6.2 Confidence Intervals for Means and Proportions

Confidence intervals help estimate population parameters like means and proportions. They provide a range of values likely to contain the true parameter, based on sample data and desired confidence level.

Sample size affects interval precision. Larger samples yield narrower intervals, increasing estimate accuracy. Formulas for means and proportions differ slightly, but both balance confidence level with margin of error.

Confidence Intervals for Population Parameters

Confidence intervals for population means

• Confidence interval formula for population mean expands point estimate ± (critical value × standard error)
• Known variance (σ² known) uses $\bar{x} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}}$ with z-distribution for critical value
• Unknown variance (σ² unknown) employs $\bar{x} \pm t_{\alpha/2, n-1} \frac{s}{\sqrt{n}}$ with t-distribution for critical value
• Interpretation conveys probability of containing true population parameter within confidence level (1 - α)
• Interval width affected by sample size, confidence level, and population standard deviation

Formulas for population proportion intervals

• Confidence interval for population proportion calculated as $\hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$
• Formula requires large sample size (n > 30) and np ≥ 5 and n(1-p) ≥ 5
• Small populations need finite population correction factor
• Alternative methods for small samples include Wilson score interval and Exact (Clopper-Pearson) interval

Sample Size and Precision

Margin of error and sample size

• Margin of error represents half-width of confidence interval
• Calculated for means as $E = z_{\alpha/2} \frac{\sigma}{\sqrt{n}}$ or $E = t_{\alpha/2, n-1} \frac{s}{\sqrt{n}}$
• Determined for proportions using $E = z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$
• Sample size for means (known σ) found through $n = (\frac{z_{\alpha/2}\sigma}{E})^2$
• Sample size for means (unknown σ) computed via $n = (\frac{t_{\alpha/2, n-1}s}{E})^2$
• Sample size for proportions derived from $n = \frac{z_{\alpha/2}^2 \hat{p}(1-\hat{p})}{E^2}$
• Sample size influenced by desired confidence level, acceptable margin of error, and estimated population variability
• Unknown population parameters require iterative process

Intervals for means vs proportions

• Both utilize point estimate ± (critical value × standard error) with similar confidence level interpretation
• Precision and confidence level share inverse relationship
• Means use z or t-distribution while proportions always use z-distribution for critical values
• Standard error for means based on standard deviation, proportions on $\sqrt{\hat{p}(1-\hat{p})}$
• Means have no strict minimum sample size, proportions require np ≥ 5 and n(1-p) ≥ 5
• Means relatively robust to non-normality for large samples, proportions less affected by extreme values
• Means apply to continuous data, proportions to categorical data (binary outcomes)