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

26 April, 2016 - 09:20
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The actual test begins by considering two hypotheses. They are called the null hypothesis and the alternate hypothesis. These hypotheses contain opposing viewpoints.

Ho: 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.

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

Example 5.1

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

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

Example 5.2

We want to test whether the mean grade point average in American colleges is different from 2.0 (out of 4.0).

H_{0}: \mu =2.0\ \ \ \ \ H_{\alpha}: \mu \neq 2.0

Example 5.3

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

Ho: \mu \geqslant 5\ \ \ \ \ H_{\alpha}: \mu < 5

Example 5.4

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%.

Ho: p =0.066\ \ \ \ \ H_{\alpha}: p>0.066

Since the null and alternate hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data.

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

Mathematical Symbols Used in Ho and Ha:



equal (=)

not equal (\neq) or greater than (>) or less than (<)

greater than or equal to (≥)

less than (<)

less than or equal to (≤)

more than (>)

Note: Ho always has a symbol with an equal in it. Ha 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.