Introduction

Statistical inference is the process of using data analysis to infer properties of an underlying distribution of probability. It uses sample data to make decisions about populations.

Key concepts include:

• Drawing conclusions from data
• Estimating population parameters
• Testing hypotheses about populations
• Quantifying uncertainty

Hypothesis Testing

Hypothesis testing is a method of statistical inference used to decide whether data sufficiently supports a particular hypothesis about a population parameter.

The main steps are:

• Formulate null and alternative hypotheses
• Choose a significance level (α)
• Calculate test statistic
• Determine p-value
• Make a decision to reject or fail to reject the null hypothesis

Confidence Intervals

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the value of an unknown population parameter.

For example, a 95% confidence interval means that if we repeated the sampling process many times, approximately 95% of the intervals would contain the true population parameter.

Point Estimation

Point estimation is the process of providing a single value (a point estimate) as an estimate of an unknown population parameter based on sample data.

Common estimators include:

• Sample mean for population mean
• Sample proportion for population proportion
• Sample variance for population variance

Maximum Likelihood Estimation

Maximum likelihood estimation (MLE) is a method of estimating the parameters of a probability distribution by maximizing the likelihood function.

The likelihood function represents the probability of observing the given data as a function of the unknown parameters.

Bayesian Inference

Bayesian inference is a method that uses probability to represent uncertainty in statistical inference. It incorporates prior beliefs about parameters and updates them based on observed data.

Key components:

• Prior distribution
• Likelihood of the data
• Posterior distribution (updated beliefs)

Applications

Statistical inference is widely used in various fields:

• Quality control in manufacturing
• Clinical trials and medical research
• Market research and surveys
• Environmental monitoring
• Financial analysis and risk assessment