One common use of statistics is the testing of scientific hypotheses. First, the investigator forms a **research hypothesis** that states an expectation to be tested. Then the investigator derives a statement that is the opposite of the research hypothesis. This statement is called the **null hypothesis** (in notation: *H* _{0}). It is the null hypothesis that is actually tested, not the research hypothesis. If the null hypothesis can be rejected, that is taken as evidence in favor of the research hypothesis (also called the **alternative hypothesis,** *H *_{a} in notation). Because individual tests are rarely conclusive, it is usually not said that the research hypothesis has been “proved,” only that it has been supported.

An example of a research hypothesis comparing two groups might be the following:

Fourth‐graders in Elmwood School perform differently in math than fourth‐graders in Lancaster School. This could be measured by comparing the means of these groups.

Or in notation: *H *_{a} : μ _{1} ≠ μ _{2}

or sometimes: *H *_{a} : μ _{1} – μ _{2} ≠ 0

The null hypothesis would be:

Fourth‐graders in Elmwood School perform the same in math as fourth‐graders in Lancaster School.

In notation: *H* _{0}: μ _{1} = μ _{2}

or: *H* _{0}: μ _{1} – μ _{2} = 0

Some research hypotheses are more specific than that, predicting not only a difference but a difference in a particular direction. These are often described as one‐sided tests:

Fourth‐graders in Elmwood School are *better* in math than fourth‐graders in Lancaster School.

In notation: *H *_{a} : μ _{1} > μ _{2}

or: *H *_{a} : μ _{1} – μ _{2} > 0