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 test of hypothesis , a sample of data is used to decide whether to reject or not to reject a hypothesis about the probability distribution from which the sample was extracted.
The hypothesis is called the null hypothesis, or simply "the null".
Table of contents
How is the null hypothesis tested, example 1 - proportion of defective items, measurement, test statistic, critical region, interpretation, example 2 - reliability of a production plant, rejection and failure to reject, not rejecting and accepting are not the same thing, failure to reject can be due to lack of power, rejections are easier to interpret, but be careful, takeaways - how to (and not to) formulate a null hypothesis, more examples, more details, best practices in science, keep reading the glossary.
Formulating null hypotheses and subjecting them to statistical testing is one of the workhorses of the scientific method.
Scientists in all fields make conjectures about the phenomena they study, translate them into null hypotheses and gather data to test them.
This process resembles a trial:
the defendant (the null hypothesis) is accused of being guilty (wrong);
evidence (data) is gathered in order to prove the defendant guilty (reject the null);
if there is evidence beyond any reasonable doubt, the defendant is found guilty (the null is rejected);
otherwise, the defendant is found not guilty (the null is not rejected).
Keep this analogy in mind because it helps to better understand statistical tests, their limitations, use and misuse, and frequent misinterpretation.
Before collecting the data:
we decide how to summarize the relevant characteristics of the sample data in a single number, the so-called test statistic ;
we derive the probability distribution of the test statistic under the hypothesis that the null is true (the data is regarded as random; therefore, the test statistic is a random variable);
we decide what probability of incorrectly rejecting the null we are willing to tolerate (the level of significance , or size of the test ); the level of significance is typically a small number, such as 5% or 1%.
we choose one or more intervals of values (collectively called rejection region) such that the probability that the test statistic falls within these intervals is equal to the desired level of significance; the rejection region is often a tail of the distribution of the test statistic (one-tailed test) or the union of the left and right tails (two-tailed test).
Then, the data is collected and used to compute the value of the test statistic.
A decision is taken as follows:
if the test statistic falls within the rejection region, then the null hypothesis is rejected;
otherwise, it is not rejected.
We now make two examples of practical problems that lead to formulate and test a null hypothesis.
A new method is proposed to produce light bulbs.
The proponents claim that it produces less defective bulbs than the method currently in use.
To check the claim, we can set up a statistical test as follows.
We keep the light bulbs on for 10 consecutive days, and then we record whether they are still working at the end of the test period.
The probability that a light bulb produced with the new method is still working at the end of the test period is the same as that of a light bulb produced with the old method.
100 light bulbs are tested:
50 of them are produced with the new method (group A)
the remaining 50 are produced with the old method (group B).
The final data comprises 100 observations of:
an indicator variable which is equal to 1 if the light bulb is still working at the end of the test period and 0 otherwise;
a categorical variable that records the group (A or B) to which each light bulb belongs.
We use the data to compute the proportions of working light bulbs in groups A and B.
The proportions are estimates of the probabilities of not being defective, which are equal for the two groups under the null hypothesis.
We then compute a z-statistic (see here for details) by:
taking the difference between the proportion in group A and the proportion in group B;
standardizing the difference:
we subtract the expected value (which is zero under the null hypothesis);
we divide by the standard deviation (it can be derived analytically).
The distribution of the z-statistic can be approximated by a standard normal distribution .
We decide that the level of confidence must be 5%. In other words, we are going to tolerate a 5% probability of incorrectly rejecting the null hypothesis.
The critical region is the right 5%-tail of the normal distribution, that is, the set of all values greater than 1.645 (see the glossary entry on critical values if you are wondering how this value was obtained).
If the test statistic is greater than 1.645, then the null hypothesis is rejected; otherwise, it is not rejected.
A rejection is interpreted as significant evidence that the new production method produces less defective items; failure to reject is interpreted as insufficient evidence that the new method is better.
A production plant incurs high costs when production needs to be halted because some machinery fails.
The plant manager has decided that he is not willing to tolerate more than one halt per year on average.
If the expected number of halts per year is greater than 1, he will make new investments in order to improve the reliability of the plant.
A statistical test is set up as follows.
The reliability of the plant is measured by the number of halts.
The number of halts in a year is assumed to have a Poisson distribution with expected value equal to 1 (using the Poisson distribution is common in reliability testing).
The manager cannot wait more than one year before taking a decision.
There will be a single datum at his disposal: the number of halts observed during one year.
The number of halts is used as a test statistic. By assumption, it has a Poisson distribution under the null hypothesis.
The manager decides that the probability of incorrectly rejecting the null can be at most 10%.
A Poisson random variable with expected value equal to 1 takes values:
larger than 1 with probability 26.42%;
larger than 2 with probability 8.03%.
Therefore, it is decided that the critical region will be the set of all values greater than or equal to 3.
If the test statistic is strictly greater than or equal to 3, then the null is rejected; otherwise, it is not rejected.
A rejection is interpreted as significant evidence that the production plant is not reliable enough (the average number of halts per year is significantly larger than tolerated).
Failure to reject is interpreted as insufficient evidence that the plant is unreliable.
This section discusses the main problems that arise in the interpretation of the outcome of a statistical test (reject / not reject).
When the test statistic does not fall within the critical region, then we do not reject the null hypothesis.
Does this mean that we accept the null? Not really.
In general, failure to reject does not constitute, per se, strong evidence that the null hypothesis is true .
Remember the analogy between hypothesis testing and a criminal trial. In a trial, when the defendant is declared not guilty, this does not mean that the defendant is innocent. It only means that there was not enough evidence (not beyond any reasonable doubt) against the defendant.
In turn, lack of evidence can be due:
either to the fact that the defendant is innocent ;
or to the fact that the prosecution has not been able to provide enough evidence against the defendant, even if the latter is guilty .
This is the very reason why courts do not declare defendants innocent, but they use the locution "not guilty".
In a similar fashion, statisticians do not say that the null hypothesis has been accepted, but they say that it has not been rejected.
To better understand why failure to reject does not in general constitute strong evidence that the null hypothesis is true, we need to use the concept of statistical power .
The power of a test is the probability (calculated ex-ante, i.e., before observing the data) that the null will be rejected when another hypothesis (called the alternative hypothesis ) is true.
Let's consider the first of the two examples above (the production of light bulbs).
In that example, the null hypothesis is: the probability that a light bulb is defective does not decrease after introducing a new production method.
Let's make the alternative hypothesis that the probability of being defective is 1% smaller after changing the production process (assume that a 1% decrease is considered a meaningful improvement by engineers).
How much is the ex-ante probability of rejecting the null if the alternative hypothesis is true?
If this probability (the power of the test) is small, then it is very likely that we will not reject the null even if it is wrong.
If we use the analogy with criminal trials, low power means that most likely the prosecution will not be able to provide sufficient evidence, even if the defendant is guilty.
Thus, in the case of lack of power, failure to reject is almost meaningless (it was anyway highly likely).
This is why, before performing a test, it is good statistical practice to compute its power against a relevant alternative .
If the power is found to be too small, there are usually remedies. In particular, statistical power can usually be increased by increasing the sample size (see, e.g., the lecture on hypothesis tests about the mean ).
As we have explained above, interpreting a failure to reject the null hypothesis is not always straightforward. Instead, interpreting a rejection is somewhat easier.
When we reject the null, we know that the data has provided a lot of evidence against the null. In other words, it is unlikely (how unlikely depends on the size of the test) that the null is true given the data we have observed.
There is an important caveat though. The null hypothesis is often made up of several assumptions, including:
the main assumption (the one we are testing);
other assumptions (e.g., technical assumptions) that we need to make in order to set up the hypothesis test.
For instance, in Example 2 above (reliability of a production plant), the main assumption is that the expected number of production halts per year is equal to 1. But there is also a technical assumption: the number of production halts has a Poisson distribution.
It must be kept in mind that a rejection is always a joint rejection of the main assumption and all the other assumptions .
Therefore, we should always ask ourselves whether the null has been rejected because the main assumption is wrong or because the other assumptions are violated.
In the case of Example 2 above, is a rejection of the null due to the fact that the expected number of halts is greater than 1 or is it due to the fact that the distribution of the number of halts is very different from a Poisson distribution?
When we suspect that a rejection is due to the inappropriateness of some technical assumption (e.g., assuming a Poisson distribution in the example), we say that the rejection could be due to misspecification of the model .
The right thing to do when these kind of suspicions arise is to conduct so-called robustness checks , that is, to change the technical assumptions and carry out the test again.
In our example, we could re-run the test by assuming a different probability distribution for the number of halts (e.g., a negative binomial or a compound Poisson - do not worry if you have never heard about these distributions).
If we keep obtaining a rejection of the null even after changing the technical assumptions several times, the we say that our rejection is robust to several different specifications of the model .
What are the main practical implications of everything we have said thus far? How does the theory above help us to set up and test a null hypothesis?
