ECO1007 - Estimation Single Population Confidence Interval for μ (σ² Known) Assumptions: Population variance σ² is known Population is normally distributed or population is not normally distributed, we have large sample, then x̄ ~ N(μ, σ²/n) Confidence interval estimate: x̄ - zα/2 * (σ / sqrt(n)) < μ < x̄ + zα/2 * (σ / sqrt(n)) Margin of Error: The confidence interval can also be written as x̄ ± ME where ME is called the margin of error ME = zα/2 * (σ / sqrt(n)) The interval width, w, is equal to twice the margin of error Margin of error can be reduced if population standard deviation can be reduced (↓) Sample size is increased (↑↑) confidence level is decreased (1-α ↓) Confidence Interval for μ (σ² Unknown) Confidence interval when σ is known were constructed using the fact that the random variable ((x̄ - μ) / (σ / sqrt(n))) has a standard normal distribution. If the population standard deviation σ is unknown, we can substitute the sample standard deviation s. This introduces extra uncertainty, since s varies from sample to sample Student's t Distribution Consider a random sample of n observations with mean x̄ and standard deviation s from a normally distributed population with mean μ Then the variable t = (x̄ - μ) / (s / sqrt(n)) follows the Student's t distribution with (n-1) degrees of freedom. The t value depends on degrees of freedom (d.f.). Number of observations that are free to vary after sample mean has been calculated. d.f. =