Margin of Error and Confidence Intervals

Margin of Error (E)

• Defined as the portion after the plus-minus in statistical formulas.
• In a normal distribution, the margin of error (E) is calculated as ( Z \times \frac{\sigma}{\sqrt{n}} ).
• In a t-distribution, it becomes ( T \times \frac{S}{\sqrt{n}} ).
• E represents error and is a key component in determining confidence intervals.

Impact of Margin of Error on Confidence Intervals

• Confidence Interval: Range between a lower and a higher value.
• Effect of larger Margin of Error:
  • Expands the confidence interval range, making cutoff values further apart.
  • Larger intervals are less precise, which is generally negative as precision is reduced.
• Preference for Smaller Range:
  • Smaller range is more precise, but precision may reduce confidence.

Confidence and Interval Size

• 98% vs. 95% Confidence Intervals:
  • Increasing confidence level increases the interval range.
  • Larger confidence levels lead to broader ranges with less precision.
• Trade-off:
  • Reducing interval size increases precision but decreases confidence.
  • Balance needed between precision and confidence level.

Visualizing Confidence Intervals

• Adjusting the cutoff values affects interval size and confidence.
• Example: A very precise mean with 0% confidence is not useful.

Factors Influencing Confidence Intervals

• Formula Components: Z, (\sigma), and n.
• Control over n:
  • Increasing the sample size (n) can adjust the interval size and margin of error.
• Sigma ((\sigma)): Cannot be changed as it is a property of the data.

Calculating Sample Size (n) for Desired Margin of Error

• Objective: Specify a desired margin of error (E) and solve for n.
• Formula Rearrangement:
  • From ( E = \frac{Z \times \sigma}{\sqrt{n}} ), rearrange to solve for n:
  • ( n = \left( \frac{Z \times \sigma}{E} \right)^2 )
• This formula helps determine the necessary sample size to achieve a specified margin of error while maintaining a given confidence level.

Conclusion

• Understanding the trade-off between precision and confidence is crucial in statistical analysis.
• The formula for n allows one to design studies that achieve desired error margins.