What we said can be summarized in the following guiding principles:
A test of hypothesis is like a criminal trial and you are the prosecutor . You want to find evidence that the defendant (the null hypothesis) is guilty. Your job is not to prove that the defendant is innocent. If you find yourself hoping that the defendant is found not guilty (i.e., the null is not rejected) then something is wrong with the way you set up the test. Remember: you are the prosecutor.
Compute the power of your test against one or more relevant alternative hypotheses. Do not run a test if you know ex-ante that it is unlikely to reject the null when the alternative hypothesis is true.
Beware of technical assumptions that you add to the main assumption you want to test. Make robustness checks in order to verify that the outcome of the test is not biased by model misspecification.
More examples of null hypotheses and how to test them can be found in the following lectures.
Where the example is found | Null hypothesis |
---|---|
The mean of a normal distribution is equal to a certain value | |
The variance of a normal distribution is equal to a certain value | |
A vector of parameters estimated by MLE satisfies a set of linear or non-linear restrictions | |
A regression coefficient is equal to a certain value |
The lecture on Hypothesis testing provides a more detailed mathematical treatment of null hypotheses and how they are tested.
This lecture on the null hypothesis was featured in Stanford University's Best practices in science .
Previous entry: Normal equations
Next entry: Parameter
Please cite as:
Taboga, Marco (2021). "Null hypothesis", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/glossary/null-hypothesis.
Most of the learning materials found on this website are now available in a traditional textbook format.
Null hypothesis is used to make decisions based on data and by using statistical tests. Null hypothesis is represented using H o and it states that there is no difference between the characteristics of two samples. Null hypothesis is generally a statement of no difference. The rejection of null hypothesis is equivalent to the acceptance of the alternate hypothesis.
Let us learn more about null hypotheses, tests for null hypotheses, the difference between null hypothesis and alternate hypothesis, with the help of examples, FAQs.
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Null hypothesis states that there is no significant difference between the observed characteristics across two sample sets. Null hypothesis states the observed population parameters or variables is the same across the samples. The null hypothesis states that there is no relationship between the sample parameters, the independent variable, and the dependent variable. The term null hypothesis is used in instances to mean that there is no differences in the two means, or that the difference is not so significant.
If the experimental outcome is the same as the theoretical outcome then the null hypothesis holds good. But if there are any differences in the observed parameters across the samples then the null hypothesis is rejected, and we consider an alternate hypothesis. The rejection of the null hypothesis does not mean that there were flaws in the basic experimentation, but it sets the stage for further research. Generally, the strength of the evidence is tested against the null hypothesis.
Null hypothesis and alternate hypothesis are the two approaches used across statistics. The alternate hypothesis states that there is a significant difference between the parameters across the samples. The alternate hypothesis is the inverse of null hypothesis. An important reason to reject the null hypothesis and consider the alternate hypothesis is due to experimental or sampling errors.
The two important approaches of statistical interference of null hypothesis are significance testing and hypothesis testing. The null hypothesis is a theoretical hypothesis and is based on insufficient evidence, which requires further testing to prove if it is true or false.
The aim of significance testing is to provide evidence to reject the null hypothesis. If the difference is strong enough then reject the null hypothesis and accept the alternate hypothesis. The testing is designed to test the strength of the evidence against the hypothesis. The four important steps of significance testing are as follows.
If the p-value is lesser than the significance level α, then the null hypothesis is rejected. And if the p-value is greater than the significance level α, then the null hypothesis is accepted.
Hypothesis testing takes the parameters from the sample and makes a derivation about the population. A hypothesis is an educated guess about a sample, which can be tested either through an experiment or an observation. Initially, a tentative assumption is made about the sample in the form of a null hypothesis.
There are four steps to perform hypothesis testing. They are:
There are often errors in the process of testing the hypothesis. The two important errors observed in hypothesis testing is as follows.
The difference between null hypothesis and alternate hypothesis can be understood through the following points.
☛ Related Topics
The following topics help in a better understanding of the null hypothesis.
Example 1: A medical experiment and trial is conducted to check if a particular drug can serve as the vaccine for Covid-19, and can prevent from occurrence of Corona. Write the null hypothesis and the alternate hypothesis for this situation.
The given situation refers to a possible new drug and its effectiveness of being a vaccine for Covid-19 or not. The null hypothesis (H o ) and alternate hypothesis (H a ) for this medical experiment is as follows.
Example 2: The teacher has prepared a set of important questions and informs the student that preparing these questions helps in scoring more than 60% marks in the board exams. Write the null hypothesis and the alternate hypothesis for this situation.
The given situation refers to the teacher who has claimed that her important questions helps to score more than 60% marks in the board exams. The null hypothesis(H o ) and alternate hypothesis(H a ) for this situation is as follows.
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Faqs on null hypothesis, what is null hypothesis in maths.
Null hypothesis is used in statistics and it states if there is any significant difference between the two samples. The acceptance of null hypothesis mean that there is no significant difference between the two samples. And the rejection of null hypothesis means that the two samples are different, and we need to accept the alternate hypothesis. The null hypothesis statement is represented as H 0 and the alternate hypothesis is represented as H a .
The null hypothesis is broadly tested using two methods. The null hypothesis can be tested using significance testing and hypothesis testing.Broadly the test for null hypothesis is performed across four stages. First the null hypothesis is identified, secondly the null hypothesis is defined. Next a suitable test is used to test the hypothesis, and finally either the null hypothesis or the alternate hypothesis is accepted.
The null hypothesis is accepted or rejected based on the result of the hypothesis testing. The p value is found and the significance level is defined. If the p-value is lesser than the significance level α, then the null hypothesis is rejected. And if the p-value is greater than the significance level α, then the null hypothesis is accepted.
The null hypothesis states that there is no significant difference between the two samples, and the alternate hypothesis states that there is a significant difference between the two samples. The null hypothesis is referred using H o and the alternate hypothesis is referred using H a . As per null hypothesis the observed variables and parameters are the same across the samples, but as per alternate hypothesis there is a significant difference between the observed variables and parameters across the samples.
A few quick examples of null hypothesis are as follows.
In mathematics, Statistics deals with the study of research and surveys on the numerical data. For taking surveys, we have to define the hypothesis. Generally, there are two types of hypothesis. One is a null hypothesis, and another is an alternative hypothesis .
In probability and statistics, the null hypothesis is a comprehensive statement or default status that there is zero happening or nothing happening. For example, there is no connection among groups or no association between two measured events. It is generally assumed here that the hypothesis is true until any other proof has been brought into the light to deny the hypothesis. Let us learn more here with definition, symbol, principle, types and example, in this article.
Table of contents:
The null hypothesis is a kind of hypothesis which explains the population parameter whose purpose is to test the validity of the given experimental data. This hypothesis is either rejected or not rejected based on the viability of the given population or sample . In other words, the null hypothesis is a hypothesis in which the sample observations results from the chance. It is said to be a statement in which the surveyors wants to examine the data. It is denoted by H 0 .
In statistics, the null hypothesis is usually denoted by letter H with subscript ‘0’ (zero), such that H 0 . It is pronounced as H-null or H-zero or H-nought. At the same time, the alternative hypothesis expresses the observations determined by the non-random cause. It is represented by H 1 or H a .
The principle followed for null hypothesis testing is, collecting the data and determining the chances of a given set of data during the study on some random sample, assuming that the null hypothesis is true. In case if the given data does not face the expected null hypothesis, then the outcome will be quite weaker, and they conclude by saying that the given set of data does not provide strong evidence against the null hypothesis because of insufficient evidence. Finally, the researchers tend to reject that.
Here, the hypothesis test formulas are given below for reference.
The formula for the null hypothesis is:
H 0 : p = p 0
The formula for the alternative hypothesis is:
H a = p >p 0 , < p 0 ≠ p 0
The formula for the test static is:
Remember that, p 0 is the null hypothesis and p – hat is the sample proportion.
Also, read:
There are different types of hypothesis. They are:
Simple Hypothesis
It completely specifies the population distribution. In this method, the sampling distribution is the function of the sample size.
Composite Hypothesis
The composite hypothesis is one that does not completely specify the population distribution.
Exact Hypothesis
Exact hypothesis defines the exact value of the parameter. For example μ= 50
Inexact Hypothesis
This type of hypothesis does not define the exact value of the parameter. But it denotes a specific range or interval. For example 45< μ <60
Sometimes the null hypothesis is rejected too. If this hypothesis is rejected means, that research could be invalid. Many researchers will neglect this hypothesis as it is merely opposite to the alternate hypothesis. It is a better practice to create a hypothesis and test it. The goal of researchers is not to reject the hypothesis. But it is evident that a perfect statistical model is always associated with the failure to reject the null hypothesis.
