as an estimate of an unknown population parameter. A common point estimator is the sample mean ((bar{x})), which is used to estimate the population mean ((mu)).

[ bar{x} = frac{1}{n} sum_{i=1}^{n} x_i ]

where x_i denotes individual sample observations, and n is the sample size.

Confidence interval

Unlike point estimation, interval estimation provides a range of values that are accepted with a certain level of confidence to contain the parameter.

For example, the confidence interval for a population mean can be calculated using the sample mean and standard deviation.

[ CI = bar{x} pm Z_{frac{alpha}{2}} times frac{sigma}{sqrt{n}} ]

where (bar{x}) is the sample mean, Z_{frac{alpha}{2}} is the Z-value from the standard normal distribution for the desired confidence level, (sigma) is the sample standard deviation, and n is the sample size.

Lower bound Upper bound

The above line shows the confidence interval.

Hypothesis testing

Hypothesis testing is a formal process of testing our ideas about the world using statistics. It is a method of making judgements using data, whether from a controlled experiment or an observational study (not controlled).

Steps of hypothesis testing

  1. State the hypotheses: This includes the null hypothesis (H_0) which indicates no effect or the status quo, and the alternative hypothesis (H_a) that you want to test.
  2. Choose the significance level ((alpha)): Commonly used values are 0.05, 0.01, and 0.1.
  3. Calculate the test statistic: This involves using statistical formulas to find a value that can help you decide whether to reject the null hypothesis.
  4. Determine the p-value: The probability of observing a given value in your data, or an even more extreme value, under the assumption that the null hypothesis is true.
  5. Draw a conclusion: If the p-value is less than or equal to the significance level, reject the null hypothesis in favor of the alternative hypothesis. Otherwise, do not reject the null hypothesis.

Example: Suppose you are testing whether a new teaching method is more effective than a traditional method.

  • H_0: The new method is not more effective (mean score difference = 0).
  • H_a: The new method is more effective (mean score difference > 0).

Suppose your test statistic follows a normal distribution and you get a p-value of 0.03. With a significance level of (alpha = 0.05), since 0.03 < 0.05, you reject H_0 in favor of H_a. Therefore, this data provides enough evidence to conclude that the new teaching method is more effective.

Commonly used tests and distributions

Various tests and statistical distributions are commonly used in statistical inference.

Normal distribution

The normal distribution is a continuous probability distribution that is symmetric on either side of the mean. It is widely used because of the central limit theorem, which states that the sum of independent random variables is normally distributed, regardless of their original distribution.

Meaning - +

T-test

T-tests are used to determine whether there are significant differences between the means of two groups. They are typically used when the data follows a normal distribution and when the population variance is unknown. Common types include:

  • One-sample t-test: This tests whether the mean of a single group is significantly different from a known or hypothesized value.
  • Independent two-sample t-test: Compares the means of two independent groups.
  • Pearson's correlation t-test: Tests for a linear relationship between two variables.

Conclusion

Statistical inference is an essential part of analyzing data and drawing conclusions about a population. With techniques such as estimation and hypothesis testing, we can understand more about our data and make informed decisions based on it. While the methods can be complex, the fundamental principles of creating samples, estimating population parameters, and making decisions through hypothesis testing remain consistent. With practice and understanding, statistical inference can be a powerful tool in a mathematician's toolkit.


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Statistical Inference

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