Null and Alternative Hypotheses

jkesler

Hypothesis Testing with One Sample

31 Null and Alternative Hypotheses

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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: It is a statement of no difference between the variables—they are not related. This can often be considered the status quo and as a result if you cannot accept the null it requires some action.

H₁: The alternative hypothesis: It is a claim about the population that is
contradictory to H₀ and what we conclude when we reject H₀. This is usually what the researcher is trying to prove.

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 “fail to reject H₀” if the sample information is insufficient to reject the null hypothesis.

Mathematical Symbols Used in H₀ and H₁:

H₀	H₁
equal (=)	not equal (≠) or greater than (>) or less than (<)

Table 8.1

Note

H₀ always has an equal symbol in it. H₁ never has a symbol with an equal in it. The choice of symbol depends on the wording of the hypothesis test. This practice is acceptable because we only make the decision to reject or not reject the null hypothesis.

Example 8.1

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

From the statement above, identify the Null and Alternative Hypothesis.

Solution 8.1

Claim (in symbols): p ≤ 0.30

Opposite of Claim: p > 0.30

Looking at the Claim and the Opposite, we first find the one that has an equal sign. In this example, it is the Claim because that is where we see the “≤” (it is read “less than or equal to” which is why we say it has the “equal” in it). This is what we convert into our null hypothesis H₀. So p ≤ 0.30 becomes p = 0.30.

H₀: p = 0.30

Looking again at the Claim and Opposite, find the the one that does not have an equal sign. In this example, it is the Opposite. This becomes our Alternative Hypothesis.

H₁: p > 0.30

It will become important later on to remember whether the Null Hypothesis or the Alternative Hypothesis came from the Claim, so it is a good practice to associate them somehow. In the table below, I put the Claim and Opposite in the first column. Then, in the second column, I put the Null Hypothesis H₀ in the first row across from the Claim, because H₀ “came from” the Claim (ie the claim had the equal sign in it).

Claim: p ≤ 0.30	H₀: p = 0.30
Opposite: p > 0.30	H₁: p > 0.30

Try It 8.1

A medical trial is conducted to test a drug company’s claim that a new medicine reduces cholesterol by 25%. State the null and alternative hypotheses.

Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the claim, opposite, null and alternative hypotheses. Also fill in the blanks indicating where the null and alternative hypotheses would go, indicating their correspondence with the claim and the opposite.

Claim: p __ 0.25	*H_{_}*: p __ 0.25
Opposite: p __ 0.25	*H_{_}*: p __ 0.25

Example 8.2

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

Solution 8.2

Claim: μ ≠ 2.0	H₁: μ ≠ 2.0
Opposite: μ = 2.0	H₀: μ = 2.0

Notice that this time, as opposed to Example 8.1, our null hypothesis corresponds to the Opposite. This is because the claim did not have an equal sign in it; in fact, it had a “not equal” sign because the original statement used the phrase “different from”, which suggests “not equal”.

Try It 8.2

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 claim, opposite, null and alternative hypotheses. Also fill in the blanks indicating where the null and alternative hypotheses would go, indicating their correspondence with the claim and the opposite.

Claim: μ __ 66	*H_{_}*: μ __ 66
Opposite: μ __ 66	*H_{_}*: μ __ 66

Example 8.3

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:

H₀: μ ≥ 5

H₁: μ < 5

Solution 8.3

Claim: μ < 5	H₁: μ < 5
Opposite: μ ≥ 5	H₀: μ = 5

Notice that in this example, we had to convert the symbol in our “Opposite” statement from “≥” to an equal symbol in our Null Hypothesis.

Try It 8.3

We want to test if it takes fewer than 45 minutes to teach a lesson plan. State the null and alternative hypotheses.

Create the table identifying the claim, opposite, null hypothesis and alternative hypothesis. Put the null and alternative hypothesis in the correct row to show correspondence with the claim and opposite.

Example 8.4

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%. State the null and alternative hypotheses.

H₀: p ≤ 0.066

H₁: p > 0.066

Solution 8.4

Claim: p > 0.066	H₁: p > 0.066
Opposite: p ≤ 0.066	H₀: p = 0.066

Notice that this time, as opposed to Example 8.1, our null hypothesis corresponds to the Opposite. This is because the claim did not have an equal sign in it; in fact, it had a “not equal” sign because the original statement used the phrase “different from”, which suggests “not equal”.

Try It 8.4

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.

Create the table identifying the claim, opposite, null hypothesis and alternative hypothesis. Put the null and alternative hypothesis in the correct row to show correspondence with the claim and opposite.

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Introductory Statistics with Google Sheets by jkesler is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.