🧰Engineering Applications of Statistics Unit 4 – Sampling & Estimation in Statistics

Key Concepts

• Population refers to the entire group of individuals, objects, or events of interest in a statistical study
• Sample is a subset of the population selected for analysis and inference about the population characteristics
• Sampling involves selecting a representative subset of the population to draw conclusions about the whole population
• Parameters are the true values of population characteristics (mean, proportion, standard deviation) usually unknown
• Statistics are the values calculated from sample data used to estimate the corresponding population parameters
• Sampling distribution describes the distribution of a statistic obtained from repeated sampling of the same size from a population
• Central Limit Theorem states that for large sample sizes, the sampling distribution of the sample mean approximates a normal distribution regardless of the population distribution shape
• Standard error measures the variability of a statistic (sample mean or proportion) from one sample to another

Types of Sampling

• Simple random sampling ensures each member of the population has an equal chance of being selected
  • Requires a complete list of population members (sampling frame)
  • Can be done with replacement (selected member is put back into the population for possible reselection) or without replacement
• Stratified sampling divides the population into homogeneous subgroups (strata) based on a specific characteristic and then randomly samples from each stratum
  • Ensures representation of all important subgroups in the sample
  • Improves precision of estimates for each stratum and the overall population
• Cluster sampling involves dividing the population into clusters (naturally occurring groups), randomly selecting some clusters, and including all members of chosen clusters in the sample
  • Useful when a complete list of population members is not available but clusters are identifiable
  • Reduces costs associated with data collection across a wide geographic area
• Systematic sampling selects members from an ordered sampling frame by starting at a randomly chosen point and then picking every kth element thereafter
  • Easier to implement than simple random sampling but may introduce bias if there is a hidden pattern in the ordering
• Convenience sampling selects members based on their easy accessibility or availability (mall intercepts, online surveys)
  • Least reliable method as the sample may not be representative of the population
  • Useful for pilot studies or when randomization is not feasible

Sample Size Determination

• Sample size is a crucial factor in determining the precision and reliability of estimates and the power of statistical tests
• Larger sample sizes generally lead to more precise estimates and higher power but also increase costs and time
• Factors influencing sample size include:
  • Desired level of precision (margin of error)
  • Confidence level (commonly 95%)
  • Population variability (more variability requires larger samples)
  • Population size (has a lesser impact when the population is large relative to the sample size)
  • Expected response rate (nonresponse requires a larger initial sample)
• Sample size can be determined using formulas, tables, or software based on the estimation problem (means, proportions) and study design
  • For estimating a population mean with a continuous outcome, the formula is: $n = \frac{Z^2 \sigma^2}{E^2}$ where $Z$ is the critical value from the standard normal distribution (e.g., 1.96 for 95% confidence), $\sigma$ is the population standard deviation, and $E$ is the desired margin of error
  • For estimating a population proportion with a categorical outcome, the formula is: $n = \frac{Z^2 p(1-p)}{E^2}$ where $p$ is the anticipated population proportion
• Adjustments to the calculated sample size may be needed to account for expected nonresponse, multiple comparisons, or clustering effects

Point Estimation

• Point estimation involves using sample data to calculate a single value (statistic) as an estimate of a population parameter
• Common point estimators include:
  • Sample mean ($\bar{x}$) estimates the population mean ($\mu$)
  • Sample proportion ($\hat{p}$) estimates the population proportion ($p$)
  • Sample variance ($s^2$) and standard deviation ($s$) estimate the population variance ($\sigma^2$) and standard deviation ($\sigma$)
• Desirable properties of point estimators are:
  • Unbiasedness: the expected value of the estimator equals the true parameter value
  • Efficiency: the estimator has the smallest variance among all unbiased estimators
  • Consistency: as the sample size increases, the estimator converges to the true parameter value
• Maximum likelihood estimation (MLE) is a general approach for obtaining point estimators with desirable properties
  • Involves finding the parameter values that maximize the likelihood function (the joint probability of observing the sample data)
• Method of moments estimation equates sample moments (mean, variance) to their population counterparts to solve for parameter estimates

