In statistics, a hypothesis is a statement regarding a characteristic of one or more populations. Hypothesis testing is the procedure we use, relying on sample evidence and probability, to test these statements. The process begins by defining two opposing hypotheses. The null hypothesis, denoted as $H_0$, is a statement of no change, no effect, or no difference. It is assumed to be true until evidence indicates otherwise. Conversely, the alternative hypothesis, denoted as $H_1$, is the statement we are actively trying to find evidence to support.
Depending on the alternative hypothesis, tests are categorized into three types:
- Two-tailed test: Tests if a parameter is simply "not equal" to a value ($H_1$: parameter $\ne$ some value).
- Left-tailed test: Tests if a parameter is "less than" a value ($H_1$: parameter $<$ some value).
- Right-tailed test: Tests if a parameter is "greater than" a value ($H_1$: parameter $>$ some value).
Understanding the Margin of Error: Type I vs. Type II
Whenever you test a hypothesis, there are four possible outcomes, two of which are correct decisions and two of which are errors.
- A Type I error occurs if you reject the null hypothesis when it is actually true. The probability of making this error is called the level of significance, denoted by $\alpha$.
- A Type II error happens if you do not reject the null hypothesis when the alternative hypothesis is true.
Interestingly, these errors are inversely related; as the probability of making a Type I error increases, the probability of a Type II error decreases, and vice versa.
Stating Your Conclusion
A crucial rule in statistics is that we never say we "accept" the null hypothesis. Because we do not have access to the entire population, we cannot be absolutely certain of the exact parameter value stated in the null hypothesis. Instead, we simply state that we "do not reject" the null hypothesis. It is very similar to the legal system, where a defendant is declared "not guilty" rather than definitively "innocent".