Stating Hypotheses

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: H₀). 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.

  Or in notation: Ha:μ1 ≠ μ2 or sometimes: Ha:μ1 − μ2 ≠ 0

The null hypothesis would be:

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

  In notation: H0:μ1 = μ2 or alternatively: H0:μ1 − μ2 = 0

Some research hypotheses are more specific than that, predicting not only a difference but a difference in a particular direction:

• Fourth-graders in Elmwood School are better in math than fourth-graders in Lancaster School.

  In notation: Ha:μ1 > μ2 or alternatively: Ha:μ1 − μ2 > 0

The accompanying null hypothesis must be equally specific so that all possibilities are covered:

• Fourth-graders in Elmwood School are worse in math, or equal to, fourth-graders in Lancaster School.

  In notation: H0:μ1 ≤ μ2 or alternatively: H0:μ1 − μ2 ≤ 0

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