Chapter 8: Hypothesis Testing with One Sample
8.1 Null and Alternative Hypotheses
Learning Objectives
By the end of this section, the student should be able to:
- Describe hypothesis testing in general and in practice.
Hypothesis Testing
The actual test begins by considering two hypotheses. They are called the null hypothesis and the alternative hypothesis. These hypotheses contain opposing viewpoints.
[latex]H_{0}[/latex]: The null hypothesis: It is a statement about the population that either is believed to be true or is used to put forth an argument unless it can be shown to be incorrect beyond a reasonable doubt.
[latex]H_{a}[/latex]: The alternative hypothesis: It is a claim about the population that is
contradictory to [latex]H_{0}[/latex] and what we conclude when we reject [latex]H_{0}[/latex].
Since the null and alternative 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 [latex]H_{0}[/latex]” if the sample information favors the alternative hypothesis or “do not reject [latex]H_{0}[/latex]” or “decline to reject [latex]H_{0}[/latex]” if the sample information is insufficient to reject the null hypothesis.
Mathematical Symbols Used in [latex]H_{0}[/latex] and [latex]H_{a}[/latex]:
| [latex]H_{0}[/latex] | [latex]H_{a}[/latex] |
|---|---|
| equal (=) | not equal (≠) or greater than (>) or less than (<) |
| greater than or equal to (≥) | less than (<) |
| less than or equal to (≤) | more than (>) |
Note
[latex]H_{0}[/latex] always has a symbol with an equal in it. [latex]H_{a}[/latex] 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 alternative hypothesis. This practice is acceptable because we only make the decision to reject or not reject the null hypothesis.
[latex]H_{0}[/latex]: No more than 30% of the registered voters in Santa Clara County voted in the primary election. [latex]p \le 30[/latex]
[latex]H_{a}[/latex]: More than 30% of the registered voters in Santa Clara County voted in the primary election. [latex]p > 30[/latex]
Example
A medical trial is conducted to test whether or not a new medicine reduces cholesterol by 25%. State the null and alternative hypotheses.
Solution
[latex]H_{0}[/latex]: The drug reduces cholesterol by 25%. [latex]p = 0.25[/latex]
[latex]H_{a}[/latex]: The drug does not reduce cholesterol by 25%. [latex]p \neq 0.25[/latex]
Your turn!
We want to test whether the mean GPA of students in American colleges is different from 2.0 (out of 4.0). The null and alternative hypotheses are:
Solution
[latex]H_{0}[/latex]: [latex]\mu = 2.0[/latex]
Ha: [latex]\mu \neq 2.0[/latex]
Exercises
Example
We want to test whether the mean height of eighth graders is 66 inches. State the null and alternative hypotheses. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.
Solution
- [latex]H_{0}: \mu = 66[/latex]
- [latex]H_{a} : \mu \neq 66[/latex]
Your Turn!
We want to test if college students take less than five years to graduate from college, on the average. The null and alternative hypotheses are:
Solution
[latex]H_{0}: \mu \ge 5[/latex]
[latex]H_{a}: \mu \lt 5[/latex]
Example
We want to test if it takes fewer than 45 minutes to teach a lesson plan. State the null and alternative hypotheses. Fill in the correct symbol [latex]( =, \neq, \ge, \lt, \le, >)[/latex] for the null and alternative hypotheses.
Solution
1. [latex]H_{0}: \mu \ge 45[/latex]
2. [latex]H_{a}: \mu \lt 45[/latex]
Your Turn!
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%. State the null and alternative hypotheses.
Solution
[latex]H_{0}: p \le 0.066[/latex]
[latex]H_{a}: p > 0.066[/latex]
Example
On a state driver’s test, about 40% pass the test on the first try. We want to test if more than 40% pass on the first try. Fill in the correct symbol [latex](=, \neq, \ge, \lt, \le, >)[/latex] for the null and alternative hypotheses.
Solution
- [latex]H_{0}: p = 0.40[/latex]
- [latex]H_{a}: p > 0.40[/latex]
References
Data from the National Institute of Mental Health. Available online at http://www.nimh.nih.gov/publicat/depression.cfm.
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 H0) and the contradictory statement is called the alternative hypothesis (notation Ha).