Lecture Notes: Sample Size and Margin of Error

Introduction

• Topic: Sample size in statistical studies.
• Approach: Using technology instead of algebra for calculations.

Importance of Statistics

• Statistics is pervasive in everyday life.
• Used in polls, advertising, healthcare, etc.
• Margin of error is a common term in statistical results.

Margin of Error

• Formula: Margin of Error = Z* × Standard Error (or Standard Deviation).
  • Z* is the cut-off point for confidence level.
  • Standard Error is used if P is not known.
• Margin of error changes inversely with sample size.
  • Larger sample size = smaller margin of error.

Determining Sample Size

• Conservative Approach: Use P-hat = 0.5 if unknown.
  • This yields the largest error, ensuring responsible estimates.
• Factors Influencing Margin of Error:
  • Sample size is the most adjustable factor.
  • Larger sample size = smaller margin of error, but increases cost.
  • Consider the context (e.g., life-critical vs. non-critical situations).

Calculating Sample Size

• Rearranging the Formula:
  • Formula for sample size: N = (P-hat(1-P-hat) × (Z*/E)^2).
  • E is the desired margin of error.
• Technology Tools:
  • Tools like the Dana Center's can compute required sample size with desired error margin without algebra.*

Examples and Applications

• Manager's Survey Example:

  • Estimating student purchases at a university bookstore.
  • Use P-hat = 0.5, E = 0.08, and a confidence level of 0.95.
  • Resulting sample size with technology: N = 151.

• Ecology Survey Example:

  • Estimating pesticide presence in bees.
  • Use P-hat = 0.5, E = 0.05, confidence level 0.95.
  • Resulting sample size: N = 385.

• Biotech Company Example:

  • Testing influenza test accuracy.
  • Use P = 0.97 (prior knowledge), E = 0.01, confidence level 0.95.
  • Resulting sample size: N = 1118.

• Auto Parts Example:

  • Testing airbag defects.
  • Use P = 0.02, E = 0.01, confidence level 0.99.
  • Resulting sample size: N = 1301.

Key Takeaways

• Adjusting sample size is the most practical way to control margin of error.
• Larger sample sizes are costly but reduce error significantly.
• Use technology to simplify sample size calculations.
• For unknown proportions, a P-hat of 0.5 is a safe default.
• Understanding the interplay between margin of error, confidence level, and sample size is crucial for accurate statistical analysis.