Sampling techniques are crucial for gathering representative data. From simple random sampling to convenience sampling, each method has its strengths and weaknesses. Understanding these techniques helps researchers choose the best approach for their study.

The distribution of sample statistics is key to making inferences about populations. It's influenced by factors like sample size and population variability. Knowing these concepts helps us grasp how sample data relates to the broader population.

Sampling Techniques

Types of sampling techniques

• Simple Random Sampling ensures every individual has equal selection chance using random number generators or tables, providing unbiased method
• Stratified Sampling divides population into subgroups (strata) then samples from each stratum ensuring representation of all subgroups (socioeconomic classes, age groups)
• Cluster Sampling divides population into clusters, randomly selects entire clusters, and samples all individuals in chosen clusters (schools, neighborhoods)
• Systematic Sampling selects every nth item from a list using sampling interval (every 10th customer, every 5th house)
• Convenience Sampling uses non-probability method selecting easily accessible subjects (students on campus, shoppers at mall)

Distribution of Sample Statistics

Sampling distribution fundamentals

• Sampling distribution describes statistic variability over many samples (mean heights from multiple samples)
• Relationship to population distribution governed by Central Limit Theorem sampling distribution approaches normal distribution as sample size increases
• Key properties include sampling distribution mean equaling population parameter and standard error decreasing with larger sample sizes

Statistics of sampling distributions

• Mean of sampling distribution for sample mean $\mu_{\bar{x}} = \mu$ and for sample proportion $\mu_{\hat{p}} = p$
• Standard error (SE) for sample mean $SE_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$ and for sample proportion $SE_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$
• Other formulas include variance of sample mean $\sigma^2_{\bar{x}} = \frac{\sigma^2}{n}$ and SE of difference between means $SE_{\bar{x}_1 - \bar{x}_2} = \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}$

Factors affecting sampling distributions

• Sample size inversely affects standard error larger samples lead to narrower sampling distributions (n=30 vs n=1000)
• Population variability increases sampling distribution spread greater variance in population leads to wider sampling distribution
• Sampling fraction (ratio of sample to population size) affects SE when sampling without replacement
• Estimator properties including bias and efficiency influence sampling distribution characteristics
• Underlying population distribution affects sampling distribution shape for small samples less influential for large samples due to Central Limit Theorem (normal, skewed, bimodal)