Statistics and Probability

Statistics and Probability

1. Role of Statistics in Research

   • Functions: Analysis of events, theory establishment, practical applications.
   • Tools: Techniques for data collection and analysis.

2. Experimental Research

   • Influenced by random or stochastic factors.
   • Characteristics: Control of phenomenon, observation, and interaction.

3. Statistical Regularity and Predictions

   • Development: 18th century, regularity in statistical populations.
   • Concept: Emergent regularity in large repetitions.

4. Statistical Method

   • Objective: Relationships between variables, laws or causes
   • Process: Observation, data collection, analysis, deduction.
     • CONTEXT CASE STUDY
       • Statistical Inference
         • Hypothesis Testing: Using information from the sample to answer questions about the population.
           • PROBABILITY
             • Origin of Probability Theory
               • Historical Context: Probability theory's origins are linked with games of chance.
               • Historical Context: Probability theory's origins are linked with games of chance.
             • Application in Modern Context
               • Data Collection: Obtaining real-world data, often unstructured and messy.
                 • • Statistical Analysis
                     • Populations: Often inaccessible or impossible to study all elements.
                     • Increase in Study Quality: More precise observations and measurements with fewer individuals.
                     • Sample Selection: Inclusion and exclusion criteria to reduce population heterogeneity.
               • Data Organization: The first step in analysis is organizing data for study.
         • Estimation: Using characteristics of a sample to infer about the population.
           • • POPULATION, SAMPLE AND INDIVIDUAL
               • Sample
                 • Definition: A subgroup of individuals selected from the population fo
                   • Example: COVID-19 patients from the same city treated in a hospital during a specific period.
                   • Purpose: To represent the population in a study, allowing for inferences about the larger group.
               • Population
                 • Definition: A set of individuals with defining characteristics of their group.
                   • Example: All COVID-19 patients in a city.
                   • Characteristics: Shared attributes specific to the group, like location and condition.
               • Individual
                 • Definition: The minimum unit of analysis within a population or sample.
                   • Examples: A patient, a family unit, etc.
                   • Note: An individual can refer to a person, group of people, animals, or things, depending on the context of the study.
       • Statistical Methods
         • Inductive: Characterizing the population based on knowledge derived from a sample.
           • • LAW OF LARGE NUMBERS
               • Probability and Relative Frequency
                 • Relationship: Probability is closely related to the relative frequency of events.
               • Weak Law of Large Numbers (Bernoulli's Theorem)
                 • Statement: If an event has a probability p in a single trial, and several independent trials are conducted under the same conditions, the most probable proportion of occurrences of the event approaches
                 • Historical Precedence: Preceded the Strong Law.
               • Laws of Large Numbers
                 • Purpose: Establish how relative frequencies of an event converge to its probability.
                 • Types of Convergence: "Weak" or "Strong".
         • Deductive: Characterizing each possible sample based on knowledge of the population.
           • • SAMPLING TYPES
               • Criteria for Sampling
                 • Feasibility: Practicality and achievability of the sampling process.
                 • Accuracy: Precision in reflecting the population's characteristics
               • Characteristics of a Good Sample
                 • Representative: Must contain the essential characteristics of the population in similar proportions.
                 • Adequate Size: Determined based on study requirements like prevalence, differences in means or proportions, correlations, or risk factors.
                 • Non-probabilistic Sampling: Selection process is subjective or arbitrary, without known probabilities for each unit's selection. Examples include convenience sampling and others.
       • Application of Statistics and Probability in Real Projects
         • Project Examples: Building a particle accelerator like the Large Hadron Collider at CERN, a hydroelectric power plant, organizing a birthday party, or planning a family vacation trip.
           • • SIMPLE LINEAR REGRESSION
               • Objective of Simple Linear Regression
                 • Purpose: To explain the relationship between a dependent (response) variable and an independent (explanatory) variable.
                 • Example: Explaining the relationship between a response variable Y and a single explanatory variable X.
               • Regression Techniques
                 • Approach: Regression of Y on X to find a function approximating a cloud of points (Xi,Yi)

5. Descriptive and Inferential Statistics

   • Inferential: Use of samples to make decisions about populations and relationships between variables.
   • Descriptive: Summarization of data in indicators and graphics.

6. Statistics as a Science

   • Objectives: Deduce laws, make forecasts, draw conclusions, make decisions.
   • Definition: Systematization and analysis of data with variability.