Module 17 Wrap-up: Estimating a Population Proportion

Key Concepts

• Sample Proportion (P̂): A point estimate for the population proportion (P).
• Population Proportion (P): The true proportion of a characteristic within a population.
• Sample Variability: Sample proportions vary, requiring interval estimates.

Confidence Interval

• Purpose: Provides a range of values to estimate the population proportion.
• Conditions for Use:
  • Sample must be randomly selected.
  • At least 10 observed successes and 10 observed failures are in the sample.

Interpreting Confidence Intervals

• Confidence Level: Determines how confident we are that the interval includes the population proportion.
  • Example: 95% confidence level means we are 95% confident the true proportion is within the interval.
  • Long Run Interpretation: 95% of random samples will generate intervals containing the population proportion; 5% will not.

Confidence Interval Formula

• Formula:
  • [ P̂ \pm \text{Margin of Error} ]
• Margin of Error: Related to the confidence level:
  • For 95% confidence level, margin of error is approximately two standard errors.
  • Z-score Related to Confidence Level:
    • 95% Confidence: 1.96
    • 90% Confidence: 1.645
    • 99% Confidence: 2.58

Factors Affecting Confidence Interval

• Confidence Levels and Sample Sizes:
  • Lower confidence levels and higher sample sizes lead to narrower intervals.
  • Narrower intervals mean smaller errors.
  • Larger sample sizes reduce error and create more precise intervals.

Conclusion

• Desirable Outcome: High levels of confidence and larger sample sizes help ensure intervals accurately estimate the population proportion.
• Preferred Approach: Using larger sample sizes to achieve narrower confidence intervals while maintaining high confidence levels.