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.
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Published on November 8, 2019 by Rebecca Bevans . Revised on June 22, 2023.
Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics . It is most often used by scientists to test specific predictions, called hypotheses, that arise from theories.
There are 5 main steps in hypothesis testing:
Though the specific details might vary, the procedure you will use when testing a hypothesis will always follow some version of these steps.
Step 1: state your null and alternate hypothesis, step 2: collect data, step 3: perform a statistical test, step 4: decide whether to reject or fail to reject your null hypothesis, step 5: present your findings, other interesting articles, frequently asked questions about hypothesis testing.
After developing your initial research hypothesis (the prediction that you want to investigate), it is important to restate it as a null (H o ) and alternate (H a ) hypothesis so that you can test it mathematically.
The alternate hypothesis is usually your initial hypothesis that predicts a relationship between variables. The null hypothesis is a prediction of no relationship between the variables you are interested in.
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For a statistical test to be valid , it is important to perform sampling and collect data in a way that is designed to test your hypothesis. If your data are not representative, then you cannot make statistical inferences about the population you are interested in.
There are a variety of statistical tests available, but they are all based on the comparison of within-group variance (how spread out the data is within a category) versus between-group variance (how different the categories are from one another).
If the between-group variance is large enough that there is little or no overlap between groups, then your statistical test will reflect that by showing a low p -value . This means it is unlikely that the differences between these groups came about by chance.
Alternatively, if there is high within-group variance and low between-group variance, then your statistical test will reflect that with a high p -value. This means it is likely that any difference you measure between groups is due to chance.
Your choice of statistical test will be based on the type of variables and the level of measurement of your collected data .
Based on the outcome of your statistical test, you will have to decide whether to reject or fail to reject your null hypothesis.
In most cases you will use the p -value generated by your statistical test to guide your decision. And in most cases, your predetermined level of significance for rejecting the null hypothesis will be 0.05 – that is, when there is a less than 5% chance that you would see these results if the null hypothesis were true.
In some cases, researchers choose a more conservative level of significance, such as 0.01 (1%). This minimizes the risk of incorrectly rejecting the null hypothesis ( Type I error ).
The results of hypothesis testing will be presented in the results and discussion sections of your research paper , dissertation or thesis .
In the results section you should give a brief summary of the data and a summary of the results of your statistical test (for example, the estimated difference between group means and associated p -value). In the discussion , you can discuss whether your initial hypothesis was supported by your results or not.
In the formal language of hypothesis testing, we talk about rejecting or failing to reject the null hypothesis. You will probably be asked to do this in your statistics assignments.
However, when presenting research results in academic papers we rarely talk this way. Instead, we go back to our alternate hypothesis (in this case, the hypothesis that men are on average taller than women) and state whether the result of our test did or did not support the alternate hypothesis.
If your null hypothesis was rejected, this result is interpreted as “supported the alternate hypothesis.”
These are superficial differences; you can see that they mean the same thing.
You might notice that we don’t say that we reject or fail to reject the alternate hypothesis . This is because hypothesis testing is not designed to prove or disprove anything. It is only designed to test whether a pattern we measure could have arisen spuriously, or by chance.
If we reject the null hypothesis based on our research (i.e., we find that it is unlikely that the pattern arose by chance), then we can say our test lends support to our hypothesis . But if the pattern does not pass our decision rule, meaning that it could have arisen by chance, then we say the test is inconsistent with our hypothesis .
If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.
Methodology
Research bias
Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.
A hypothesis states your predictions about what your research will find. It is a tentative answer to your research question that has not yet been tested. For some research projects, you might have to write several hypotheses that address different aspects of your research question.
A hypothesis is not just a guess — it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).
Null and alternative hypotheses are used in statistical hypothesis testing . The null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship.
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Bevans, R. (2023, June 22). Hypothesis Testing | A Step-by-Step Guide with Easy Examples. Scribbr. Retrieved September 3, 2024, from https://www.scribbr.com/statistics/hypothesis-testing/
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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 .
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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.
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5.2 - writing hypotheses.
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 (\(H_a\)).
When writing hypotheses there are three things that we need to know: (1) the parameter that we are testing (2) the direction of the test (non-directional, right-tailed or left-tailed), and (3) the value of the hypothesized parameter.
Hypotheses are always written in terms of population parameters (e.g., \(p\) and \(\mu\)). The tables below display all of the possible hypotheses for the parameters that we have learned thus far. Note that the null hypothesis always includes the equality (i.e., =).
