7.1 Point and Interval Estimation

Estimation techniques are crucial for understanding population characteristics from sample data. Point estimates, like sample means and proportions, provide single values for parameters, while confidence intervals offer ranges likely containing true values.

Biased and unbiased estimators impact accuracy, with unbiased ones having expected values equal to true parameters. Sample size affects precision, balancing confidence levels and margins of error. These methods help researchers draw meaningful conclusions from limited data.

Understanding Estimation Techniques

Point estimation for population parameters

• Point estimation uses a single value to estimate a population parameter based on sample data
• Common population parameters include mean (μ), proportion (p), and variance (σ²)
• Calculation methods for point estimates:
  1. Sample mean ($\bar{x}$) estimates population mean: $\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$
  2. Sample proportion ($\hat{p}$) estimates population proportion: $\hat{p} = \frac{x}{n}$
  3. Sample variance ($s^2$) estimates population variance: $s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}$
• Good point estimators exhibit consistency, efficiency, and sufficiency

Biased vs unbiased estimators

• Unbiased estimators have expected value equal to true population parameter (sample mean, sample proportion)
• Biased estimators have expected value different from true parameter (sample variance using n instead of n-1)
• Bias formula: $Bias(\hat{\theta}) = E(\hat{\theta}) - \theta$, where $\hat{\theta}$ is estimator and $\theta$ is true parameter
• Properties include variance and mean squared error (MSE)
• Bias-variance trade-off impacts estimator performance

Confidence intervals for means and proportions

• Confidence interval provides range likely containing true population parameter with specified confidence level
• Components: point estimate and margin of error
• Population mean (known σ): $\bar{x} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}}$
• Population mean (unknown σ): $\bar{x} \pm t_{\alpha/2, n-1} \frac{s}{\sqrt{n}}$
• Population proportion: $\hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$
• Wider intervals indicate less precision, narrower intervals more precision
• Higher confidence levels result in wider intervals

Sample size for interval precision

• Factors: desired confidence level, acceptable margin of error, population variability
• Population mean: $n = (\frac{z_{\alpha/2} \sigma}{E})^2$, where E is desired margin of error
• Population proportion: $n = \frac{z_{\alpha/2}^2 p(1-p)}{E^2}$, use $p = 0.5$ for conservative estimate
• Finite population correction factor adjusts sample size for small populations
• Trade-offs between precision, cost, and time influence sample size determination