6.6 Inferential statistics

Inferential statistics is a powerful tool for drawing conclusions about populations based on sample data. It provides methods for estimating parameters, testing hypotheses, and quantifying uncertainty in statistical analyses.

From probability distributions to hypothesis testing, inferential statistics offers a range of techniques for making informed decisions. Understanding concepts like confidence intervals, p-values, and statistical power is crucial for interpreting research findings and designing effective studies.

Foundations of inferential statistics

• Inferential statistics forms the backbone of data-driven decision-making in mathematics and scientific research
• Allows mathematicians to draw conclusions about larger populations based on smaller, representative samples
• Provides tools for estimating parameters, testing hypotheses, and quantifying uncertainty in statistical analyses

Population vs sample

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• Population encompasses all individuals or items of interest in a study
• Sample represents a subset of the population selected for analysis
• Random sampling ensures each member of the population has an equal chance of selection
• Stratified sampling divides the population into subgroups before sampling (age groups, income levels)

Parameters vs statistics

• Parameters describe characteristics of entire populations (μ for population mean, σ for population standard deviation)
• Statistics serve as estimates of population parameters based on sample data ($\bar{x}$ for sample mean, s for sample standard deviation)
• Sampling distribution connects sample statistics to population parameters
• Standard error measures the variability of a statistic across different samples

Sampling methods

• Simple random sampling gives each member of the population an equal chance of selection
• Systematic sampling selects every nth item from a population list
• Cluster sampling divides the population into clusters and randomly selects entire clusters
• Convenience sampling uses easily accessible subjects but may introduce bias

Probability distributions

• Probability distributions model the likelihood of different outcomes in random processes
• Essential for understanding variability and making predictions in statistical analyses
• Form the basis for many inferential techniques, including hypothesis testing and confidence intervals

Normal distribution

• Bell-shaped, symmetric curve characterized by mean (μ) and standard deviation (σ)
• 68-95-99.7 rule describes data distribution within 1, 2, and 3 standard deviations of the mean
• Z-scores standardize normal distributions, allowing comparisons across different scales
• Central Limit Theorem states that means of large samples approximate a normal distribution

t-distribution

• Similar to normal distribution but with heavier tails
• Used when sample size is small or population standard deviation is unknown
• Degrees of freedom determine the shape of the t-distribution
• Approaches normal distribution as sample size increases

Chi-square distribution

• Always positive and right-skewed
• Used for goodness-of-fit tests and tests of independence
• Shape determined by degrees of freedom
• Approaches normal distribution as degrees of freedom increase

Confidence intervals

• Provide a range of plausible values for population parameters
• Quantify uncertainty in parameter estimates
• Help researchers make informed decisions based on sample data
• Balance precision and confidence in statistical inference

Margin of error

• Represents the maximum expected difference between the sample statistic and population parameter
• Calculated as the product of the critical value and standard error
• Decreases as sample size increases
• Affects the width of confidence intervals

Confidence level

• Probability that the true population parameter falls within the confidence interval
• Common levels include 90%, 95%, and 99%
• Higher confidence levels result in wider intervals
• Trade-off between confidence and precision in parameter estimation

Sample size considerations

• Larger samples generally lead to narrower confidence intervals
• Power analysis helps determine appropriate sample sizes for desired precision
• Cost and feasibility constraints may limit sample size in practice
• Balancing statistical power and practical limitations in study design

Hypothesis testing

• Formal process for evaluating claims about population parameters
• Allows researchers to make decisions based on sample data
• Involves comparing observed results to expected outcomes under null hypothesis
• Crucial for scientific inquiry and evidence-based decision making

Null vs alternative hypotheses

• Null hypothesis (H0) assumes no effect or difference in the population
• Alternative hypothesis (Ha) proposes a specific effect or difference
• One-tailed tests specify direction of effect (greater than or less than)
• Two-tailed tests consider effects in both directions

Type I and Type II errors

• Type I error occurs when rejecting a true null hypothesis (false positive)
• Type II error involves failing to reject a false null hypothesis (false negative)
• α (alpha) represents the probability of Type I error (significance level)
• β (beta) denotes the probability of Type II error (1 - power)

p-values and significance levels

• p-value measures the probability of obtaining results as extreme as observed, assuming null hypothesis is true
• Significance level (α) sets the threshold for rejecting the null hypothesis
• Common significance levels include 0.05 and 0.01
• Researchers reject H0 when p-value < α

Statistical tests

• Various tests designed to evaluate specific types of hypotheses
• Selection depends on research question, data type, and sample characteristics
• Parametric tests assume normally distributed data
• Non-parametric tests used for non-normal distributions or ordinal data

t-tests

• Compare means between two groups or one group against a known value
• Independent samples t-test used for two separate groups
• Paired samples t-test applied to before-and-after measurements on same subjects
• One-sample t-test compares sample mean to hypothesized population mean

ANOVA

• Analysis of Variance compares means across three or more groups
• One-way ANOVA examines effect of one independent variable on dependent variable
• Two-way ANOVA investigates effects of two independent variables and their interaction
• F-statistic used to assess overall significance of group differences

