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5.2.4 Determining the Sample Size

Introduction

  • Objective: This section discusses how to calculate the necessary sample size to achieve a desired margin of error in confidence intervals.
  • Importance: Choosing the right sample size is crucial for ensuring the reliability and accuracy of the confidence intervals derived from sample data.

Sample Size Determination

  • Theoretical Basis: The sample size affects the margin of error and the confidence interval width. Larger samples lead to narrower confidence intervals, providing more precise estimates of the population parameters.

Methodology

  • Known Sigma (σ):
  • When σ is known, the sample size \( n \) can be determined using the formula: [ n = \left(\frac{z_{\alpha/2} \sigma}{E}\right)^2 ] where \( E \) is the desired margin of error, \( \sigma \) is the population standard deviation, and \( z_{\alpha/2} \) is the critical value from the normal distribution corresponding to the desired confidence level.
  • Unknown Sigma:
  • In cases where σ is not known, an estimate of σ can be obtained from previous studies or a pilot study.
  • Alternatively, a rough estimate can be calculated using the range of the data divided by 4 as a surrogate for σ.

Conclusion

  • Implications: Properly determining the sample size before data collection is essential for conducting robust statistical analysis.
  • Future Directions: The methods discussed provide a framework for researchers and business practitioners to design studies that accurately reflect the characteristics of the population, supporting sound decision-making based on statistical data.
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