Testing Hypotheses
A hypothesis test involves testing a claim, or null hypothesis $$H_0$$, about a parameter against an alternative, $$H_1$$. A decision to reject $$H_0$$, or not reject $$H_0$$, uses sample evidence to calculate a test statistic which is judged against a critical value. $$H_0$$ is maintained unless it is made untenable by sample evidence. Rejecting $$H_0$$ when we should not is a Type I error.
The probability (we are prepared to accept) of making a Type I error is called the significance level, $$\alpha$$, and yields the critical value. The tail probability that corresponds to the observed sample data is the $$p$$-value of the test. If the $$p$$-value is smaller than $$\alpha$$ the value of the test statistic is said to be statistically significant.
Not rejecting $$H_0$$ when we should is a Type II error, with probability $$\beta$$ . The power of a hypothesis test is $$1-\beta$$.