Interval estimation and confidence intervals are key tools for understanding population parameters based on sample data. They provide a range of likely values for unknown parameters, helping researchers make informed decisions despite uncertainty.

Confidence intervals balance precision and reliability, with wider intervals offering more confidence but less specificity. Sample size, confidence level, and population variability all affect interval width, influencing the conclusions we can draw from our data.

Confidence Intervals for Parameters

Constructing Confidence Intervals

• Confidence intervals are ranges of values derived from sample statistics that likely contain the value of an unknown population parameter
• The general form of a confidence interval is point estimate ± margin of error
  • The margin of error is calculated as the critical value (z or t) multiplied by the standard error
• To construct a confidence interval for a population mean (μ) when the population standard deviation (σ) is known, use a z-interval: $X̄ ± z(σ/√n)$
  • $X̄$ is the sample mean
  • $z$ is the critical value from the standard normal distribution
  • $σ$ is the population standard deviation
  • $n$ is the sample size
• To construct a confidence interval for a population mean (μ) when the population standard deviation (σ) is unknown, use a t-interval: $X̄ ± t(s/√n)$
  • $X̄$ is the sample mean
  • $t$ is the critical value from the Student's t-distribution with n-1 degrees of freedom
  • $s$ is the sample standard deviation
  • $n$ is the sample size
• To construct a confidence interval for a population proportion (p), use the formula: $p̂ ± z√(p̂(1-p̂)/n)$
  • $p̂$ is the sample proportion
  • $z$ is the critical value from the standard normal distribution
  • $n$ is the sample size

Assumptions and Requirements

• Z-intervals assume that the population standard deviation (σ) is known
• T-intervals are used when σ is unknown and must be estimated from the sample data using the sample standard deviation (s)
• Confidence intervals for proportions use the standard normal (z) distribution to determine the critical value
  • The sampling distribution of proportions is approximately normal for large sample sizes (np ≥ 10 and n(1-p) ≥ 10)
• The width of a confidence interval for a proportion depends on the sample size (n) and the sample proportion (p̂)
  • Intervals are widest when p̂ is close to 0.5 and narrowest when p̂ is close to 0 or 1

Interpreting Confidence Intervals

Understanding Confidence Intervals

• A confidence interval provides a range of plausible values for the population parameter based on the sample data and the chosen confidence level
• The confidence level (95%, 99%, etc.) represents the probability that the method used to construct the interval will produce an interval containing the true population parameter if the process is repeated many times
  • A 95% confidence interval does not mean that there is a 95% probability that the true parameter lies within the interval
  • The true parameter is either inside the interval or not; the confidence level refers to the reliability of the method
• Narrower confidence intervals provide more precise estimates of the population parameter, while wider intervals indicate greater uncertainty

Comparing Confidence Intervals

• If two confidence intervals overlap, it suggests that there may not be a significant difference between the two population parameters
• Non-overlapping intervals suggest a significant difference
• Overlapping confidence intervals do not necessarily imply that there is no significant difference between the parameters
  • Formal hypothesis testing should be used to make definitive conclusions about the difference between parameters

Sample Size, Confidence Level, and Interval Width

Sample Size and Interval Width

• As the sample size (n) increases, the width of the confidence interval decreases, providing a more precise estimate of the population parameter
  • A larger sample size reduces the standard error
• The relationship between sample size and interval width is inversely proportional
  • Doubling the sample size reduces the interval width by a factor of $√2$
  • Halving the sample size increases the interval width by a factor of $√2$

Confidence Level and Interval Width

• As the confidence level increases (from 90% to 95% to 99%, etc.), the width of the confidence interval increases
  • Higher confidence levels require a larger margin of error to ensure that the interval captures the true parameter more often
• The relationship between confidence level and interval width is directly proportional
  • Increasing the confidence level widens the interval
  • Decreasing the confidence level narrows the interval

Confidence Interval Properties: Comparison

Z-intervals vs. T-intervals

• Z-intervals and t-intervals are both used to estimate population means but differ in their assumptions and the distribution used to determine the critical value
• T-intervals are generally wider than z-intervals for the same sample size and confidence level
  • The t-distribution has heavier tails than the standard normal distribution, accounting for the additional uncertainty in estimating the population standard deviation

Confidence Intervals for Proportions

• Confidence intervals for proportions use the standard normal (z) distribution to determine the critical value
  • The sampling distribution of proportions is approximately normal for large sample sizes (np ≥ 10 and n(1-p) ≥ 10)
• The width of a confidence interval for a proportion depends on the sample size (n) and the sample proportion (p̂)
  • Intervals are widest when p̂ is close to 0.5 (maximum variability)
  • Intervals are narrowest when p̂ is close to 0 or 1 (minimum variability)