7.4 Sample Size Determination

Determining the right sample size is crucial for accurate statistical estimates. It's all about balancing precision with practicality. Too small, and your results might be unreliable. Too large, and you're wasting resources.

For means and proportions, specific formulas help calculate ideal sample sizes. These consider factors like desired confidence level, margin of error, and population variability. Remember, larger samples generally mean more precise estimates, but there's always a trade-off with time and cost.

Sample Size Determination for Confidence Intervals

Sample size for population mean

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• Calculate sample size for estimating population mean using formula:
  • $n = (\frac{z_{\alpha/2}\sigma}{E})^2$
    • $n$ represents sample size
    • $z_{\alpha/2}$ is critical value from standard normal distribution based on confidence level (1.96 for 95% confidence)
    • $\sigma$ denotes population standard deviation
    • $E$ signifies desired margin of error (±0.05 for ±5%)
• Substitute sample standard deviation $s$ for $\sigma$ if population standard deviation unknown
• Increasing sample size narrows confidence intervals and improves precision of estimates
• Maintaining same margin of error at higher confidence levels necessitates larger sample sizes (99% confidence requires ~2.5 times sample size of 95% confidence)

Sample size for population proportion

• Determine sample size for estimating population proportion with formula:
  • $n = (\frac{z_{\alpha/2}}{E})^2 \hat{p}(1-\hat{p})$
    • $n$ denotes sample size
    • $z_{\alpha/2}$ represents critical value from standard normal distribution based on confidence level (1.96 for 95% confidence)
    • $E$ is desired margin of error (±0.03 for ±3%)
    • $\hat{p}$ signifies estimated population proportion
• Use $\hat{p} = 0.5$ for most conservative (largest) sample size when no prior information about population proportion available
• Larger sample sizes yield narrower confidence intervals and more precise estimates
• Higher confidence levels demand larger sample sizes to preserve same margin of error (99% confidence requires ~1.7 times sample size of 95% confidence for proportions)

Relationships in interval estimation

• Sample size, margin of error, and confidence level interconnected in interval estimation
  • Increasing sample size decreases margin of error at constant confidence level
  • Raising confidence level increases margin of error at fixed sample size
  • Preserving desired margin of error at higher confidence levels requires larger sample sizes
• Researchers must consider trade-offs between sample size, margin of error, and confidence level
  • Balance precision and reliability of estimates with available resources (time, budget)
  • Prioritize precision for critical estimates, allow wider margins for less crucial parameters

Sample size in business applications

1. Identify research question and parameter of interest (mean sales, proportion of defective items)
2. Specify desired margin of error and confidence level based on required precision and reliability (±2% at 95% confidence for critical metrics, ±5% at 90% confidence for exploratory research)
3. Estimate population standard deviation or proportion using prior data or pilot study (historical sales data, industry benchmarks)
4. Calculate required sample size using appropriate formula for mean or proportion
5. Adjust sample size for practical considerations:
   • Expected response rate (anticipate 30% response for customer surveys)
   • Stratification or clustering in sampling design (oversample underrepresented segments)
   • Finite population correction for small populations (adjust formula when sampling >5% of population)
6. Collect data from random sample of determined size and construct confidence interval for parameter (95% confidence interval for mean customer satisfaction score)