5.2 Interval estimation and confidence intervals

Interval estimation provides a range of plausible values for population parameters, offering advantages over point estimation. It accounts for sampling variability, provides precision measures, and offers confidence levels. This approach is crucial in decision-making across various fields, from financial forecasting to quality control.

Constructing confidence intervals involves different formulas depending on the parameter being estimated and available information. Sample size determination is a critical aspect, considering factors like desired confidence level, acceptable margin of error, and population variability. Understanding these relationships helps balance precision and resource allocation in statistical decision-making.

Understanding Interval Estimation

Interval vs point estimation

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• Interval estimation provides range of plausible values for population parameter expressed as confidence interval
• Point estimation gives single value as estimate of population parameter
• Interval estimation advantages account for sampling variability, provide measure of precision, offer level of confidence in estimate
• Applications in decision-making include risk assessment (financial forecasting), quality control (manufacturing tolerances), market research (consumer preferences)

Construction of confidence intervals

• Confidence interval for population mean (known standard deviation)
  • Formula: $\bar{x} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}}$
  • Range containing true population mean with specified probability
• Confidence interval for population mean (unknown standard deviation)
  • Uses t-distribution
  • Formula: $\bar{x} \pm t_{\alpha/2, n-1} \frac{s}{\sqrt{n}}$
• Confidence interval for population proportion
  • Formula: $\hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$
  • Used for binary data or percentages (voter preferences, product defects)
• Confidence interval for population variance
  • Based on chi-square distribution
  • Formula: $[\frac{(n-1)s^2}{\chi^2_{\alpha/2, n-1}}, \frac{(n-1)s^2}{\chi^2_{1-\alpha/2, n-1}}]$

Sample Size and Confidence Level Relationships

Sample size determination

• Factors affecting sample size include desired confidence level, acceptable margin of error, population variability
• Sample size calculation for estimating population mean
  • Formula: $n = (\frac{z_{\alpha/2}\sigma}{E})^2$
  • E represents margin of error
• Sample size calculation for estimating population proportion
  • Formula: $n = \frac{z^2_{\alpha/2}p(1-p)}{E^2}$
  • Use $p = 0.5$ for conservative estimate
• Practical considerations involve budget constraints (research costs), time limitations (survey duration), population accessibility (remote populations)

Factors affecting confidence intervals

• Confidence level represents probability interval contains true population parameter (90%, 95%, 99%)
• Increasing confidence level results in wider confidence interval, requires larger sample size for same precision
• Increasing sample size leads to narrower confidence interval, improved precision of estimate
• Trade-offs in statistical decision-making balance precision and resource allocation, consider practical vs statistical significance
• Margin of error equals half-width of confidence interval, inversely proportional to square root of sample size