Chi-square tests

• Chi-square goodness-of-fit test compares observed frequencies to expected frequencies
• Chi-square test of independence examines relationship between two categorical variables
• Degrees of freedom calculated based on number of categories
• Assumptions include independent observations and expected frequencies > 5 in each cell

Regression analysis

• Models relationships between variables
• Allows prediction of dependent variable based on independent variable(s)
• Quantifies strength and direction of associations
• Widely used in economics, social sciences, and natural sciences

Simple linear regression

• Models relationship between one independent variable (X) and one dependent variable (Y)
• Equation: Y = β0 + β1X + ε, where β0 is y-intercept and β1 is slope
• Least squares method minimizes sum of squared residuals
• R-squared measures proportion of variance in Y explained by X

Multiple regression

• Extends simple linear regression to include multiple independent variables
• Equation: Y = β0 + β1X1 + β2X2 + ... + βkXk + ε
• Partial regression coefficients represent effect of each X while controlling for others
• Adjusted R-squared accounts for number of predictors in model

Correlation coefficients

• Pearson's r measures strength and direction of linear relationship between two variables
• Values range from -1 (perfect negative correlation) to +1 (perfect positive correlation)
• Spearman's rho used for ordinal data or non-linear relationships
• Point-biserial correlation applied when one variable is dichotomous

Bayesian inference

• Alternative approach to statistical inference based on Bayes' theorem
• Incorporates prior knowledge or beliefs into statistical analysis
• Allows for updating of probabilities as new evidence becomes available
• Gaining popularity in fields such as machine learning and data science

Bayes' theorem

• Fundamental principle of Bayesian statistics
• Expresses posterior probability in terms of prior probability and likelihood
• Formula: P(A|B) = [P(B|A) * P(A)] / P(B)
• Enables calculation of conditional probabilities

Prior vs posterior probabilities

• Prior probability represents initial belief or knowledge before observing data
• Likelihood function describes probability of observed data given different parameter values
• Posterior probability combines prior and likelihood to update beliefs based on evidence
• Iterative process allows for continuous updating as new data becomes available

Bayesian vs frequentist approaches

• Frequentist methods focus on long-run probabilities of events
• Bayesian approach incorporates subjective probabilities and prior knowledge
• Frequentist inference uses fixed parameters and random data
• Bayesian inference treats parameters as random variables and data as fixed

Sampling distributions

• Theoretical distributions of sample statistics
• Describe variability of statistics across repeated sampling
• Crucial for understanding precision of parameter estimates
• Form basis for many inferential techniques

Central limit theorem

• States that sampling distribution of means approaches normal distribution as sample size increases
• Applies regardless of underlying population distribution (with some exceptions)
• Enables use of normal distribution for inference about population means
• Generally considered applicable when n ≥ 30

Standard error

• Measures variability of a sample statistic
• Calculated as standard deviation of sampling distribution
• Decreases as sample size increases
• Used in calculation of confidence intervals and test statistics

Sampling variability

• Refers to differences in statistics across different samples from same population
• Affected by sample size, population variability, and sampling method
• Larger samples generally lead to less sampling variability
• Understanding sampling variability crucial for interpreting statistical results

Effect size and power

• Effect size quantifies magnitude of observed effects or relationships
• Statistical power represents probability of detecting a true effect
• Both concepts essential for designing studies and interpreting results
• Help researchers distinguish between statistical and practical significance

Cohen's d

• Standardized measure of effect size for comparing two group means
• Calculated as difference between means divided by pooled standard deviation
• Interpretations: small (0.2), medium (0.5), large (0.8)
• Allows comparison of effects across different scales or studies

Statistical power

• Probability of correctly rejecting false null hypothesis (1 - β)
• Influenced by effect size, sample size, and significance level
• Conventionally, power of 0.8 (80%) considered adequate
• Power analysis helps determine sample size needed to detect meaningful effects

Sample size determination

• Involves balancing statistical power, effect size, and practical constraints
• Larger samples increase power but may be costly or impractical
• A priori power analysis estimates required sample size before study
• Post hoc power analysis calculates achieved power after study completion

Advanced inferential techniques

• More sophisticated methods for complex research questions
• Often require specialized software and advanced statistical knowledge
• Address limitations of traditional inferential approaches
• Expanding rapidly with advances in computing power and data availability

Bootstrapping

• Resampling technique for estimating sampling distributions
• Involves repeatedly drawing samples with replacement from original data
• Useful when theoretical sampling distributions are unknown or assumptions violated
• Provides robust estimates of standard errors and confidence intervals

Meta-analysis

• Statistical method for combining results from multiple studies
• Increases statistical power and precision of effect size estimates
• Accounts for between-study variability and publication bias
• Widely used in medicine, psychology, and other fields for synthesizing research findings

Multivariate analysis

• Analyzes relationships among multiple variables simultaneously
• Includes techniques such as MANOVA, factor analysis, and discriminant analysis
• Accounts for correlations among dependent variables
• Allows for more comprehensive understanding of complex phenomena