Confidence Intervals for Proportions

Overview

This lecture explains how to calculate one-population proportion confidence intervals, including the relevant formulas, example calculation, and the required assumptions for accuracy.

Confidence Interval Basics

• A confidence interval estimates a population parameter by giving a range (two numbers) believed to contain the true value.
• The general form is: sample statistic ± margin of error.
• The margin of error represents potential difference between the sample statistic and the population parameter.

Calculating the Margin of Error

• Historically, the empirical rule used ±2 standard deviations for a 95% confidence interval.
• This evolved to using the z-score for different confidence levels: margin of error = z-score × standard error.
• Typical z-scores: 90% = 1.645, 95% = 1.96, 99% = 2.576.

Standard Error for Proportions

• For proportions, standard error = sqrt [P̂ × (1 − P̂)/n], where P̂ is the sample proportion and n is the sample size.

Example: Calculating a Confidence Interval

• Sample: 108 students, 4 smoked (P̂ = 0.037).
• Standard error: sqrt [0.037 × (1 − 0.037) / 108] ≈ 0.018.
• Margin of error for 90% confidence: 1.645 × 0.018 ≈ 0.030.
• Confidence interval: 0.037 ± 0.030, giving (0.007, 0.067) or 0.7% to 6.7%.

Interpreting and Assumptions

• The computed interval means 90% confidence that the true proportion is within 0.7% and 6.7%.
• Formula accuracy depends on:
  • Random/representative sampling
  • Independent individuals
  • At least 10 successes and 10 failures in the sample (normality assumption)
• In the example, the "at least 10 successes" condition was not met, so the formula's accuracy is questionable.

Key Terms & Definitions

• Confidence Interval — Range believed to contain the population parameter, based on sample data.
• Margin of Error — Maximum expected difference between sample statistic and true parameter.
• Sample Proportion (P̂) — Number of successes divided by total sample size.
• Standard Error — Estimate of the standard deviation of the sampling distribution.
• Z-score (Critical Value) — Number of standard deviations from the mean for a given confidence level.

Action Items / Next Steps

• Review how confidence intervals change with different confidence levels and sample sizes.
• Practice identifying when the assumptions for a confidence interval are satisfied.
• Use technology for calculations and focus on interpreting confidence intervals.