greater than (>) less than (<)
H 0 always has a symbol with an equal in it. H a 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.
H 0 : No more than 30% of the registered voters in Santa Clara County voted in the primary election. p ≤ 30
H a : More than 30% of the registered voters in Santa Clara County voted in the primary election. p > 30
A medical trial is conducted to test whether or not a new medicine reduces cholesterol by 25%. State the null and alternative hypotheses.
H 0 : The drug reduces cholesterol by 25%. p = 0.25
H a : The drug does not reduce cholesterol by 25%. p ≠ 0.25
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:
H 0 : μ = 2.0
H a : μ ≠ 2.0
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. H 0 : μ __ 66 H a : μ __ 66
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:
H 0 : μ ≥ 5
H a : μ < 5
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. H 0 : μ __ 45 H a : μ __ 45
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.
H 0 : p ≤ 0.066
H a : p > 0.066
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 (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses. H 0 : p __ 0.40 H a : p __ 0.40
In a hypothesis test , sample data is evaluated in order to arrive at a decision about some type of claim. If certain conditions about the sample are satisfied, then the claim can be evaluated for a population. In a hypothesis test, we: Evaluate the null hypothesis , typically denoted with H 0 . The null is not rejected unless the hypothesis test shows otherwise. The null statement must always contain some form of equality (=, ≤ or ≥) Always write the alternative hypothesis , typically denoted with H a or H 1 , using less than, greater than, or not equals symbols, i.e., (≠, >, or <). If we reject the null hypothesis, then we can assume there is enough evidence to support the alternative hypothesis. Never state that a claim is proven true or false. Keep in mind the underlying fact that hypothesis testing is based on probability laws; therefore, we can talk only in terms of non-absolute certainties.
H 0 and H a are contradictory.
by Marco Taboga , PhD
In a statistical test, observed data is used to decide whether or not to reject a restriction on the data-generating probability distribution.
The assumption that the restriction is true is called null hypothesis , while the statement that the restriction is not true is called alternative hypothesis.
A correct specification of the alternative hypothesis is essential to decide between one-tailed and two-tailed tests.
Table of contents
Choice between one-tailed and two-tailed tests, the critical region, the interpretation of the rejection, the interpretation must be coherent with the alternative hypothesis.
More details, keep reading the glossary.
In order to fully understand the concept of alternative hypothesis, we need to remember the essential elements of a statistical inference problem:
we observe a sample drawn from an unknown probability distribution;
in principle, any valid probability distribution could have generated the sample;
however, we usually place some a priori restrictions on the set of possible data-generating distributions;
A couple of simple examples follow.
When we conduct a statistical test, we formulate a null hypothesis as a restriction on the statistical model.
The alternative hypothesis is
The alternative hypothesis is used to decide whether a test should be one-tailed or two-tailed.
The null hypothesis is rejected if the test statistic falls within a critical region that has been chosen by the statistician.
The critical region is a set of values that may comprise:
only the left tail of the distribution or only the right tail (one-tailed test);
both the left and the right tail (two-tailed test).
The choice of the critical region depends on the alternative hypothesis. Let us see why.
The interpretation is different depending on the tail of the distribution in which the test statistic falls.
The choice between a one-tailed or a two-tailed test needs to be done in such a way that the interpretation of a rejection is always coherent with the alternative hypothesis.
When we deal with the power function of a test, the term "alternative hypothesis" has a special meaning.
We conclude with a caveat about the interpretation of the outcome of a test of hypothesis.
The interpretation of a rejection of the null is controversial.
According to some statisticians, rejecting the null is equivalent to accepting the alternative.
However, others deem that rejecting the null does not necessarily imply accepting the alternative. In fact, it is possible to think of situations in which both hypotheses can be rejected. Let us see why.
According to the conceptual framework illustrated by the images above, there are three possibilities:
the null is true;
the alternative is true;
neither the null nor the alternative is true because the true data-generating distribution has been excluded from the statistical model (we say that the model is mis-specified).
If we are in case 3, accepting the alternative after a rejection of the null is an incorrect decision. Moreover, a second test in which the alternative becomes the new null may lead us to another rejection.
You can find more details about the alternative hypothesis in the lecture on Hypothesis testing .
Previous entry: Almost sure
Next entry: Binomial coefficient
Please cite as:
Taboga, Marco (2021). "Alternative hypothesis", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/glossary/alternative-hypothesis.
Most of the learning materials found on this website are now available in a traditional textbook format.
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The null and alternative hypotheses are two competing claims that researchers weigh evidence for and against using a statistical test: Null hypothesis (H0): There’s no effect in the population. Alternative hypothesis (Ha or H1): There’s an effect in the population.
H a —, the alternative hypothesis: a claim about the population that is contradictory to H 0 and what we conclude when we reject H 0. 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.
In statistics, alternative hypothesis is often denoted as Ha or H1. Hypotheses are formulated to compare in a statistical hypothesis test. In the domain of inferential statistics, two rival hypotheses can be compared by explanatory power and predictive power .
There are two types of alternative hypotheses: A one-tailed hypothesis involves making a “greater than” or “less than ” statement. For example, suppose we assume the mean height of a male in the U.S. is greater than or equal to 70 inches. The null and alternative hypotheses in this case would be: Null hypothesis: µ ≥ 70 inches.
The first step in conducting a hypothesis test is to write the hypothesis statements that are going to be tested. For each test you will have a null hypothesis (\(H_0\)) and an alternative hypothesis (\(H_a\)).
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 \(H_{0}\)) and the contradictory statement is called the alternative hypothesis (notation \(H_{a}\)).
The alternative hypothesis is simply the reverse of the null hypothesis, and there are three options, depending on where we expect the difference to lie. Thus, our alternative hypothesis is the mathematical way of stating our research question.
Describe hypothesis testing in general and in practice. The actual test begins by considering two hypotheses. They are called the null hypothesis and the alternative hypothesis. These hypotheses contain opposing viewpoints.
It is the uninteresting hypothesis--the boring hypothesis. Usually, it is the hypothesis that assumes no difference. It is the opposite of your research hypothesis. The alternative hypothesis--that is, the research hypothesis--is the idea, phenomenon, observation that you want to prove.
Definition. When we conduct a statistical test, we formulate a null hypothesis as a restriction on the statistical model. Denote by the true data-generating distribution. The null hypothesis can be expressed as where is a subset of . The alternative hypothesis is.