Null and Alternative Hypotheses

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

H0, the - null hypothesis: a statement of no difference between sample means or proportions or no difference between a sample mean or proportion and a population mean or proportion. In other words, the difference equals 0.

Ha - , the alternative hypothesis: a claim about the population that is contradictory to H0 and what we conclude when we reject H0.

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 H0 if the sample information favors the alternative hypothesis or do not reject H0 or decline to reject H0 if the sample information is insufficient to reject the null hypothesis.

Mathematical Symbols Used in H0 and Ha:

H0 Ha
equal (=) not equal (≠) or greater than (>) or less than (<)
greater than or equal to (≥) less than (<)
less than or equal to (≤) more than (>)

Table 9.1

Note
H0 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 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.

Example 9.1
H0 : No more than 30 percent of the registered voters in Santa Clara County voted in the primary election. p ≤ 30
Ha : More than 30 percent of the registered voters in Santa Clara County voted in the primary election. p > 30


Try It 9.1
A medical trial is conducted to test whether or not a new medicine reduces cholesterol by 25 percent. State the null and alternative hypotheses.


Example 9.2
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 the following:
H0 : μ = 2.0
Ha : μ ≠ 2.0


Try It 9.2
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.
  1. H0 : μ __ 66
  2. Ha : μ __ 66

Example 9.3
We want to test if college students take fewer than five years to graduate from college, on the average. The null and alternative hypotheses are the following:
H0 : μ ≥ 5
Ha : μ < 5


Try It 9.3
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 ( =, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

  1. H0 : μ __ 45
  2. Ha : μ __ 45

Example 9.4
An article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third of the students pass. The same article stated that 6.6 percent of U.S. students take advanced placement exams and 4.4 percent pass. Test if the percentage of U.S. students who take advanced placement exams is more than 6.6 percent. State the null and alternative hypotheses.
H0 : p ≤ 0.066
Ha : p > 0.066


Try It 9.4
On a state driver's test, about 40 percent pass the test on the first try. We want to test if more than 40 percent pass on the first try. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.
  1. H0 : p __ 0.40
  2. Ha : p __ 0.40

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

Source: OpenStax, https://openstax.org/books/statistics/pages/9-introduction
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