Test statistical hypotheses, including Type I and Type II errors.

A pricing actuary suspects a new underwriting rule has cut the claim rate below the historical 8%. The data could lie either way. Hypothesis testing is the framework that turns "I think rates dropped" into a defensible decision with calibrated error rates.

Setting up the hypotheses. You frame the question as two competing claims about a parameter $\theta$. The null hypothesis $H_0$ is the baseline (often equality, e.g., $\theta=\theta_0$). The alternative $H_1$ is what you would conclude with sufficient evidence. The alternative can be one-sided ($\theta>\theta_0$ or $\theta<\theta_0$) or two-sided ($\theta\ne\theta_0$).

KEY: Always state $H_0$ and $H_1$ before looking at the data. Choosing the alternative after seeing the sample inflates the true Type I rate.

The two error types. Decisions split four ways. Reject $H_0$ when it is true: Type I error, probability $\alpha$. Fail to reject $H_0$ when it is false: Type II error, probability $\beta$. The probability of correctly rejecting a false $H_0$ is the power, $1-\beta$.

Significance level. You pick $\alpha$ (commonly 0.10, 0.05, 0.01) before testing. It is the maximum Type I rate you tolerate.

Common mistakes

• Treating "fail to reject" as "accept $H_0$." The data simply did not muster enough evidence; the null could still be wrong. Phrase conclusions accordingly.
• Using $\alpha/2$ for a one-sided test. One-sided alternatives put the full $\alpha$ in a single tail. Splitting it doubles the Type I rate or doubles the required evidence by mistake.
• Confusing p-value with $P(H_0 \text{ true})$. The p-value is computed assuming $H_0$. It says nothing about the prior probability that $H_0$ holds.

Bottom line

• $H_0$ is the status-quo claim you can reject; $H_1$ is what you adopt if evidence is strong enough. You never accept $H_0$, you only fail to reject it.
• Type I error ($\alpha$) = rejecting a true $H_0$; Type II error ($\beta$) = failing to reject a false $H_0$. Power equals $1-\beta$.
• Reject when the test statistic falls in the rejection region, or equivalently when the p-value is at or below $\alpha$.
• Lowering $\alpha$ raises $\beta$ for a fixed sample size. Increasing $n$ is the only way to shrink both at once.

Exam shortcut

When the problem asks for the p-value of a two-sided test, compute the one-tail probability and double it; for a one-sided test, do not double. When a question asks "what is the probability of a Type I error," the answer is $\alpha$ by construction, not anything derived from the data. For sample-size and power problems, memorize the formula $n = \sigma^2 (z_{1-\alpha} + z_{1-\beta})^2 / (\mu_0 - \mu_1)^2$; it collapses both error rates into one clean expression.

The full lesson (about 2,554 words, 17 min read) adds 8 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• B3

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