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Inferential Statistics: Confidence Intervals and t-Tests

Confidence Intervals

A confidence interval (CI) provides a range of plausible values for a population parameter, based on sample data. A 95% CI is constructed so that if we repeated the sampling process many times, approximately 95% of those intervals would contain the true population parameter. The formula for a 95% CI for the population mean μ when σ is known: CI = x̄ ± z*(σ/√n), where z* = 1.96 for 95% confidence and n is sample size. When σ is unknown (the usual case), we use the t-distribution: CI = x̄ ± t*(s/√n), where t* is the critical t-value with n−1 degrees of freedom. Important interpretation: a 95% CI of (47.2, 52.8) does NOT mean 'there is a 95% probability the true mean is in this interval.' The true mean is fixed — either it's in this specific interval or it isn't. The 95% refers to the process: 95% of intervals constructed this way will capture the true mean. Wider CIs indicate less precision (smaller n, larger variance). Narrower CIs indicate more precision. Increasing n by a factor of 4 halves the CI width (because √n appears in the denominator).

Inferential Statistics: Confidence Intervals and t-Tests

Confidence Intervals

A confidence interval (CI) provides a range of plausible values for a population parameter, based on sample data. A 95% CI is constructed so that if we repeated the sampling process many times, approximately 95% of those intervals would contain the true population parameter. The formula for a 95% CI for the population mean μ when σ is known: CI = x̄ ± z*(σ/√n), where z* = 1.96 for 95% confidence and n is sample size. When σ is unknown (the usual case), we use the t-distribution: CI = x̄ ± t*(s/√n), where t* is the critical t-value with n−1 degrees of freedom. Important interpretation: a 95% CI of (47.2, 52.8) does NOT mean 'there is a 95% probability the true mean is in this interval.' The true mean is fixed — either it's in this specific interval or it isn't. The 95% refers to the process: 95% of intervals constructed this way will capture the true mean. Wider CIs indicate less precision (smaller n, larger variance). Narrower CIs indicate more precision. Increasing n by a factor of 4 halves the CI width (because √n appears in the denominator).