C

Key Concepts of Confidence Intervals

Confidence Intervals Overview

  • Sampling theory is key for estimating population parameters from sample data, resulting in uncertainty in estimates.
  • Confidence intervals (CIs) quantify this uncertainty by stating a range within which a parameter is expected to lie with a certain level of confidence (e.g., 95%).

Constructing Confidence Intervals

  • For a population mean \mu with standard deviation \sigma:
    • Sample mean is \bar{X} from N participants.
    • According to the Central Limit Theorem, the sampling distribution of the mean is approximately normal.
    • 95% confidence interval can be constructed as:
      (\bar{X} - 1.96 \times SEM, \bar{X} + 1.96 \times SEM)
    • Standard Error of the Mean (SEM) is defined as: SEM = \frac{\sigma}{\sqrt{N}}

Adjusting for Sample Standard Deviation

  • Often, the true population standard deviation \sigma is not known, requiring the use of the sample standard deviation s instead.
  • This necessitates the use of the t-distribution, especially with small sample sizes, which leads to a larger multiplier (e.g., t_{N-1}).

Sample Size Impact

  • As sample size N increases, confidence intervals tend to be narrower; with small N, confidence intervals are wider due to increased uncertainty about estimates.
    • Example for N = 10: Multiplier = 2.26 (wider CI) vs. N = 300: Multiplier ≈ 1.96.

Interpreting Confidence Intervals

  • A CI communicates that there is a specified probability (e.g., 95%) that the true population mean falls within the interval derived from sample data.
  • For instance, a sample mean \bar{X} = 100.14 with a confidence interval (98.85, 100.43) means:
    • We are 95% confident the true mean lies within this range.
    • Reflects the consistency across different potential samples drawn from the population.

Long-Run Interpretation

  • Confidence intervals are defined within the context of repeated sampling:
    • Over many samples, 95% of CIs constructed should contain the true population mean.
    • Each individual CI either contains or does not contain the true mean.
  • This principle is visually illustrated with sample means and CIs in empirical data, reinforcing the concept of repeatability and uncertainty in estimates.