Interval Estimation

• Interval estimation provides a range of plausible values for a population parameter with a specified level of confidence
• Confidence intervals (CIs) are the most common form of interval estimation
  • Consist of a lower and upper limit calculated from sample data and a confidence level (e.g., 95%)
  • Interpretation: if repeated samples were taken and CIs constructed for each, the specified proportion (e.g., 95%) of those intervals would contain the true parameter value
• General form of a CI: point estimate ± margin of error
  • Margin of error depends on the desired confidence level, sample variability, and sample size
• CIs can be one-sided (lower or upper bound only) or two-sided (both bounds)
• CIs for means assume normally distributed data or a large enough sample size for the Central Limit Theorem to apply
• CIs for proportions require a large enough sample size (usually $np \geq 10$ and $n(1-p) \geq 10$) and a normal approximation to the binomial distribution
• Factors affecting the width of a CI:
  • Confidence level: higher confidence leads to wider intervals
  • Sample size: larger samples produce narrower intervals
  • Population variability: more variability results in wider intervals

Confidence Intervals

• CI for a population mean ($\mu$) with known population standard deviation ($\sigma$): $\bar{x} \pm Z_{\alpha/2} \frac{\sigma}{\sqrt{n}}$ where $Z_{\alpha/2}$ is the critical value from the standard normal distribution corresponding to the desired confidence level
• CI for a population mean ($\mu$) with unknown population standard deviation (use sample standard deviation $s$ as an estimate): $\bar{x} \pm t_{\alpha/2, n-1} \frac{s}{\sqrt{n}}$ where $t_{\alpha/2, n-1}$ is the critical value from the t-distribution with $n-1$ degrees of freedom
• CI for a population proportion ($p$): $\hat{p} \pm Z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$ where $\hat{p}$ is the sample proportion
• CIs for the difference between two means or two proportions follow a similar format, using the appropriate standard error and critical value
• CIs can be used for hypothesis testing by checking whether a hypothesized value falls within the interval
  • If the hypothesized value is outside the CI, it is rejected at the corresponding significance level
  • If the hypothesized value is inside the CI, it cannot be rejected at that significance level

Estimation Errors and Biases

• Sampling error occurs due to the variability inherent in selecting a sample from a population
  • Larger samples tend to have smaller sampling errors
  • Quantified by the standard error of the estimator
• Nonsampling error arises from sources other than the sampling process, such as:
  • Measurement error: inaccurate measurements or responses
  • Coverage error: the sampling frame does not include all members of the target population
  • Nonresponse error: differences between respondents and nonrespondents lead to biased estimates
• Selection bias occurs when the sampling method systematically favors certain members of the population over others
  • Example: voluntary response samples often overrepresent individuals with strong opinions or interests
• Undercoverage bias arises when certain segments of the population are inadequately represented in the sample
  • Example: telephone surveys may exclude households without landlines
• Response bias happens when respondents provide inaccurate or misleading answers due to factors such as social desirability, question wording, or interviewer effects
• Nonresponse bias occurs when those who respond to a survey differ systematically from those who do not respond
• Strategies to minimize biases:
  • Use probability sampling methods to ensure representativeness
  • Validate the sampling frame against the target population
  • Design clear and neutral questions to minimize response bias
  • Follow up with nonrespondents to encourage participation and assess potential differences
  • Weight the sample data to adjust for known discrepancies between the sample and population demographics

Real-World Applications

• Quality control: sampling is used to monitor the quality of products or processes in manufacturing settings
  • Example: a company producing light bulbs may randomly test a sample of bulbs from each batch to ensure they meet specifications for brightness and longevity
• Public opinion polls: surveys are conducted to gauge public sentiment on various issues or to predict election outcomes
  • Example: a news organization commissions a poll of likely voters to estimate support for different candidates or policies
• Clinical trials: medical researchers use sampling to test the safety and efficacy of new drugs or treatments
  • Example: a pharmaceutical company conducts a randomized controlled trial with a sample of patients to compare a new medication against a placebo or existing treatment
• Environmental monitoring: scientists use sampling to assess the health of ecosystems or the levels of pollutants in air, water, or soil
  • Example: a government agency collects water samples from a river at various locations to estimate the concentration of contaminants and their sources
• Auditing: financial auditors use sampling techniques to verify the accuracy of accounting records or detect fraud
  • Example: an auditor selects a random sample of transactions from a company's ledger to check for errors or irregularities
• Market research: businesses use surveys or focus groups to gather information about consumer preferences, satisfaction, or behavior
  • Example: a car manufacturer surveys a sample of recent buyers to assess their experience with the vehicle and identify areas for improvement
• Educational assessment: schools or testing organizations use sampling to evaluate student learning or the effectiveness of curricula
  • Example: a state education department administers standardized tests to a representative sample of students to measure achievement gaps and progress over time