Confidence Intervals and Hypothesis Testing for MAS-I

Confidence Intervals

A (1-alpha)% confidence interval provides a range of plausible values for a parameter. For a normal mean with known variance: x-bar +/- z_{alpha/2} * sigma/sqrt(n). With unknown variance, use the t-distribution: x-bar +/- t_{alpha/2,n-1} * s/sqrt(n). For proportions: p-hat +/- z_{alpha/2} * sqrt(p-hat*(1-p-hat)/n). For variance: (n-1)s^2/chi^2_{alpha/2} to (n-1)s^2/chi^2_{1-alpha/2}. Interpret correctly: the interval is random, the parameter is fixed, and 95% confidence means 95% of such intervals contain the true parameter.

Hypothesis Testing Framework

Hypothesis tests evaluate evidence against a null hypothesis H0. The test statistic, computed from data, determines whether to reject H0. Type I error (alpha) is rejecting a true H0. Type II error (beta) is failing to reject a false H0. Power = 1 minus beta. The p-value is the probability of observing results as extreme as the data under H0. Common tests include z-tests, t-tests, chi-squared tests, and F-tests. Exam MAS-I requires setting up hypotheses, computing test statistics, finding p-values, and interpreting results in actuarial contexts.