Confidence Intervals and Hypothesis Testing for MAS-I

Confidence Intervals for Means

A confidence interval provides a range of plausible values for an unknown parameter based on observed data. For a normal population mean with known variance sigma^2, the (1-alpha)% confidence interval is x-bar +/- z_{alpha/2} * sigma/sqrt(n), where z_{alpha/2} is the upper alpha/2 quantile of the standard normal distribution. For a 95% confidence interval, z_{0.025} = 1.96. The width of the interval decreases as n increases, reflecting the increased precision from larger samples.

When the population variance is unknown (the more common situation), replace sigma with the sample standard deviation s and use the t-distribution: x-bar +/- t_{alpha/2, n-1} * s/sqrt(n). The t-distribution has heavier tails than the normal, producing wider intervals that account for the additional uncertainty in estimating sigma. As n grows large, the t-distribution converges to the normal, and the two intervals become essentially identical for n > 30 or so.

Confidence Intervals for Proportions and Variances

For a population proportion p estimated by p-hat from n Bernoulli trials, the approximate (1-alpha)% confidence interval is p-hat +/- z_{alpha/2} * sqrt(p-hat * (1 - p-hat) / n). This approximation relies on the CLT and works well when both np-hat and n(1-p-hat) are at least 10. For small samples or extreme proportions, exact methods based on the Binomial distribution are preferred.

For the variance of a normal population, the (1-alpha)% confidence interval is ((n-1)*s^2 / chi^2_{alpha/2, n-1}, (n-1)*s^2 / chi^2_{1-alpha/2, n-1}), where chi^2 denotes quantiles of the chi-squared distribution with n-1 degrees of freedom. Note the reversal: the larger chi-squared quantile appears in the denominator of the lower bound. This interval is not symmetric around s^2 because the chi-squared distribution is skewed.

Interpreting Confidence Intervals

The correct interpretation of a 95% confidence interval is that the procedure, if repeated many times, would produce intervals containing the true parameter 95% of the time. The parameter is fixed; the interval is random. Once a specific interval is calculated, the parameter either is or is not in that interval, and we do not know which. Saying "there is a 95% probability that the parameter is in this interval" is technically incorrect in the frequentist framework, though it captures the practical spirit.

The width of a confidence interval reflects precision. A narrower interval indicates more precise estimation. Width depends on three factors: the sample size n (larger n gives narrower intervals), the variability in the data (more variable data gives wider intervals), and the confidence level (higher confidence gives wider intervals). These trade-offs are important in sample size determination for actuarial studies.

Hypothesis Testing Framework

Hypothesis testing evaluates evidence against a null hypothesis H0 in favor of an alternative hypothesis H1. The null hypothesis is the "status quo" or "no effect" claim, while the alternative represents the claim you are trying to establish. Tests can be one-sided (H1: theta > theta_0 or H1: theta < theta_0) or two-sided (H1: theta != theta_0).

The procedure involves computing a test statistic from the data and comparing it to a critical value or computing a p-value. The test statistic measures how far the observed data is from what would be expected under H0. For testing a normal mean with known variance, the z-test statistic is Z = (x-bar - mu_0) / (sigma/sqrt(n)). For unknown variance, the t-test statistic is T = (x-bar - mu_0) / (s/sqrt(n)), which follows a t-distribution with n-1 degrees of freedom under H0.

Type I Error, Type II Error, and Power

Type I error (alpha) is the probability of rejecting H0 when it is true: a false positive. The significance level of the test is the maximum Type I error probability, chosen before looking at the data. Common choices are alpha = 0.05 and alpha = 0.01. Type II error (beta) is the probability of failing to reject H0 when it is false: a false negative. The power of a test is 1 - beta, the probability of correctly rejecting a false H0.

Power depends on the true parameter value (larger departures from H0 are easier to detect), the sample size (larger n increases power), the significance level (larger alpha increases power but also increases Type I error), and the variability in the data. Power analysis is important for planning actuarial studies: you need enough data to have a reasonable chance of detecting effects that matter practically.

P-Values

The p-value is the probability of observing a test statistic as extreme as (or more extreme than) the observed value, assuming H0 is true. A small p-value indicates strong evidence against H0. The decision rule using p-values is: reject H0 if the p-value is less than alpha. This is equivalent to comparing the test statistic to the critical value but provides more information because it quantifies the strength of evidence.

For a two-sided z-test with observed statistic z = 2.15, the p-value is 2 * P(Z > 2.15) = 2 * 0.0158 = 0.0316. Since 0.0316 < 0.05, we reject H0 at the 5% level but not at the 1% level. The p-value gives a continuous measure of evidence rather than a binary accept/reject decision.

Common Pitfalls

A frequent mistake is confusing the confidence level with the probability that the parameter is in a specific computed interval. Another common error involves one-sided versus two-sided tests: the p-value for a one-sided test is half the two-sided p-value (when the observed statistic is in the direction of the alternative). Students also sometimes state conclusions incorrectly. When you fail to reject H0, you do not "accept" H0; you simply lack sufficient evidence to reject it.

In hypothesis testing, mixing up the direction of the inequality in the alternative hypothesis can lead to incorrect p-value calculations. Always verify that the rejection region matches the alternative hypothesis before computing the p-value.

Study Tips

Exam MAS-I requires fluency in constructing confidence intervals and conducting hypothesis tests. Practice the complete workflow: state hypotheses, compute the test statistic, find the p-value or critical value, make a decision, and state the conclusion in context. Know the formulas for z-tests, t-tests, chi-squared tests for variance, and F-tests for comparing variances. Be prepared to compute sample sizes needed to achieve specified confidence interval widths or power levels. Understand the duality between confidence intervals and hypothesis tests: a (1-alpha)% confidence interval contains all parameter values that would not be rejected at significance level alpha.