3.3 Estimation and inference in simple random sampling

Simple random sampling is a cornerstone of survey research. It allows us to make inferences about entire populations based on smaller samples. This chapter dives into the nitty-gritty of how we use sample data to estimate population characteristics.

We'll explore key concepts like point estimation, confidence intervals, and hypothesis testing. These tools help us quantify uncertainty and make informed decisions about population parameters using sample data. Understanding these techniques is crucial for interpreting survey results accurately.

Estimation

Population Parameters and Sample Statistics

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• Population parameter represents a characteristic of the entire population
• Sample statistic serves as an estimate of the population parameter based on a subset of the population
• Point estimation involves using a single value from the sample to estimate the population parameter
• Unbiased estimator produces estimates that are correct on average when repeated over many samples
• Finite population correction factor adjusts estimates when sampling from small populations

Estimator Properties and Corrections

• Bias measures the difference between an estimator's expected value and the true population parameter
• Consistency describes an estimator's tendency to converge on the true parameter as sample size increases
• Efficiency compares the variance of different estimators for the same parameter
• Robustness indicates an estimator's ability to perform well under various conditions or with outliers present
• Sufficiency determines if an estimator uses all relevant information from the sample

Sampling Distribution

Characteristics of Sampling Distributions

• Sampling distribution represents the distribution of a sample statistic over all possible samples
• Standard error quantifies the variability of the sampling distribution
• Sampling distribution shape often approximates a normal distribution for large sample sizes
• Sample size affects the spread of the sampling distribution (larger samples lead to narrower distributions)
• Sampling with or without replacement influences the sampling distribution (affects independence of observations)

Central Limit Theorem and Its Applications

• Central Limit Theorem states that the sampling distribution of the mean approaches a normal distribution as sample size increases
• Applies to many statistics beyond the mean (proportions, totals)
• Enables the use of normal distribution properties for inference, even when the population distribution is non-normal
• Requires sufficiently large sample sizes (typically n ≥ 30) for most practical applications
• Facilitates the construction of confidence intervals and hypothesis tests for population parameters

Confidence Intervals

Constructing and Interpreting Confidence Intervals

• Confidence interval provides a range of plausible values for the population parameter
• Margin of error represents the maximum likely difference between the sample statistic and population parameter
• Precision refers to the width of the confidence interval (narrower intervals indicate higher precision)
• Level of confidence determines the probability that the interval contains the true population parameter
• Confidence interval formula: \text{[Point Estimate](https://www.fiveableKeyTerm:Point_Estimate)} \pm (\text{Critical Value} \times \text{Standard Error})

Factors Affecting Confidence Intervals

• Sample size influences the width of the confidence interval (larger samples lead to narrower intervals)
• Population variability affects the precision of estimates (more variable populations require larger samples)
• Confidence level impacts the width of the interval (higher confidence levels result in wider intervals)
• Sampling method can affect the standard error and thus the confidence interval width
• Non-response and measurement errors can introduce bias and affect the accuracy of confidence intervals

Inference

Hypothesis Testing Fundamentals

• Hypothesis testing evaluates claims about population parameters based on sample data
• Null hypothesis (H₀) represents the status quo or no effect
• Alternative hypothesis (H₁ or Hₐ) represents the claim to be tested
• Test statistic quantifies the difference between the sample statistic and the hypothesized parameter value
• p-value measures the probability of obtaining the observed results if the null hypothesis is true

Sample Size Determination and Power Analysis

• Sample size determination calculates the number of observations needed to achieve desired precision or power
• Factors affecting sample size include desired precision, confidence level, and population variability
• Power analysis assesses the probability of correctly rejecting a false null hypothesis
• Effect size quantifies the magnitude of the difference or relationship being tested
• Trade-offs between sample size, power, and effect size guide study design decisions