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.
Statistics By Jim
Making statistics intuitive
By Jim Frost 6 Comments
The null hypothesis in statistics states that there is no difference between groups or no relationship between variables. It is one of two mutually exclusive hypotheses about a population in a hypothesis test.
In every study or experiment, researchers assess an effect or relationship. This effect can be the effectiveness of a new drug, building material, or other intervention that has benefits. There is a benefit or connection that the researchers hope to identify. Unfortunately, no effect may exist. In statistics, we call this lack of an effect the null hypothesis. Researchers assume that this notion of no effect is correct until they have enough evidence to suggest otherwise, similar to how a trial presumes innocence.
In this context, the analysts don’t necessarily believe the null hypothesis is correct. In fact, they typically want to reject it because that leads to more exciting finds about an effect or relationship. The new vaccine works!
You can think of it as the default theory that requires sufficiently strong evidence to reject. Like a prosecutor, researchers must collect sufficient evidence to overturn the presumption of no effect. Investigators must work hard to set up a study and a data collection system to obtain evidence that can reject the null hypothesis.
Related post : What is an Effect in Statistics?
Null hypotheses start as research questions that the investigator rephrases as a statement indicating there is no effect or relationship.
Does the vaccine prevent infections? | The vaccine does not affect the infection rate. |
Does the new additive increase product strength? | The additive does not affect mean product strength. |
Does the exercise intervention increase bone mineral density? | The intervention does not affect bone mineral density. |
As screen time increases, does test performance decrease? | There is no relationship between screen time and test performance. |
After reading these examples, you might think they’re a bit boring and pointless. However, the key is to remember that the null hypothesis defines the condition that the researchers need to discredit before suggesting an effect exists.
Let’s see how you reject the null hypothesis and get to those more exciting findings!
So, you want to reject the null hypothesis, but how and when can you do that? To start, you’ll need to perform a statistical test on your data. The following is an overview of performing a study that uses a hypothesis test.
The first step is to devise a research question and the appropriate null hypothesis. After that, the investigators need to formulate an experimental design and data collection procedures that will allow them to gather data that can answer the research question. Then they collect the data. For more information about designing a scientific study that uses statistics, read my post 5 Steps for Conducting Studies with Statistics .
After data collection is complete, statistics and hypothesis testing enter the picture. Hypothesis testing takes your sample data and evaluates how consistent they are with the null hypothesis. The p-value is a crucial part of the statistical results because it quantifies how strongly the sample data contradict the null hypothesis.
When the sample data provide sufficient evidence, you can reject the null hypothesis. In a hypothesis test, this process involves comparing the p-value to your significance level .
Reject the null hypothesis when the p-value is less than or equal to your significance level. Your sample data favor the alternative hypothesis, which suggests that the effect exists in the population. For a mnemonic device, remember—when the p-value is low, the null must go!
When you can reject the null hypothesis, your results are statistically significant. Learn more about Statistical Significance: Definition & Meaning .
Conversely, when the p-value is greater than your significance level, you fail to reject the null hypothesis. The sample data provides insufficient data to conclude that the effect exists in the population. When the p-value is high, the null must fly!
Note that failing to reject the null is not the same as proving it. For more information about the difference, read my post about Failing to Reject the Null .
That’s a very general look at the process. But I hope you can see how the path to more exciting findings depends on being able to rule out the less exciting null hypothesis that states there’s nothing to see here!
Let’s move on to learning how to write the null hypothesis for different types of effects, relationships, and tests.
Related posts : How Hypothesis Tests Work and Interpreting P-values
The null hypothesis varies by the type of statistic and hypothesis test. Remember that inferential statistics use samples to draw conclusions about populations. Consequently, when you write a null hypothesis, it must make a claim about the relevant population parameter . Further, that claim usually indicates that the effect does not exist in the population. Below are typical examples of writing a null hypothesis for various parameters and hypothesis tests.