The null hypothesis says there is no correlation between the measured event (the dependent variable) and the independent variable. We don’t have to believe that the null hypothesis is true to test it. On the contrast, you will possibly assume that there is a connection between a set of variables ( dependent and independent).
The null hypothesis is rejected using the P-value approach. If the P-value is less than or equal to the α, there should be a rejection of the null hypothesis in favour of the alternate hypothesis. In case, if P-value is greater than α, the null hypothesis is not rejected.
Now, let us discuss the difference between the null hypothesis and the alternative hypothesis.
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1 | The null hypothesis is a statement. There exists no relation between two variables | Alternative hypothesis a statement, there exists some relationship between two measured phenomenon |
2 | Denoted by H | Denoted by H |
3 | The observations of this hypothesis are the result of chance | The observations of this hypothesis are the result of real effect |
4 | The mathematical formulation of the null hypothesis is an equal sign | The mathematical formulation alternative hypothesis is an inequality sign such as greater than, less than, etc. |
Here, some of the examples of the null hypothesis are given below. Go through the below ones to understand the concept of the null hypothesis in a better way.
If a medicine reduces the risk of cardiac stroke, then the null hypothesis should be “the medicine does not reduce the chance of cardiac stroke”. This testing can be performed by the administration of a drug to a certain group of people in a controlled way. If the survey shows that there is a significant change in the people, then the hypothesis is rejected.
Few more examples are:
1). Are there is 100% chance of getting affected by dengue?
Ans: There could be chances of getting affected by dengue but not 100%.
2). Do teenagers are using mobile phones more than grown-ups to access the internet?
Ans: Age has no limit on using mobile phones to access the internet.
3). Does having apple daily will not cause fever?
Ans: Having apple daily does not assure of not having fever, but increases the immunity to fight against such diseases.
4). Do the children more good in doing mathematical calculations than grown-ups?
Ans: Age has no effect on Mathematical skills.
In many common applications, the choice of the null hypothesis is not automated, but the testing and calculations may be automated. Also, the choice of the null hypothesis is completely based on previous experiences and inconsistent advice. The choice can be more complicated and based on the variety of applications and the diversity of the objectives.
The main limitation for the choice of the null hypothesis is that the hypothesis suggested by the data is based on the reasoning which proves nothing. It means that if some hypothesis provides a summary of the data set, then there would be no value in the testing of the hypothesis on the particular set of data.
What is meant by the null hypothesis.
In Statistics, a null hypothesis is a type of hypothesis which explains the population parameter whose purpose is to test the validity of the given experimental data.
Hypothesis testing is defined as a form of inferential statistics, which allows making conclusions from the entire population based on the sample representative.
The null hypothesis is either accepted or rejected in terms of the given data. If P-value is less than α, then the null hypothesis is rejected in favor of the alternative hypothesis, and if the P-value is greater than α, then the null hypothesis is accepted in favor of the alternative hypothesis.
The importance of the null hypothesis is that it provides an approximate description of the phenomena of the given data. It allows the investigators to directly test the relational statement in a research study.
If the result of the chi-square test is bigger than the critical value in the table, then the data does not fit the model, which represents the rejection of the null hypothesis.
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Null Hypothesis , often denoted as H 0, is a foundational concept in statistical hypothesis testing. It represents an assumption that no significant difference, effect, or relationship exists between variables within a population. It serves as a baseline assumption, positing no observed change or effect occurring. The null is t he truth or falsity of an idea in analysis.
In this article, we will discuss the null hypothesis in detail, along with some solved examples and questions on the null hypothesis.
Table of Content
Null hypothesis symbol, formula of null hypothesis, types of null hypothesis, null hypothesis examples, principle of null hypothesis, how do you find null hypothesis, null hypothesis in statistics, null hypothesis and alternative hypothesis, null hypothesis and alternative hypothesis examples, null hypothesis – practice problems.
Null Hypothesis in statistical analysis suggests the absence of statistical significance within a specific set of observed data. Hypothesis testing, using sample data, evaluates the validity of this hypothesis. Commonly denoted as H 0 or simply “null,” it plays an important role in quantitative analysis, examining theories related to markets, investment strategies, or economies to determine their validity.
Null Hypothesis represents a default position, often suggesting no effect or difference, against which researchers compare their experimental results. The Null Hypothesis, often denoted as H 0 asserts a default assumption in statistical analysis. It posits no significant difference or effect, serving as a baseline for comparison in hypothesis testing.
The null Hypothesis is represented as H 0 , the Null Hypothesis symbolizes the absence of a measurable effect or difference in the variables under examination.
Certainly, a simple example would be asserting that the mean score of a group is equal to a specified value like stating that the average IQ of a population is 100.
The Null Hypothesis is typically formulated as a statement of equality or absence of a specific parameter in the population being studied. It provides a clear and testable prediction for comparison with the alternative hypothesis. The formulation of the Null Hypothesis typically follows a concise structure, stating the equality or absence of a specific parameter in the population.
H 0 : μ 1 = μ 2
This asserts that there is no significant difference between the means of two populations or groups.
H 0 : p 1 − p 2 = 0
This suggests no significant difference in proportions between two populations or conditions.
H 0 : σ 1 = σ 2
This states that there’s no significant difference in variances between groups or populations.
H 0 : Variables are independent
This asserts that there’s no association or relationship between categorical variables.
Null Hypotheses vary including simple and composite forms, each tailored to the complexity of the research question. Understanding these types is pivotal for effective hypothesis testing.
The Equality Null Hypothesis, also known as the Simple Null Hypothesis, is a fundamental concept in statistical hypothesis testing that assumes no difference, effect or relationship between groups, conditions or populations being compared.
In some studies, the focus might be on demonstrating that a new treatment or method is not significantly worse than the standard or existing one.
The concept of a superiority null hypothesis comes into play when a study aims to demonstrate that a new treatment, method, or intervention is significantly better than an existing or standard one.
In certain statistical tests, such as chi-square tests for independence, the null hypothesis assumes no association or independence between categorical variables.
In tests like ANOVA (Analysis of Variance), the null hypothesis suggests that there’s no difference in population means across different groups.
The principle of the null hypothesis is a fundamental concept in statistical hypothesis testing. It involves making an assumption about the population parameter or the absence of an effect or relationship between variables.
In essence, the null hypothesis (H 0 ) proposes that there is no significant difference, effect, or relationship between variables. It serves as a starting point or a default assumption that there is no real change, no effect or no difference between groups or conditions.
The null hypothesis is usually formulated to be tested against an alternative hypothesis (H 1 or H [Tex]\alpha [/Tex] ) which suggests that there is an effect, difference or relationship present in the population.
Rejecting the Null Hypothesis occurs when statistical evidence suggests a significant departure from the assumed baseline. It implies that there is enough evidence to support the alternative hypothesis, indicating a meaningful effect or difference. Null Hypothesis rejection occurs when statistical evidence suggests a deviation from the assumed baseline, prompting a reconsideration of the initial hypothesis.
Identifying the Null Hypothesis involves defining the status quotient, asserting no effect and formulating a statement suitable for statistical analysis.
The Null Hypothesis is rejected when statistical tests indicate a significant departure from the expected outcome, leading to the consideration of alternative hypotheses. It occurs when statistical evidence suggests a deviation from the assumed baseline, prompting a reconsideration of the initial hypothesis.
In statistical hypothesis testing, researchers begin by stating the null hypothesis, often based on theoretical considerations or previous research. The null hypothesis is then tested against an alternative hypothesis (Ha), which represents the researcher’s claim or the hypothesis they seek to support.
The process of hypothesis testing involves collecting sample data and using statistical methods to assess the likelihood of observing the data if the null hypothesis were true. This assessment is typically done by calculating a test statistic, which measures the difference between the observed data and what would be expected under the null hypothesis.
In the realm of hypothesis testing, the null hypothesis (H 0 ) and alternative hypothesis (H₁ or Ha) play critical roles. The null hypothesis generally assumes no difference, effect, or relationship between variables, suggesting that any observed change or effect is due to random chance. Its counterpart, the alternative hypothesis, asserts the presence of a significant difference, effect, or relationship between variables, challenging the null hypothesis. These hypotheses are formulated based on the research question and guide statistical analyses.
The null hypothesis (H 0 ) serves as the baseline assumption in statistical testing, suggesting no significant effect, relationship, or difference within the data. It often proposes that any observed change or correlation is merely due to chance or random variation. Conversely, the alternative hypothesis (H 1 or Ha) contradicts the null hypothesis, positing the existence of a genuine effect, relationship or difference in the data. It represents the researcher’s intended focus, seeking to provide evidence against the null hypothesis and support for a specific outcome or theory. These hypotheses form the crux of hypothesis testing, guiding the assessment of data to draw conclusions about the population being studied.
