📉Statistical Methods for Data Science Unit 4 – Sampling and Estimation in Data Science

Key Concepts and Terminology

• Population refers to the entire group of individuals, objects, or events of interest in a study
• Sample is a subset of the population selected for analysis and inference
• Sampling is the process of selecting a representative subset of the population for study
• Sampling frame is a list or database that represents the entire population from which a sample is drawn
• Sampling bias occurs when the sample selected does not accurately represent the population, leading to skewed results
• Sampling error is the difference between the sample statistic and the true population parameter
• Probability sampling involves selecting a sample using random methods, giving each member of the population a known chance of being selected
• Non-probability sampling relies on non-random methods to select a sample, often based on convenience or judgment

Types of Sampling Methods

• Simple random sampling (SRS) selects a sample from the population such that each member has an equal chance of being chosen
  • Requires a complete list of the population (sampling frame) and can be time-consuming for large populations
• Stratified sampling divides the population into homogeneous subgroups (strata) and selects a random sample from each stratum
  • Ensures representation of important subgroups and can improve precision of estimates
• Cluster sampling involves dividing the population into clusters (naturally occurring groups) and randomly selecting a subset of clusters to sample
  • Useful when a complete list of the population is not available or when the population is geographically dispersed
• Systematic sampling selects every kth element from the population after randomly choosing a starting point
  • Easy to implement but may introduce bias if there is a hidden pattern in the population
• Convenience sampling selects a sample based on ease of access or availability (non-probability method)
  • Prone to bias and may not be representative of the population
• Snowball sampling relies on initial participants to recruit additional participants from their networks (non-probability method)
  • Useful for hard-to-reach or hidden populations but may introduce bias

Probability Theory Fundamentals

• Probability is a measure of the likelihood of an event occurring, expressed as a number between 0 and 1
• Sample space is the set of all possible outcomes of an experiment or random process
• Event is a subset of the sample space, representing a specific outcome or set of outcomes
• Probability distribution is a function that assigns probabilities to each possible outcome in the sample space
  • Discrete probability distributions (binomial, Poisson) are used for countable outcomes
  • Continuous probability distributions (normal, exponential) are used for measurable outcomes
• Independence occurs when the occurrence of one event does not affect the probability of another event
• Conditional probability is the probability of an event occurring given that another event has already occurred
• Bayes' theorem describes the relationship between conditional probabilities and can be used to update probabilities based on new evidence

Point Estimation Techniques

• Point estimation involves using sample data to calculate a single value (point estimate) that serves as a "best guess" for an unknown population parameter
• Method of moments estimator equates sample moments (mean, variance) to their population counterparts and solves for the parameter
  • Easy to calculate but may not always be the most efficient or robust estimator
• Maximum likelihood estimator (MLE) selects the parameter value that maximizes the likelihood function, given the observed data
  • Asymptotically efficient and consistent but may be computationally intensive
• Bayesian estimator incorporates prior knowledge about the parameter in the form of a prior distribution and updates it with observed data to obtain a posterior distribution
  • Allows for the incorporation of subjective information but requires the specification of a prior distribution
• Unbiased estimator is an estimator whose expected value is equal to the true population parameter
  • Desirable property but not always achievable or the most important consideration
• Consistent estimator converges in probability to the true population parameter as the sample size increases
  • Important for ensuring the reliability of estimates in large samples

Interval Estimation and Confidence Intervals

• Interval estimation provides a range of plausible values for an unknown population parameter, rather than a single point estimate
• Confidence interval is an interval estimate that has a specified probability (confidence level) of containing the true population parameter
  • Commonly used confidence levels are 90%, 95%, and 99%
• Margin of error is the half-width of the confidence interval and represents the maximum expected difference between the point estimate and the true population parameter
  • Smaller margins of error indicate more precise estimates
• Factors affecting the width of a confidence interval include sample size, variability in the data, and the desired confidence level
  • Larger sample sizes, lower variability, and lower confidence levels lead to narrower intervals
• Interpreting confidence intervals involves understanding that the interval represents a range of plausible values for the population parameter, not the probability of the parameter falling within the interval
  • Confidence level refers to the long-run proportion of intervals that would contain the true parameter if the sampling process were repeated many times

Bias and Variance in Sampling

• Bias is the systematic deviation of an estimator from the true population parameter
  • Can arise from non-representative sampling, measurement errors, or flawed estimation methods
• Variance is a measure of the variability or dispersion of an estimator around its expected value
  • Reflects the inherent randomness in the sampling process and the estimator's sensitivity to sample fluctuations
• Bias-variance tradeoff is the relationship between an estimator's bias and variance, where reducing one often leads to an increase in the other
  • Unbiased estimators may have high variance, while biased estimators with lower variance can sometimes be preferable
• Sampling bias can be minimized through careful design and implementation of sampling methods, such as using probability sampling and ensuring a representative sampling frame
• Variance can be reduced by increasing the sample size, using more efficient estimators, or employing techniques like stratified sampling
• Mean squared error (MSE) is a measure that combines both bias and variance to assess the overall quality of an estimator
  • MSE = Bias^2 + Variance

Sample Size Determination

• Sample size determination is the process of calculating the minimum number of observations needed to achieve a desired level of precision or statistical power
• Factors influencing sample size include the variability in the population, the desired margin of error, confidence level, and the type of analysis being conducted
  • More variable populations, smaller margins of error, higher confidence levels, and more complex analyses generally require larger sample sizes
• Power analysis is used to determine the sample size needed to detect a specific effect size with a given level of significance and power
  • Effect size is the magnitude of the difference or relationship being studied
  • Significance level (alpha) is the probability of rejecting a true null hypothesis (Type I error)
  • Power is the probability of correctly rejecting a false null hypothesis (1 - Type II error)
• Sample size calculators and formulas are available for various study designs and analysis methods, such as means, proportions, and regression
• Practical considerations in sample size determination include budget constraints, time limitations, and the availability of participants
  • Adaptive designs and sequential analysis can be used to adjust the sample size during the study based on interim results

Real-world Applications and Case Studies

• Market research uses sampling to gather information about consumer preferences, product satisfaction, and market trends
  • Stratified sampling can ensure representation of different demographic groups
  • Cluster sampling can be used to survey geographically dispersed populations
• Public opinion polls employ sampling techniques to gauge public sentiment on various issues, candidates, or policies
  • Random digit dialing (RDD) is a form of SRS used to select phone numbers for telephone surveys
  • Weighting techniques are used to adjust for non-response bias and ensure the sample is representative of the population
• Quality control in manufacturing relies on sampling to monitor the quality of products and identify defects
  • Acceptance sampling involves selecting a sample from a batch and deciding whether to accept or reject the entire batch based on the sample results
  • Sequential sampling allows for the adjustment of sample size based on the observed quality levels
• Clinical trials use sampling and estimation to evaluate the safety and efficacy of new medical treatments
  • Randomized controlled trials (RCTs) employ random assignment to treatment and control groups to minimize bias
  • Interim analyses and adaptive designs can be used to modify the sample size or stop the trial early based on the observed treatment effects
• Environmental monitoring uses sampling to assess the quality of air, water, and soil resources
  • Stratified sampling can be used to ensure coverage of different regions or land use types
  • Composite sampling involves combining multiple samples to reduce the number of analyses needed while still providing representative results