Research Question | Is the population mean different from \( \mu_{0} \)? | Is the population mean greater than \(\mu_{0}\)? | Is the population mean less than \(\mu_{0}\)? |
---|---|---|---|
Null Hypothesis, \(H_{0}\) | \(\mu=\mu_{0} \) | \(\mu=\mu_{0} \) | \(\mu=\mu_{0} \) |
Alternative Hypothesis, \(H_{a}\) | \(\mu\neq \mu_{0} \) | \(\mu> \mu_{0} \) | \(\mu<\mu_{0} \) |
Type of Hypothesis Test | Two-tailed, non-directional | Right-tailed, directional | Left-tailed, directional |
Research Question | Is there a difference in the population? | Is there a mean increase in the population? | Is there a mean decrease in the population? |
---|---|---|---|
Null Hypothesis, \(H_{0}\) | \(\mu_d=0 \) | \(\mu_d =0 \) | \(\mu_d=0 \) |
Alternative Hypothesis, \(H_{a}\) | \(\mu_d \neq 0 \) | \(\mu_d> 0 \) | \(\mu_d<0 \) |
Type of Hypothesis Test | Two-tailed, non-directional | Right-tailed, directional | Left-tailed, directional |
Research Question | Is the population proportion different from \(p_0\)? | Is the population proportion greater than \(p_0\)? | Is the population proportion less than \(p_0\)? |
---|---|---|---|
Null Hypothesis, \(H_{0}\) | \(p=p_0\) | \(p= p_0\) | \(p= p_0\) |
Alternative Hypothesis, \(H_{a}\) | \(p\neq p_0\) | \(p> p_0\) | \(p< p_0\) |
Type of Hypothesis Test | Two-tailed, non-directional | Right-tailed, directional | Left-tailed, directional |
Research Question | Are the population means different? | Is the population mean in group 1 greater than the population mean in group 2? | Is the population mean in group 1 less than the population mean in groups 2? |
---|---|---|---|
Null Hypothesis, \(H_{0}\) | \(\mu_1=\mu_2\) | \(\mu_1 = \mu_2 \) | \(\mu_1 = \mu_2 \) |
Alternative Hypothesis, \(H_{a}\) | \(\mu_1 \ne \mu_2 \) | \(\mu_1 \gt \mu_2 \) | \(\mu_1 \lt \mu_2\) |
Type of Hypothesis Test | Two-tailed, non-directional | Right-tailed, directional | Left-tailed, directional |
Research Question | Are the population proportions different? | Is the population proportion in group 1 greater than the population proportion in groups 2? | Is the population proportion in group 1 less than the population proportion in group 2? |
---|---|---|---|
Null Hypothesis, \(H_{0}\) | \(p_1 = p_2 \) | \(p_1 = p_2 \) | \(p_1 = p_2 \) |
Alternative Hypothesis, \(H_{a}\) | \(p_1 \ne p_2\) | \(p_1 \gt p_2 \) | \(p_1 \lt p_2\) |
Type of Hypothesis Test | Two-tailed, non-directional | Right-tailed, directional | Left-tailed, directional |
Research Question | Is the slope in the population different from 0? | Is the slope in the population positive? | Is the slope in the population negative? |
---|---|---|---|
Null Hypothesis, \(H_{0}\) | \(\beta =0\) | \(\beta= 0\) | \(\beta = 0\) |
Alternative Hypothesis, \(H_{a}\) | \(\beta\neq 0\) | \(\beta> 0\) | \(\beta< 0\) |
Type of Hypothesis Test | Two-tailed, non-directional | Right-tailed, directional | Left-tailed, directional |
Research Question | Is the correlation in the population different from 0? | Is the correlation in the population positive? | Is the correlation in the population negative? |
---|---|---|---|
Null Hypothesis, \(H_{0}\) | \(\rho=0\) | \(\rho= 0\) | \(\rho = 0\) |
Alternative Hypothesis, \(H_{a}\) | \(\rho \neq 0\) | \(\rho > 0\) | \(\rho< 0\) |
Type of Hypothesis Test | Two-tailed, non-directional | Right-tailed, directional | Left-tailed, directional |
COMMENTS
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.
Examples of Alternative Hypotheses. The following examples illustrate how to define the null and alternative hypotheses for different research problems. Example 1: A biologist wants to test if the mean weight of a certain population of turtle is different from the widely-accepted mean weight of 300 pounds.
The alternative hypothesis is the complement to the null hypothesis. Null and alternative hypotheses are exhaustive, meaning that together they cover every possible outcome. They are also mutually exclusive, meaning that only one can be true at a time.
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.
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 \(H_{a}\)).
Alternative Hypothesis: The mean weight loss should be greater for those on a low carbohydrate diet when compared with those on a low fat diet (population mean weight loss on a low carbohydrate diet > population mean weight loss on a low fat diet). This is a one-sided alternative hypothesis.
Example. 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: H0: μ = 2.0. Ha: μ ≠ 2.0. try it. We want to test whether the mean height of eighth graders is 66 inches. State the null and alternative hypotheses.
Step 1: State your null and alternate hypothesis. Step 2: Collect data. Step 3: Perform a statistical test. Step 4: Decide whether to reject or fail to reject your null hypothesis. Step 5: Present your findings. Other interesting articles. Frequently asked questions about hypothesis testing. Step 1: State your null and alternate hypothesis.
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 ...
Alternative Hypothesis. The statement that there is some difference in the population (s), denoted as H a or H 1. When writing hypotheses there are three things that we need to know: (1) the parameter that we are testing (2) the direction of the test (non-directional, right-tailed or left-tailed), and (3) the value of the hypothesized parameter.