Criteria | Null Hypothesis | Alternative Hypothesis |
---|---|---|
Definition | Assumes no effect or difference | Asserts a specific effect or difference |
Symbol | H | H (or Ha) |
Formulation | States equality or absence of parameter | States a specific value or relationship |
Testing Outcome | Rejected if evidence of a significant effect | Accepted if evidence supports the hypothesis |
Let’s envision a scenario where a researcher aims to examine the impact of a new medication on reducing blood pressure among patients. In this context:
Null Hypothesis (H 0 ): “The new medication does not produce a significant effect in reducing blood pressure levels among patients.”
Alternative Hypothesis (H 1 or Ha): “The new medication yields a significant effect in reducing blood pressure levels among patients.”
The null hypothesis implies that any observed alterations in blood pressure subsequent to the medication’s administration are a result of random fluctuations rather than a consequence of the medication itself. Conversely, the alternative hypothesis contends that the medication does indeed generate a meaningful alteration in blood pressure levels, distinct from what might naturally occur or by random chance.
Mathematics Maths Formulas Probability and Statistics
Example 1: A researcher claims that the average time students spend on homework is 2 hours per night.
Null Hypothesis (H 0 ): The average time students spend on homework is equal to 2 hours per night. Data: A random sample of 30 students has an average homework time of 1.8 hours with a standard deviation of 0.5 hours. Test Statistic and Decision: Using a t-test, if the calculated t-statistic falls within the acceptance region, we fail to reject the null hypothesis. If it falls in the rejection region, we reject the null hypothesis. Conclusion: Based on the statistical analysis, we fail to reject the null hypothesis, suggesting that there is not enough evidence to dispute the claim of the average homework time being 2 hours per night.
Example 2: A company asserts that the error rate in its production process is less than 1%.
Null Hypothesis (H 0 ): The error rate in the production process is 1% or higher. Data: A sample of 500 products shows an error rate of 0.8%. Test Statistic and Decision: Using a z-test, if the calculated z-statistic falls within the acceptance region, we fail to reject the null hypothesis. If it falls in the rejection region, we reject the null hypothesis. Conclusion: The statistical analysis supports rejecting the null hypothesis, indicating that there is enough evidence to dispute the company’s claim of an error rate of 1% or higher.
Q1. A researcher claims that the average time spent by students on homework is less than 2 hours per day. Formulate the null hypothesis for this claim?
Q2. A manufacturing company states that their new machine produces widgets with a defect rate of less than 5%. Write the null hypothesis to test this claim?
Q3. An educational institute believes that their online course completion rate is at least 60%. Develop the null hypothesis to validate this assertion?
Q4. A restaurant claims that the waiting time for customers during peak hours is not more than 15 minutes. Formulate the null hypothesis for this claim?
Q5. A study suggests that the mean weight loss after following a specific diet plan for a month is more than 8 pounds. Construct the null hypothesis to evaluate this statement?
The null hypothesis (H 0 ) and alternative hypothesis (H a ) are fundamental concepts in statistical hypothesis testing. The null hypothesis represents the default assumption, stating that there is no significant effect, difference, or relationship between variables. It serves as the baseline against which the alternative hypothesis is tested. In contrast, the alternative hypothesis represents the researcher’s hypothesis or the claim to be tested, suggesting that there is a significant effect, difference, or relationship between variables. The relationship between the null and alternative hypotheses is such that they are complementary, and statistical tests are conducted to determine whether the evidence from the data is strong enough to reject the null hypothesis in favor of the alternative hypothesis. This decision is based on the strength of the evidence and the chosen level of significance. Ultimately, the choice between the null and alternative hypotheses depends on the specific research question and the direction of the effect being investigated.
What does null hypothesis stands for.
The null hypothesis, denoted as H 0 , is a fundamental concept in statistics used for hypothesis testing. It represents the statement that there is no effect or no difference, and it is the hypothesis that the researcher typically aims to provide evidence against.
A null hypothesis is formed based on the assumption that there is no significant difference or effect between the groups being compared or no association between variables being tested. It often involves stating that there is no relationship, no change, or no effect in the population being studied.
In statistical hypothesis testing, if the p-value (the probability of obtaining the observed results) is lower than the chosen significance level (commonly 0.05), we reject the null hypothesis. This suggests that the data provides enough evidence to refute the assumption made in the null hypothesis.
In research, the null hypothesis represents the default assumption or position that there is no significant difference or effect. Researchers often try to test this hypothesis by collecting data and performing statistical analyses to see if the observed results contradict the assumption.
The null hypothesis (H0) is the default assumption that there is no significant difference or effect. The alternative hypothesis (H1 or Ha) is the opposite, suggesting there is a significant difference, effect or relationship.
Rejecting the null hypothesis implies that there is enough evidence in the data to support the alternative hypothesis. In simpler terms, it suggests that there might be a significant difference, effect or relationship between the groups or variables being studied.
Formulating a null hypothesis often involves considering the research question and assuming that no difference or effect exists. It should be a statement that can be tested through data collection and statistical analysis, typically stating no relationship or no change between variables or groups.
The null hypothesis is commonly symbolized as H 0 in statistical notation.
The null hypothesis serves as a starting point for hypothesis testing, enabling researchers to assess if there’s enough evidence to reject it in favor of an alternative hypothesis.
Rejecting the null hypothesis implies that there is sufficient evidence to support an alternative hypothesis, suggesting a significant effect or relationship between variables.
Various statistical tests, such as t-tests or chi-square tests, are employed to evaluate the validity of the Null Hypothesis in different scenarios.
Similar reads.
The alternative hypothesis.
Adam Hayes, Ph.D., CFA, is a financial writer with 15+ years Wall Street experience as a derivatives trader. Besides his extensive derivative trading expertise, Adam is an expert in economics and behavioral finance. Adam received his master's in economics from The New School for Social Research and his Ph.D. from the University of Wisconsin-Madison in sociology. He is a CFA charterholder as well as holding FINRA Series 7, 55 & 63 licenses. He currently researches and teaches economic sociology and the social studies of finance at the Hebrew University in Jerusalem.
A null hypothesis is a type of statistical hypothesis that proposes that no statistical significance exists in a set of given observations. Hypothesis testing is used to assess the credibility of a hypothesis by using sample data. Sometimes referred to simply as the “null,” it is represented as H 0 .
The null hypothesis, also known as “the conjecture,” is used in quantitative analysis to test theories about markets, investing strategies, and economies to decide if an idea is true or false.
Alex Dos Diaz / Investopedia
A gambler may be interested in whether a game of chance is fair. If it is, then the expected earnings per play come to zero for both players. If it is not, then the expected earnings are positive for one player and negative for the other.
To test whether the game is fair, the gambler collects earnings data from many repetitions of the game, calculates the average earnings from these data, then tests the null hypothesis that the expected earnings are not different from zero.
If the average earnings from the sample data are sufficiently far from zero, then the gambler will reject the null hypothesis and conclude the alternative hypothesis—namely, that the expected earnings per play are different from zero. If the average earnings from the sample data are near zero, then the gambler will not reject the null hypothesis, concluding instead that the difference between the average from the data and zero is explainable by chance alone.
A null hypothesis can only be rejected, not proven.
The null hypothesis assumes that any kind of difference between the chosen characteristics that you see in a set of data is due to chance. For example, if the expected earnings for the gambling game are truly equal to zero, then any difference between the average earnings in the data and zero is due to chance.
Analysts look to reject the null hypothesis because doing so is a strong conclusion. This requires evidence in the form of an observed difference that is too large to be explained solely by chance. Failing to reject the null hypothesis—that the results are explainable by chance alone—is a weak conclusion because it allows that while factors other than chance may be at work, they may not be strong enough for the statistical test to detect them.
An important point to note is that we are testing the null hypothesis because there is an element of doubt about its validity. Whatever information that is against the stated null hypothesis is captured in the alternative (alternate) hypothesis (H 1 ).
For the examples below, the alternative hypothesis would be:
In other words, the alternative hypothesis is a direct contradiction of the null hypothesis.
Here is a simple example: A school principal claims that students in her school score an average of seven out of 10 in exams. The null hypothesis is that the population mean is not 7.0. To test this null hypothesis, we record marks of, say, 30 students ( sample ) from the entire student population of the school (say, 300) and calculate the mean of that sample.
We can then compare the (calculated) sample mean to the (hypothesized) population mean of 7.0 and attempt to reject the null hypothesis. (The null hypothesis here—that the population mean is not 7.0—cannot be proved using the sample data. It can only be rejected.)
Take another example: The annual return of a particular mutual fund is claimed to be 8%. Assume that the mutual fund has been in existence for 20 years. The null hypothesis is that the mean return is not 8% for the mutual fund. We take a random sample of annual returns of the mutual fund for, say, five years (sample) and calculate the sample mean. We then compare the (calculated) sample mean to the (claimed) population mean (8%) to test the null hypothesis.