Related posts : Descriptive vs. Inferential Statistics and Populations, Parameters, and Samples in Inferential Statistics
T-tests and ANOVA assess the differences between group means. For these tests, the null hypothesis states that there is no difference between group means in the population. In other words, the experimental conditions that define the groups do not affect the mean outcome. Mu (µ) is the population parameter for the mean, and you’ll need to include it in the statement for this type of study.
For example, an experiment compares the mean bone density changes for a new osteoporosis medication. The control group does not receive the medicine, while the treatment group does. The null states that the mean bone density changes for the control and treatment groups are equal.
Proportions tests assess the differences between group proportions. For these tests, the null hypothesis states that there is no difference between group proportions. Again, the experimental conditions did not affect the proportion of events in the groups. P is the population proportion parameter that you’ll need to include.
For example, a vaccine experiment compares the infection rate in the treatment group to the control group. The treatment group receives the vaccine, while the control group does not. The null states that the infection rates for the control and treatment groups are equal.
Some studies assess the relationship between two continuous variables rather than differences between groups.
In these studies, analysts often use either correlation or regression analysis . For these tests, the null states that there is no relationship between the variables. Specifically, it says that the correlation or regression coefficient is zero. As one variable increases, there is no tendency for the other variable to increase or decrease. Rho (ρ) is the population correlation parameter and beta (β) is the regression coefficient parameter.
For example, a study assesses the relationship between screen time and test performance. The null states that there is no correlation between this pair of variables. As screen time increases, test performance does not tend to increase or decrease.
For all these cases, the analysts define the hypotheses before the study. After collecting the data, they perform a hypothesis test to determine whether they can reject the null hypothesis.
The preceding examples are all for two-tailed hypothesis tests. To learn about one-tailed tests and how to write a null hypothesis for them, read my post One-Tailed vs. Two-Tailed Tests .
Related post : Understanding Correlation
Neyman, J; Pearson, E. S. (January 1, 1933). On the Problem of the most Efficient Tests of Statistical Hypotheses . Philosophical Transactions of the Royal Society A . 231 (694–706): 289–337.
January 11, 2024 at 2:57 pm
Thanks for the reply.
January 10, 2024 at 1:23 pm
Hi Jim, In your comment you state that equivalence test null and alternate hypotheses are reversed. For hypothesis tests of data fits to a probability distribution, the null hypothesis is that the probability distribution fits the data. Is this correct?
January 10, 2024 at 2:15 pm
Those two separate things, equivalence testing and normality tests. But, yes, you’re correct for both.
Hypotheses are switched for equivalence testing. You need to “work” (i.e., collect a large sample of good quality data) to be able to reject the null that the groups are different to be able to conclude they’re the same.
With typical hypothesis tests, if you have low quality data and a low sample size, you’ll fail to reject the null that they’re the same, concluding they’re equivalent. But that’s more a statement about the low quality and small sample size than anything to do with the groups being equal.
So, equivalence testing make you work to obtain a finding that the groups are the same (at least within some amount you define as a trivial difference).
For normality testing, and other distribution tests, the null states that the data follow the distribution (normal or whatever). If you reject the null, you have sufficient evidence to conclude that your sample data don’t follow the probability distribution. That’s a rare case where you hope to fail to reject the null. And it suffers from the problem I describe above where you might fail to reject the null simply because you have a small sample size. In that case, you’d conclude the data follow the probability distribution but it’s more that you don’t have enough data for the test to register the deviation. In this scenario, if you had a larger sample size, you’d reject the null and conclude it doesn’t follow that distribution.
I don’t know of any equivalence testing type approach for distribution fit tests where you’d need to work to show the data follow a distribution, although I haven’t looked for one either!
February 20, 2022 at 9:26 pm
Is a null hypothesis regularly (always) stated in the negative? “there is no” or “does not”
February 23, 2022 at 9:21 pm
Typically, the null hypothesis includes an equal sign. The null hypothesis states that the population parameter equals a particular value. That value is usually one that represents no effect. In the case of a one-sided hypothesis test, the null still contains an equal sign but it’s “greater than or equal to” or “less than or equal to.” If you wanted to translate the null hypothesis from its native mathematical expression, you could use the expression “there is no effect.” But the mathematical form more specifically states what it’s testing.
