All hypothesis tests share the same basic terminology and structure.
A null hypothesis is an assertion about a population that you would like to test. It is “null” in the sense that it often represents a status quo belief, such as the absence of a characteristic or the lack of an effect. It may be formalized by asserting that a population parameter, or a combination of population parameters, has a certain value. In the example given in the Introduction to Hypothesis Tests, the null hypothesis would be that the average price of gas across the state was $1.15. This is written H0: µ = 1.15.
An alternative hypothesis is a contrasting assertion about the population that can be tested against the null hypothesis. In the example given in the Introduction to Hypothesis Tests, possible alternative hypotheses are:
H1: µ ≠ 1.15 — State average was different from $1.15 (two-tailed test)
H1: µ > 1.15 — State average was greater than $1.15 (right-tail test)
H1: µ < 1.15 — State average was less than $1.15 (left-tail test)
To conduct a hypothesis test, a random sample from the population is collected and a relevant test statistic is computed to summarize the sample. This statistic varies with the type of test, but its distribution under the null hypothesis must be known (or assumed).
The significance level of a test is a threshold of probability α agreed to before the test is conducted. A typical value of α is 0.05. If the p value of a test is less than α, the test rejects the null hypothesis. If the p value is greater than α, there is insufficient evidence to reject the null hypothesis. Note that lack of evidence for rejecting the null hypothesis is not evidence for accepting the null hypothesis. Also note that substantive “significance” of an alternative cannot be inferred from the statistical significance of a test.
The significance level α can be interpreted as the probability of rejecting the null hypothesis when it is actually true—a type I error. The distribution of the test statistic under the null hypothesis determines the probability α of a type I error. Even if the null hypothesis is not rejected, it may still be false—a type II error. The distribution of the test statistic under the alternative hypothesis determines the probability β of a type II error. Type II errors are often due to small sample sizes. The power of a test, 1 – β, is the probability of correctly rejecting a false null hypothesis.
Results of hypothesis tests are often communicated with a confidence interval. A confidence interval is an estimated range of values with a specified probability of containing the true population value of a parameter. Upper and lower bounds for confidence intervals are computed from the sample estimate of the parameter and the known (or assumed) sampling distribution of the estimator. A typical assumption is that estimates will be normally distributed with repeated sampling (as dictated by the Central Limit Theorem). Wider confidence intervals correspond to poor estimates (smaller samples); narrow intervals correspond to better estimates (larger samples). If the null hypothesis asserts the value of a population parameter, the test rejects the null hypothesis when the hypothesized value lies outside the computed confidence interval for the parameter.