For the above examples, null hypotheses are:
For the purposes of determining whether to reject the null hypothesis (abbreviated H0), said hypothesis is assumed, for the sake of argument, to be true. Then the likely range of possible values of the calculated statistic (e.g., the average score on 30 students’ tests) is determined under this presumption (e.g., the range of plausible averages might range from 6.2 to 7.8 if the population mean is 7.0).
If the sample average is outside of this range, the null hypothesis is rejected. Otherwise, the difference is said to be “explainable by chance alone,” being within the range that is determined by chance alone.
As an example related to financial markets, assume Alice sees that her investment strategy produces higher average returns than simply buying and holding a stock . The null hypothesis states that there is no difference between the two average returns, and Alice is inclined to believe this until she can conclude contradictory results.
Refuting the null hypothesis would require showing statistical significance, which can be found by a variety of tests. The alternative hypothesis would state that the investment strategy has a higher average return than a traditional buy-and-hold strategy.
One tool that can determine the statistical significance of the results is the p-value. A p-value represents the probability that a difference as large or larger than the observed difference between the two average returns could occur solely by chance.
A p-value that is less than or equal to 0.05 often indicates whether there is evidence against the null hypothesis. If Alice conducts one of these tests, such as a test using the normal model, resulting in a significant difference between her returns and the buy-and-hold returns (the p-value is less than or equal to 0.05), she can then reject the null hypothesis and conclude the alternative hypothesis.
The analyst or researcher establishes a null hypothesis based on the research question or problem they are trying to answer. Depending on the question, the null may be identified differently. For example, if the question is simply whether an effect exists (e.g., does X influence Y?), the null hypothesis could be H 0 : X = 0. If the question is instead, is X the same as Y, the H 0 would be X = Y. If it is that the effect of X on Y is positive, H 0 would be X > 0. If the resulting analysis shows an effect that is statistically significantly different from zero, the null can be rejected.
In finance , a null hypothesis is used in quantitative analysis. It tests the premise of an investing strategy, the markets, or an economy to determine if it is true or false.
For instance, an analyst may want to see if two stocks, ABC and XYZ, are closely correlated. The null hypothesis would be ABC ≠ XYZ.
Statistical hypotheses are tested by a four-step process . The first is for the analyst to state the two hypotheses so that only one can be right. The second is to formulate an analysis plan, which outlines how the data will be evaluated. The third is to carry out the plan and physically analyze the sample data. The fourth and final step is to analyze the results and either reject the null hypothesis or claim that the observed differences are explainable by chance alone.
An alternative hypothesis is a direct contradiction of a null hypothesis. This means that if one of the two hypotheses is true, the other is false.
A null hypothesis states there is no difference between groups or relationship between variables. It is a type of statistical hypothesis and proposes that no statistical significance exists in a set of given observations. “Null” means nothing.
The null hypothesis is used in quantitative analysis to test theories about economies, investing strategies, and markets to decide if an idea is true or false. Hypothesis testing assesses the credibility of a hypothesis by using sample data. It is represented as H 0 and is sometimes simply known as “the null.”
Sage Publishing. “ Chapter 8: Introduction to Hypothesis Testing ,” Page 4.
Sage Publishing. “ Chapter 8: Introduction to Hypothesis Testing ,” Pages 4 to 7.
Sage Publishing. “ Chapter 8: Introduction to Hypothesis Testing ,” Page 7.
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The average direct, indirect and total effects of environmental concern on pro-environmental behavior.
2.1. background, designing observational study as experimental, 2.2. data section, 2.3. technical point and description of the method, 2.4. sensitivity analysis.
Click here to enlarge figure
4. discussion, author contributions, data availability statement, conflicts of interest.
Item Definitions | Categories | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
I Affective dimension (worry) (treatment): Are you concerned about the environmental situation? * | nk/na | nothing | little | much | ||||||
Freq. | % | Freq. | % | Freq. | % | Freq. | % | |||
327 | 1.33 | 836 | 3.40 | 4384 | 17.84 | 19,024 | 77.42 | |||
Conative dimension (M) (mediator) (tolerance, attitudes and intentions): the sum of 2 to 9 | ||||||||||
Would you be in favor of the following environmental protection measures? | nk/na | yes | no | |||||||
Freq. | % | Freq. | % | Freq. | % | |||||
I | Mandatory, subject to a fine, the separation of household waste | 3 | 0.01 | 12,719 | 51.76 | 11,849 | 48.22 | |||
I | Regulate or restrict the abusive water consumption of each dwelling | 2 | 0.01 | 19,532 | 79.49 | 5037 | 20.50 | |||
I | Establishing an environmental tax on the most polluting fuels | 1 | 0.00 | 15,611 | 63.53 | 8959 | 36.46 | |||
I | Restrictive measures on the use of private transport | 2 | 0.01 | 11,363 | 46.25 | 13,206 | 53.75 | |||
I | Introduce an eco-tax on tourism | 3 | 0.01 | 8055 | 32.78 | 16,513 | 67.21 | |||
I | Installation of a renewable energy park (wind, solar) in your municipality despite the effect on the landscape | - | - | 17,576 | 71.53 | 6995 | 28.47 | |||
I | Pay more for the use of alternative energies | 3 | 0.01 | 5587 | 22.74 | 18,981 | 77.25 | |||
I | Reduce noise on main roads (anti-noise panels, sound-reducing paving) | 1 | 0.00 | 20,930 | 85.18 | 3640 | 14.81 | |||
Cognitive dimension (C ) (confounder): the sum of 10 and 11 | ||||||||||
I Environmental campaign: In the last year, have you been aware of any awareness-raising campaigns concerning environmental protection (water, energy, recycling, etc.)? | nk/na | yes | no | |||||||
Freq. | % | Freq. | % | Freq. | % | |||||
704 | 2.87 | 14,762 | 60.08 | 9105 | 37.06 | |||||
I Environmental problem detection: During 2007, have you detected any environmental problems in your environment? | nk/na | yes | no | |||||||
Freq. | % | Freq. | % | Freq. | % | |||||
- | - | 6436 | 26.19 | 18,135 | 73.81 | |||||
Active dimension or pro-environmental behavior (Y) (PEB, outcome): the sum of 12 to 17 | ||||||||||
Do you use any of the following products? | Never % | Sometimes % | Somewhat often % | Whenever possible % | nk/na % | |||||
I Recycled paper? | 66.22 | 18.67 | 7.85 | 5.02 | 2.24 | |||||
I Returnable packaging? | 72.02 | 14.34 | 7.45 | 4.59 | 1.60 | |||||
I Rechargeable batteries? | 49.20 | 20.93 | 16.83 | 11.62 | 1.42 | |||||
Rate the importance they attach to the following elements when buying a new product (household appliance, food product, cleaning product, etc.): | No importance % | Little importance % | Quite importance % | Very importance % | ||||||
