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Hypothesis Testing of Single Mean and Single Proportion: Null and Alternate Hypotheses

Module by: Susan Dean, Barbara Illowsky, Ph.D.

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

The actual test begins by considering two hypotheses. They are called the null hypothesis and the alternate hypothesis. These hypotheses contain opposing viewpoints.

H o H o : The null hypothesis: It is a statement about the population that will be assumed to be true unless it can be shown to be incorrect beyond a reasonable doubt.

H a H a : The alternate hypothesis: It is a claim about the population that is contradictory to HoHo and what we conclude when we reject HoHo.

Example 1

HoHo: No more than 30% of the registered voters in Santa Clara County voted in the primary election.

H a H a : More than 30% of the registered voters in Santa Clara County voted in the primary election.

Example 2

We want to test whether the average grade point average in American colleges is 2.0 (out of 4.0) or not.

H o H o : μ = 2.0 μ=2.0 Ha Ha: μ ≠ 2.0 μ≠2.0

Example 3

We want to test if college students take less than five years to graduate from college, on the average.

HoHo: μ ≥ 5 μ≥5 H a H a : μ < 5 μ 5

Example 4

In an issue of U. S. News and World Report, an article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third pass. The same article stated that 6.6% of U. S. students take advanced placement exams and 4.4 % pass. Test if the percentage of U. S. students who take advanced placement exams is more than 6.6%.

HoHo: pp=0.066=0.066 H a H a : pp > 0.066 >0.066

Since the null and alternate hypotheses are contradictory, you must examine evidence to decide which hypothesis the evidence supports. The evidence is in the form of sample data. The sample might support either the null hypothesis or the alternate hypothesis but not both.

After you have determined which hypothesis the sample supports, you make a decision. There are two options for a decision. They are "reject HoHo" if the sample information favors the alternate hypothesis or "do not reject HoHo" if the sample information favors the null hypothesis, meaning that there is not enough information to reject the null.

Mathematical Symbols Used in HoHo and HaHa:

Table 1
HoHo HaHa
equal (==) not equal (≠≠) or greater than (>>) or less than (<)
greater than or equal to (≥≥) less than (<)
less than or equal to (≤≤) more than (>>)

Note:

HoHo always has a symbol with an equal in it. HaHa never has a symbol with an equal in it. The choice of symbol depends on the wording of the hypothesis test. However, be aware that many researchers (including one of the co-authors in research work) use == in the Null Hypothesis, even with >> or << as the symbol in the Alternate Hypothesis. This practice is acceptable because we only make the decision to reject or not reject the Null Hypothesis.

Optional Collaborative Classroom Activity

Bring to class a newspaper, some news magazines, and some Internet articles . In groups, find articles from which your group can write a null and alternate hypotheses. Discuss your hypotheses with the rest of the class.

Glossary

Hypothesis:
A statement about the value of a population parameter. In case of two hypotheses, the statement assumed to be true is called the null hypothesis (notation H0H0 size 12{H rSub { size 8{0} } } {}) and the contradictory statement is called the alternate hypothesis (notation HaHa size 12{H rSub { size 8{a} } } {}).

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