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.