I Energy consumption/efficiency | 6.74 | 15.31 | 48.15 | 29.80 | ||||||
I Eco-label/eco-guarantee (organic food) | 18.96 | 29.98 | 35.70 | 15.35 | ||||||
I Local product/proximity of products | 18.71 | 27.01 | 34.65 | 19.63 |
Variables | Categories | Frequency | Cluster_1 | Cluster_2 | Cluster_3 | Cluster_4 |
---|---|---|---|---|---|---|
Are you concerned about the environmental situation? * | Much | 77.42 | 81.91 | 59.58 | 89.40 | 76.27 |
Little | 17.84 | 16.39 | 29.43 | 8.66 | 19.24 | |
nothing, nk/na | 4.73 | 1.70 | 10.99 | 1.94 | 4.48 | |
Household income [net, monthly] | 1. Less than €1100 | 26.33 | 13.58 | 34.57 | 23.98 | 29.48 |
2. From €1101 to €1800 | 27.53 | 27.72 | 26.16 | 28.86 | 27.12 | |
3. From 1801 to 2700 €. | 15.55 | 19.86 | 13.23 | 16.50 | 13.95 | |
4. More than 2700 € | 9.52 | 12.86 | 6.56 | 11.28 | 8.15 | |
5. na/nk | 21.08 | 25.97 | 19.48 | 19.37 | 21.30 | |
Household type | 1. One-person household | 18.30 | 12.86 | 23.00 | 15.65 | 20.40 |
2. Single couple | 23.54 | 19.86 | 23.19 | 23.87 | 25.55 | |
3. Parent-child household | 47.11 | 57.15 | 41.44 | 49.64 | 43.29 | |
4. Household with others | 11.05 | 10.13 | 12.37 | 10.84 | 10.77 | |
Education * | 1. University studies | 15.21 | 21.97 | 11.17 | 18.19 | 12.53 |
2. Baccalaureate and vocational education | 26.87 | 36.34 | 22.81 | 28.73 | 22.93 | |
3. Illiterate up to 1st stage secondary education | 57.62 | 41.69 | 66.02 | 53.08 | 64.54 | |
Municipality size (number of inhabitants) | 1. Provincial capitals and municipalities > 100,000 | 38.65 | 43.61 | 36.74 | 40.61 | 35.35 |
2. From 50,001 to 100,000 | 8.30 | 9.14 | 8.44 | 8.35 | 7.67 | |
3. 20,001 to 50,000 | 15.56 | 16.85 | 15.09 | 15.46 | 15.32 | |
4. 10,000 to 20,000 | 12.33 | 11.50 | 12.74 | 12.14 | 12.67 | |
5. Less than 10,000 | 25.15 | 18.89 | 26.99 | 23.44 | 28.99 | |
N | 26,689 | 4462 | 6170 | 8119 | 7938 |
Stats. | Affective | Conative | Cognitive | PEB | Income | Household Type | Education | Age | Municipality Size |
---|---|---|---|---|---|---|---|---|---|
mean | 2 | 3.81 | 0.60 | 17.83 | 1.89 | 3.08 | 2.57 | 60.12 | 2.94 |
T = 1 | - | 2.89 | 0.34 | 15.74 | 1.60 | 3.07 | 2.76 | 66.54 | 3.15 |
T = 2 | - | 3.69 | 0.50 | 17.70 | 1.94 | 3.06 | 2.58 | 58.71 | 2.94 |
T = 3 | - | 4.85 | 0.97 | 20.06 | 2.10 | 3.10 | 2.38 | 55.12 | 2.74 |
sd | 0.81 | 2.26 | 0.69 | 5.77 | 0.96 | 2.08 | 0.67 | 17.95 | 1.67 |
T = 1 | - | 2.30 | 0.55 | 6.03 | 0.90 | 2.31 | 0.53 | 17.58 | 1.72 |
T = 2 | - | 2.12 | 0.65 | 5.49 | 0.95 | 2.03 | 0.66 | 17.79 | 1.64 |
T = 3 | - | 1.91 | 0.71 | 4.89 | 0.97 | 1.88 | 0.75 | 16.55 | 1.64 |
p50 | 2 | 4 | 0 | 18 | 2 | 3 | 3 | 61 | 3 |
T = 1 | - | 3 | 0 | 16 | 1 | 2 | 3 | 72 | 3 |
T = 2 | - | 4 | 0 | 18 | 2 | 3 | 3 | 58 | 3 |
T = 3 | - | 5 | 1 | 20 | 2 | 3 | 3 | 54 | 3 |
p25 | 1 | 2 | 0 | 14 | 1 | 1 | 2 | 45 | 1 |
T = 1 | - | 1 | 0 | 11 | 1 | 1 | 3 | 53 | 1 |
T = 2 | - | 2 | 0 | 13 | 1 | 2 | 2 | 44 | 1 |
T = 3 | - | 4 | 0 | 17 | 1 | 2 | 2 | 42 | 1 |
p75 | 3 | 5 | 1 | 22 | 3 | 4 | 3 | 75 | 5 |
T = 1 | - | 5 | 1 | 20 | 2 | 4 | 3 | 80 | 5 |
T = 2 | - | 5 | 1 | 22 | 3 | 4 | 3 | 75 | 5 |
T = 3 | - | 6 | 1 | 23 | 3 | 4 | 3 | 69 | 4 |
min | 1 | 0 | 0 | 3 | 1 | 1 | 1 | 19 | 1 |
max | 3 | 8 | 2 | 38 | 4 | 8 | 3 | 98 | 5 |
N | 3300 obs., [1100 per level of environmental concern (T)] |
Variables | Gaussian Identity (OLS) | Gaussian Identity (OLS) | Ordered_logit | |||
---|---|---|---|---|---|---|
Ec. (1) | Ec. (2) | Ec. (3) | ||||
PEB (Y) | (95% Conf. Interval) | Cn (M) | (95% Conf. Interval) | Affective (T) | ( 95% Conf. Interval) | |
Conative dim. (C ) | 0.54 *** | [0.45; 0.64] | ||||
(11.22) | ||||||
Affective dim. | 1.11 *** | [0.82; 1.39] | 0.88 *** | [0.78; 0.97] | ||
(7.60) | (18.11) | |||||
Cognitive dim. (C ) | 0.64 *** | [0.32; 0.97] | 0.31 *** | [0.20; 0.42] | 1.06 *** | [0.96; 1.16] |
(3.90) | (5.75) | (20.63) | ||||
Income | 0.78 *** | [0.53; 1.02] | ||||
(6.18) | ||||||
Household_type | 0.03 | [−0.07; 0.13] | ||||
(0.57) | ||||||
Education | −0.60 *** | [−0.93; −0.27] | ||||
(−3.59) | ||||||
Age | −0.003 | [−0.01; 0.009] | ||||
(−0.53) | ||||||
Municipality size | −0.031 | [−0.15; −0.091] | ||||
(−0.50) | ||||||
_cons | 13.43 *** | [11.89; 14.97] | 1.85 *** | [1.65; 2.05] | ||
(17.08) | (18.26) | |||||
/cut1_ec.(3) | −0.16 *** | [−2.24; −0.07] | ||||
(−3.70) | ||||||
/cut2_ec.(3) | 1.39 *** | [1.30; 1.49] | ||||
(28.29) | ||||||
var(e.peb)_ec.(1) | 27.37 *** | [26.01; 29.02] | ||||
(35.80) | ||||||
var(e.cn)_ec(2) | 4.45 *** | [4.28; 4.63] | ||||
(49.24) | ||||||
N | 2638 | 3300 | 3300 |
1 | (accessed on 20 March 2024). |
2 | |
3 | |
4 | (accessed on 18 June 2018). |
5 |
Pro-Environmental Behavior | Environmental Concern as the Treatment or Exposure Variable | |||||
---|---|---|---|---|---|---|
Nothing | Little | Much | ||||
Mean | std. dev. | Mean | std. dev. | Mean | std. dev. | |
PEB [min = 3; max = 38] | 15.74 | 6.03 | 17.70 | 5.49 | 20.06 | 4.89 |
N (3300 obs.) | 1100 | 1100 | 1100 | |||
χ Pearson’s test = 486.75; Pr. = 0.000; correlate = 0.30 * |
Variables | Statistics | Conative Dimension [Mediator (M)] | Cognitive Dimension [Confounder (C )] |
---|---|---|---|
Affective dimension (T) | Pearson’s test | 474.71 (pr. = 0.000) | 495.58 (pr. = 0.000) |
Correlate | 0.35 * | 0.36 * | |
PEB (Y) (outcome) (active dimension) | Pearson’s test | 760.42 (pr. = 0.000) | 290 (pr. = 0.000) |
Correlate | 0.29 * | 0.24 * | |
Conative dimension (M) | Pearson’s test | - | 219,22 (pr. = 0.000) |
Correlate | - | 0.21 * |
Causal Effects under Different Hypothesis | DAG_1 | DAG_2 | ||||
---|---|---|---|---|---|---|
no Confounder (no Cognitive), no Covariates | with Confounder (Cognitive), no Covariates | |||||
Estimate (1) | 95% CI (2) | p-Value (3) | Estimate (4) | 95% CI (5) | p-Value (6) | |
A. Estimations under the hypothesis of simple treatment and control conditions, i. e., the lowest level of affection (1) vs. the other two conditions (average) | ||||||
ACME | 0.54 | [0.47–0.63] | <2 × 10 *** | 0.44 | [0.34–0.52] | <2 × 10 *** |
ADE | 1.60 | [1.34–1.81] | <2 × 10 *** | 1.31 | [1.05–1.52] | <2 × 10 *** |
TE | 2.14 | [1.90–2.43] | <2 × 10 *** | 1.75 | [1.52–1.96] | <2 × 10 *** |
PM | 0.25 | [0.21–0.31] | <2 × 10 *** | 0.25 | [0.19–0.30] | <2 × 10 *** |
B. Estimations under the hypothesis of the lowest value of the treatment as control (=1 = the lowest level of concern) compares to the highest level of the exposure (treatment = 3 = level of concern) | ||||||
ACME | 1.07 | [0.89–1.31] | <2 × 10 *** | 0.88 | [0.71–1.10] | <2 × 10 *** |
ADE | 3.26 | [2.79–3.72] | <2 × 10 *** | 2.65 | [2.21–3.20] | <2 × 10 *** |
TE | 4.33 | [3.89–4.82] | <2 × 10 *** | 3.54 | [3.10–4.05] | <2 × 10 *** |
PM | 0.24 | [0.20–0.31] | <2 × 10 *** | 0.25 | [0.18–0.31] | <2 × 10 *** |
C. Estimations under the hypothesis of the intermediate value of the treatment as control (=2 = intermediate level of concern) compares to the highest level of the exposure (treatment = 3 = level of concern) | ||||||
ACME | 0.53 | [0.43–0.64] | <2 × 10 *** | 0.45 | [0.37–0.56] | <2 × 10 *** |
ADE | 1.61 | [1.33–1.84] | <2 × 10 *** | 1.33 | [1.12–1.58] | <2 × 10 *** |
TE | 2.14 | [1.91–2.38] | <2 × 10 *** | 1.78 | [1.57–2.02] | <2 × 10 *** |
PM | 0.24 | [0.20–0.30] | <2 × 10 *** | 0.25 | [0.20–0.32] | <2 × 10 *** |
Sample | 3300 obs. | |||||
Adjusted for covariates: income, household type, education, age, and size of the municipality: | ||||||
