Lecture Notes on Confidence Intervals (Part 2)

Introduction to Confidence Intervals

• Presenter: Dennis Davis
• Focus: Inferences about populations from samples (Inferential Statistics)
  • Inferential Statistics: Drawing conclusions based on observed evidence and prior knowledge (e.g., smoke inferring fire).
  • Sample: Observed evidence to infer about populations using statistical knowledge.

Types of Inferences

• Confidence Intervals: This video
• Hypothesis Testing: Next video

Key Terminology and Concepts

• Point Estimate: Single value estimate of a target (population parameter)
• Interval Estimate: Range of values likely to include the target
• Confidence Interval: Interval estimate with an associated confidence level (probability)
• Confidence Level: Probability that a confidence interval includes the population parameter
• Margin of Error: Product of critical value and standard error
• Critical Value: The z-score corresponding to the confidence level
• Standard Error: Standard deviation of the sampling distribution
• Sigma (σ) Known/Unknown
• T-distribution and Confidence Intervals for Proportions

Creating Confidence Intervals

1. Point Estimate ± Margin of Error
   • Point Estimate = Sample Statistic (e.g., sample mean (\bar{x}))
   • Margin of Error = Critical Value * Standard Error of the Statistic
   • Critical Value: Typically a z-score when sigma is known
   • Standard Error: Variability or natural random variation from the mean*

Example: Golf Thought Experiment

• Point Estimate: Shot landing at 124.73 yards
• Margin of Error: 5.28 yards
• Confidence Interval: 119.45 to 130.01 yards

Formula for Confidence Intervals

• (\bar{x} ± Z_{critical} * \sigma_{\bar{x}})
• Critical Z Value corresponds to the confidence level
• Example Calculation
  • Sample Mean: 382.9 minutes
  • Standard Error: 6.1 minutes
  • Confidence Interval: 370.9 to 394.9 minutes*

Interpreting Confidence Levels

• Confidence Level of 95%: Interval includes the true mean 95% of the time
• Adjusting Confidence Levels: Changes width of the interval
  • Higher confidence = Wider interval
  • Lower confidence = Narrower interval

Applications and Calculations

• Variance and Degrees of Freedom (DF): DF = Sample size - 1
• T-distribution for unknown population standard deviation (σ)
  • T-critical Values: Larger than Z due to additional uncertainty

Proportions

• Population Proportion (p): Number of successes / Population size
• Sample Proportion (p̂): Number of successes in the sample / Sample size
• Standard Error for Proportion: (\sqrt{\frac{p̂(1-p̂)}{n}})

Example Problems

1. Calculate 90%, 95%, 98%, 99% confidence intervals
2. Contrived problem to back into population standard deviation (σ)
3. Confidence intervals with different sample sizes and confidence levels
4. Applications in political polling and statistical research

Conclusion

• Confidence intervals are crucial for statistical inference, helping to estimate population parameters based on sample data.
• They require understanding of point estimates, interval estimates, critical values, and standard errors.
• Proper interpretation and calculation of confidence intervals are foundational for effective data analysis and decision-making in statistics.

Next Steps: Hypothesis testing in the subsequent lecture.