Hypothesis Testing

Study guides: Exam VEE-MS, Exam MAS-I

Hypothesis Testing is a statistical procedure for deciding between two competing claims about a population parameter: the null hypothesis $H_0$ and the alternative hypothesis $H_1$ (or $H_a$ ).

Decision Rule: Reject $H_0$ if the test statistic falls in the rejection region (critical region); otherwise, fail to reject $H_0$ .

$\text{p-value} = P(\text{test statistic as extreme or more extreme} \mid H_0 \text{ true})$

Types of hypotheses:

Simple $H_0: \theta = \theta_0$ vs. one-sided $H_1: \theta > \theta_0$ or $H_1: \theta < \theta_0$
Two-sided $H_1: \theta \neq \theta_0$

Decision errors:

	$H_0$ True	$H_0$ False
Reject $H_0$	Type I Error ( $\alpha$ )	Correct (Power $= 1-\beta$ )
Fail to Reject $H_0$	Correct	Type II Error ( $\beta$ )

The significance level $\alpha = P(\text{Type I Error})$ is set in advance (e.g., 0.05)
Power $= 1 - \beta = P(\text{Reject } H_0 \mid H_1 \text{ true})$
Reject $H_0$ when p-value $< \alpha$

One-Sample Z-Test for a Mean A claims adjuster believes average claim size is $ $5{,}000$ . A sample of $n = 100$ claims has $\bar{x} = 5{,}200$ and $\sigma = 800$ . Test $H_0: \mu = 5{,}000$ vs. $H_1: \mu > 5{,}000$ at $\alpha = 0.05$ .

Answer $Z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}} = \frac{5{,}200 - 5{,}000}{800 / \sqrt{100}} = \frac{200}{80} = 2.50$ The critical value for a one-sided test at $\alpha = 0.05$ is $z_{0.05} = 1.645$ . Since $2.50 > 1.645$ , reject $H_0$ . There is sufficient evidence that the mean claim size exceeds $ $5{,}000$ .