D. Estimations under the hypothesis of the lowest value of the treatment as control (=1 = the lowest level of concern) compares to the highest level of the exposure (treatment = 3 = level of concern) | ||||||
ACME | 1.04 | [0.82–1.29] | <2 × 10 *** | 0.96 | [0.76–1.15] | <2 × 10 *** |
ADE | 2.60 | [2.14–3.10] | <2 × 10 *** | 2.24 | [1.63–2.78] | <2 × 10 *** |
TE | 3.64 | [3.16–4.16] | <2 × 10 *** | 3.21 | [2.63–3.76] | <2 × 10 *** |
PM | 0.28 | [0.22–0.36] | <2 × 10 *** | 0.29 | [0.23–0.39] | <2 × 10 *** |
E. Estimations under the hypothesis of the intermediate value of the treatment as control (=2 = intermediate level of concern) compares to the highest level of the exposure (treatment = 3 = level of concern) | ||||||
ACME | 0.52 | [0.42–0.63] | <2 × 10 *** | 0.47 | [0.37–0.58] | <2 × 10 *** |
ADE | 1.29 | [1.00–1.59] | <2 × 10 *** | 1.10 | [0.79–1.40] | <2 × 10 *** |
TE | 1.81 | [1.57–2.13] | <2 × 10 *** | 1.57 | [1.26–1.92] | <2 × 10 *** |
PM | 0.28 | [0.23–0.38] | <2 × 10 *** | 0.30 | [0.24–0.37] | <2 × 10 *** |
Sample | 3300 |
Causal Effects | DAG_1 | DAG_2 | ||||
---|---|---|---|---|---|---|
Estimate (1) | 95% CI (2) | p-Value (3) | Estimate (4) | 95% CI (5) | p-Value (6) | |
ACME [control] | 0.71 | [0.48–0.93] | <2 × 10 *** | 0.65 | [0.42–0.88] | <2 × 10 *** |
ACME [treated] | 0.60 | [0.47–0.78] | <2 × 10 *** | 0.55 | [0.40–0.70] | <2 × 10 *** |
ADE [control] | 1.48 | [1.10–1.94] | <2 × 10 *** | 1.32 | [0.98–1.72] | <2 × 10 *** |
ADE [treated] | 1.38 | [1.05–1.75] | <2 × 10 *** | 1.22 | [0.94–1.57] | <2 × 10 *** |
TE | 2.09 | [1.61–2.59] | <2 × 10 *** | 1.87 | [1.49–2.33] | <2 × 10 *** |
PM [control] | 0.34 | [0.26–0.42] | <2 × 10 *** | 0.35 | [0.24–0.42] | <2 × 10 *** |
PM [treated] | 0.29 | [0.23–0.36] | <2 × 10 *** | 0.29 | [0.23–0.36] | <2 × 10 *** |
ACME [average] | 0.66 | [0.48–0.86] | <2 × 10 *** | 0.60 | [0.41–0.79] | <2 × 10 *** |
ADE [average] | 1.43 | [1.08–1.85] | <2 × 10 *** | 1.27 | [0.96–1.65] | <2 × 10 *** |
PM [average] | 0.31 | [0.25–0.39] | <2 × 10 *** | 0.32 | [0.24–0.38] | <2 × 10 *** |
TMint-test for the null hypothesis, [(ACME (3)) − (ACME (1)) = 0]; N = 2629 Obs. | ||||||
(ACME (3)) − (ACME (1)) = −0.10, p-value = 0.1; alternative hypothesis: true ACME (3) − ACME (1) is not equal to 95 percent confidence interval: [−0.20; 0.018] | (ACME (3)) − (ACME (1)) = −0.09, p-value = 0.04; alternative hypothesis: true ACME (3) − ACME (1) is not equal to 95 percent confidence interval: [−0.18; −0.003] |
Covariates/ Moderators | DAG_1 | DAG_2 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
Average Effects for the Average Treatment (Worry) Level | Adjusting for Treatment (Worry) to the Highest Level (t = 3) | Modmed-Test on Average(5) | Average Effects for the Average Treatment (Worry) Level | Adjusting for Treatment (Worry) to the Highest Level (t = 3) | Modmed-Test on Average (10) | |||||
ACME (1) | ADE (2) | ACME (3) | ADE (4) | Diff Diff | ACME (6) | ADE (7) | ACME (8) | ADE (9) | Diff Diff | |
Income (mean) | 0.51 *** | 1.24 *** | 1.56 *** | 3.72 *** | Income: 0.60; [0.10] CI [0.95] = [−0.65; 0.81] ; [2.2 × 10 ] CI [0.95] = [−1.08; 3.13] | 0.46 *** | 1.10 *** | 1.40 *** | 3.27 *** | Income: 0.14; [0.24] CI [0.95] = [−0.10; 0.39] 0.99; [0.02] CI [0.95] = [−0.10; 0.39] |
1. Income_ | 0.56 *** | 1.53 *** | 1.73 *** | 4.56 *** | 0.51 *** | 1.40 *** | 1.53 *** | 4.17 *** | ||
4. Income_ | 0.39 *** | 0.60 | 1.13 *** | 1.71 | 0.34 *** | 0.38 | 1.06 *** | 1.14 *** | ||
Age (average) | 0.54 *** | 1.23 *** | 1.62 *** | 3.78 *** | Age: −0.07: [0.44] CI [0.95] = [−0.25; 0.13] 0.28; [0.30] CI [0.95] = [−0.14; 0.71] | 0.48 *** | 1.10 *** | 1.48 *** | 3.27 *** | Age: −0.06; [0.42] CI [0.95] = [−0.30; 0.14] ; [0.26] CI [0.95] = [−0.17; 0.75] |
Age | 0.54 *** | 1.14 *** | 1.58 *** | 3.58 *** | 0.47 *** | 0.93 *** | 1.43 *** | 2.96 *** | ||
Age | 0.45 *** | 1.42 *** | 1.37 *** | 4.32 *** | 0.41 *** | 1.21 | 1.20 *** | 3.71 *** | ||
Household (mean) | 0.51 *** | 1.28 *** | 1.57 *** | 3.75 *** | Household type: 0.021; [0.72] CI [0.95] = [−0.13; 0.15] 0.22; [0.32] CI [0.95] = [−0.15; 0.65] | 0.47 *** | 1.11 *** | 1.43 *** | 3.31 *** | Household type: 0.03; [0.66] CI [0.95] = [−0.13; 0.17] 0.259; [0.30] CI [0.95] = [−0.15; 0.73] |
1. Househ | 0.54 *** | 1.55 *** | 1.64 *** | 4.68 *** | 0.48 *** | 1.40 *** | 1.47 *** | 4.37 *** | ||
2. Househ | 0.52 *** | 1.31 *** | 1.57 *** | 3.94 *** | 0.48 *** | 1.17 *** | 1.45 *** | 3.56 *** | ||
3. Househ | 0.50 *** | 1.10 *** | 1.51 *** | 3.36 *** | 0.44 *** | 0.92 *** | 1.36 *** | 2.78 *** | ||
4. Househ | 0.49 *** | 0.92 *** | 1.44 *** | 2.70 *** | 0.43 *** | 0.66 * | 1.30 *** | 1.92 * | ||
Education (mean) | 0.51 *** | 1.25 *** | 1.55 *** | 3.75 *** | Education: −0.11; [0.48] CI [0.95] = [−0.38; 0.16] −0.57; [0.16] CI [0.95] = [−1.21; 0.12] −0.19; [0.10] CI [0.95] = [−0.39; 0.03] −1.22; [2.2 × 10 ] CI [0.95]= [−1.83; −0.49] −0.10; [0.24] CI [0.95] = [−0.27; 0.03] [0.06] CI [0.95] = [−0.99; 0.05] | 0.46 *** | 0.11 *** | 1.41 *** | 3.31 *** | Education: −0.07; [0.56] CI [0.95] = [−0.29; 0.13] −0.63; [0.10] CI [0.95] = [−1.39; 0.18] −0.19; [0.12] CI [0.95] = [−0.44; 0.08] −1.26; [2.2 × 10 ] CI [0.95] = [−2.02; −0.60] −0.11; [0.14] CI [0.95] = [−0.26; 0.03] −0.57; [0.02] CI [0.95] = [−1.09; −0.06] |
1. Edu_university | 0.36 *** | 0.26 | 1.06 *** | 0.91 | 0.33 *** | 0.12 | 0.99 *** | 0.31 | ||
2. Edu_Bacc. + voc. | 1.35 *** | 2.71 *** | 1.35 *** | 2.71 *** | 0.40 *** | 0.73 *** | 1.25 *** | 2.26 *** | ||
3. Edu_Illit. + prim. | 0.55 *** | 1.49 *** | 1.68 *** | 4.50 *** | 0.51 *** | 1.35 *** | 1.52 *** | 4.00 *** | ||
Municipal size (mean) | 1.56 *** | 3.80 *** | 1.56 *** | 3.83 *** | Municipality size: −0.049; [0.64] CI [0.95] = [−0.26; 0.17] 0.049; [0.86] CI [0.95] = [−0.47; 0.57] 0.001; [0.98] CI [0.95] = [−0.14; 0.13] 0.017; [0.92] CI [0.95] = [−0.43; 0.36] −0.02; [0.78] CI [0.95] = [−0.27; 0.18] ; [0.84] CI [0.95] = [−0.50; 0.49] | 0.48 *** | 1.10 *** | 1.43 *** | 3.29 *** | Municipality size: −0.03; [0.74] CI [0.95] = [−0.24; 0.16] 0.08; [0.80] CI [0.95] = [−0.53; 0.64] −0.02; [0.66] CI [0.95] = [−0.16; 0.12] 0.05; [0.84] CI [0.95] = [−0.33; 0.44] −0.03; [0.76] CI [0.95] = [−0.26; 0.15] 0.02; [0.99] CI [0.95] = [−0.55; 0.59] |
1. >100,000 | 0.48 *** | 1.29 *** | 1.48 *** | 3.97 *** | 0.44 *** | 1.13 *** | 1.33 *** | 3.44 *** | ||
2. [50,000–100,000] | 0.50 *** | 1.29 *** | 1.51 *** | 3.82 *** | 0.45 *** | 1.11 *** | 1.40 *** | 3.35 *** | ||
3. [20,000–50,000] | 0.52 *** | 1.26 *** | 1.54 *** | 3.80 *** | 0.46 *** | 1.11 *** | 1.41 *** | 3.36 *** | ||
4. [10,000–20,000] | 0.53 *** | 1.25 *** | 1.60 *** | 3.77 *** | 0.49 *** | 1.05 *** | 1.45 *** | 3.29 *** | ||
5. <10,000 | 0.55 *** | 1.22 *** | 1.63 *** | 3.70 *** | 0.51 *** | 1.05 *** | 1.51 *** | 3.19 *** | ||
Sample | 2629 |
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Hernández-Alemán, A.; Cruz-Pérez, N.; Santamarta, J.C. The Average Direct, Indirect and Total Effects of Environmental Concern on Pro-Environmental Behavior. Land 2024 , 13 , 1229. https://doi.org/10.3390/land13081229
Hernández-Alemán A, Cruz-Pérez N, Santamarta JC. The Average Direct, Indirect and Total Effects of Environmental Concern on Pro-Environmental Behavior. Land . 2024; 13(8):1229. https://doi.org/10.3390/land13081229
Hernández-Alemán, Anastasia, Noelia Cruz-Pérez, and Juan C. Santamarta. 2024. "The Average Direct, Indirect and Total Effects of Environmental Concern on Pro-Environmental Behavior" Land 13, no. 8: 1229. https://doi.org/10.3390/land13081229
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H 0 (Null Hypothesis): Population parameter =, ≤, ≥ some value. H A (Alternative Hypothesis): Population parameter <, >, ≠ some value. Note that the null hypothesis always contains the equal sign. We interpret the hypotheses as follows: Null hypothesis: The sample data provides no evidence to support some claim being made by an individual.
When your sample contains sufficient evidence, you can reject the null and conclude that the effect is statistically significant. Statisticians often denote the null hypothesis as H 0 or H A.. Null Hypothesis H 0: No effect exists in the population.; Alternative Hypothesis H A: The effect exists in the population.; In every study or experiment, researchers assess an effect or relationship.
To distinguish it from other hypotheses, the null hypothesis is written as H 0 (which is read as "H-nought," "H-null," or "H-zero"). A significance test is used to determine the likelihood that the results supporting the null hypothesis are not due to chance. A confidence level of 95% or 99% is common. Keep in mind, even if the confidence level is high, there is still a small chance the ...
The null hypothesis and the alternative hypothesis are types of conjectures used in statistical tests to make statistical inferences, which are formal methods of reaching conclusions and separating scientific claims from statistical noise.. The statement being tested in a test of statistical significance is called the null hypothesis. The test of significance is designed to assess the strength ...
Step 1: Figure out the hypothesis from the problem. The hypothesis is usually hidden in a word problem, and is sometimes a statement of what you expect to happen in the experiment. The hypothesis in the above question is "I expect the average recovery period to be greater than 8.2 weeks.". Step 2: Convert the hypothesis to math.
The null and alternative hypotheses offer competing answers to your research question. When the research question asks "Does the independent variable affect the dependent variable?": The null hypothesis ( H0) answers "No, there's no effect in the population.". The alternative hypothesis ( Ha) answers "Yes, there is an effect in the ...
Null Hypothesis Examples. "Hyperactivity is unrelated to eating sugar " is an example of a null hypothesis. If the hypothesis is tested and found to be false, using statistics, then a connection between hyperactivity and sugar ingestion may be indicated. A significance test is the most common statistical test used to establish confidence in a ...
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. If you suspect that girls take longer to get ready for school than boys, then: Alternative: girls time > boys time. Null: girls time <= boys time.
The actual test begins by considering two hypotheses.They are called the null hypothesis and the alternative hypothesis.These hypotheses contain opposing viewpoints. H 0, 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.
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): There's an effect in the population. The effect is usually the effect of the independent variable on the dependent ...
Write a research null hypothesis as a statement that the studied variables have no relationship to each other, or that there's no difference between 2 groups. Write a statistical null hypothesis as a mathematical equation, such as. μ 1 = μ 2 {\displaystyle \mu _ {1}=\mu _ {2}} if you're comparing group means.
The actual test begins by considering two hypotheses.They are called the null hypothesis and the alternative hypothesis.These hypotheses contain opposing viewpoints. \(H_0\): 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.
The null hypothesis states no relationship exists between the two variables being studied (i.e., one variable does not affect the other). The null hypothesis is the statement that a researcher or an investigator wants to disprove.
An example of the null hypothesis is that light color has no effect on plant growth. The null hypothesis (H 0) is the hypothesis that states there is no statistical difference between two sample sets. In other words, it assumes the independent variable does not have an effect on the dependent variable in a scientific experiment.
10.1 - Setting the Hypotheses: Examples. A significance test examines whether the null hypothesis provides a plausible explanation of the data. The null hypothesis itself does not involve the data. It is a statement about a parameter (a numerical characteristic of the population). These population values might be proportions or means or ...
This page titled 9.2: Null and Alternative Hypotheses is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform.
The actual test begins by considering two hypotheses.They are called the null hypothesis and the alternative hypothesis.These hypotheses contain opposing viewpoints. H 0: 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.
The null hypothesis is often made up of several assumptions, including: the main assumption (the one we are testing); other assumptions (e.g., technical assumptions) that we need to make in order to set up the hypothesis test. For instance, in Example 2 above (reliability of a production plant), the main assumption is that the expected number ...
Null hypothesis is used to make decisions based on data and by using statistical tests. Null hypothesis is represented using H o and it states that there is no difference between the characteristics of two samples. Null hypothesis is generally a statement of no difference. The rejection of null hypothesis is equivalent to the acceptance of the ...
The null hypothesis is a statement. There exists no relation between two variables: Alternative hypothesis a statement, there exists some relationship between two measured phenomenon. 2. Denoted by H 0: Denoted by H 1. 3. The observations of this hypothesis are the result of chance:
The null hypothesis ( H0. H 0. ) 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. The alternative hypothesis ( Ha. H a. ) is a claim about the population that is contradictory to H0. H 0.
Null Hypothesis, often denoted as H0, is a foundational concept in statistical hypothesis testing. It represents an assumption that no significant difference, effect, or relationship exists between variables within a population. It serves as a baseline assumption, positing no observed change or effect occurring.
Null Hypothesis: A null hypothesis is a type of hypothesis used in statistics that proposes that no statistical significance exists in a set of given observations. The null hypothesis attempts to ...
Example 1: One-Way ANOVA. Suppose we want to know whether or not three different exam prep programs lead to different mean scores on a certain exam. To test this, we recruit 30 students to participate in a study and split them into three groups. The students in each group are randomly assigned to use one of the three exam prep programs for the ...
The test.TMint rejects the null hypothesis, [(ACME (3)) - (ACME (1)) = 0], which means that average mediation effect depends on the baseline treatment status for both DAGs. ... Disclaimer/Publisher's Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not ...