It’s the alternative hypothesis that typically contains does not equal.
There are some exceptions. For example, in an equivalence test where the researchers want to show that two things are equal, the null hypothesis states that they’re not equal.
In short, the null hypothesis states the condition that the researchers hope to reject. They need to work hard to set up an experiment and data collection that’ll gather enough evidence to be able to reject the null condition.
February 15, 2022 at 9:32 am
Dear sir I always read your notes on Research methods.. Kindly tell is there any available Book on all these..wonderfull Urgent
ASC Chat is usually available at the following times ( Pacific Time):
Days | Hours (Pacific time) |
---|---|
Mon. | 9 am - 8 pm |
Tue. | 7 am - 1 pm 3 pm - 10 pm |
Wed. | 7 am - 1 pm 3 pm - 10 pm |
Thurs. | 7 am - 1 pm 2 pm - 10 pm |
Fri. | 9 am - 1 pm 3 pm - 5 pm 6 pm - 8 pm |
Sat. | 7 am - 1 pm 6 pm - 9 pm |
Sun. | 10 am - 1 pm 5 pm - 9 pm |
If there is not a coach on duty, submit your question via one of the below methods:
928-440-1325
Ask a Coach
Search our FAQs on the Academic Success Center's Ask a Coach page.
Once you have developed a clear and focused research question or set of research questions, you’ll be ready to conduct further research, a literature review, on the topic to help you make an educated guess about the answer to your question(s). This educated guess is called a hypothesis.
In research, there are two types of hypotheses: null and alternative. They work as a complementary pair, each stating that the other is wrong.
Null Hypothesis: H 0 : There is no difference in the salary of factory workers based on gender. Alternative Hypothesis : H a : Male factory workers have a higher salary than female factory workers.
Null Hypothesis : H 0 : There is no relationship between height and shoe size. Alternative Hypothesis : H a : There is a positive relationship between height and shoe size.
Null Hypothesis : H 0 : Experience on the job has no impact on the quality of a brick mason’s work. Alternative Hypothesis : H a : The quality of a brick mason’s work is influenced by on-the-job experience.
Content preview.
Arcu felis bibendum ut tristique et egestas quis:
6a.1 - introduction to hypothesis testing, basic terms section .
The first step in hypothesis testing is to set up two competing hypotheses. The hypotheses are the most important aspect. If the hypotheses are incorrect, your conclusion will also be incorrect.
The two hypotheses are named the null hypothesis and the alternative hypothesis.
The goal of hypothesis testing is to see if there is enough evidence against the null hypothesis. In other words, to see if there is enough evidence to reject the null hypothesis. If there is not enough evidence, then we fail to reject the null hypothesis.
Consider the following example where we set up these hypotheses.
A man, Mr. Orangejuice, goes to trial and is tried for the murder of his ex-wife. He is either guilty or innocent. Set up the null and alternative hypotheses for this example.
Putting this in a hypothesis testing framework, the hypotheses being tested are:
Let's set up the null and alternative hypotheses.
\(H_0\colon \) Mr. Orangejuice is innocent
\(H_a\colon \) Mr. Orangejuice is guilty
Remember that we assume the null hypothesis is true and try to see if we have evidence against the null. Therefore, it makes sense in this example to assume the man is innocent and test to see if there is evidence that he is guilty.
We want to know the answer to a research question. We determine our null and alternative hypotheses. Now it is time to make a decision.
The decision is either going to be...
Consider the following table. The table shows the decision/conclusion of the hypothesis test and the unknown "reality", or truth. We do not know if the null is true or if it is false. If the null is false and we reject it, then we made the correct decision. If the null hypothesis is true and we fail to reject it, then we made the correct decision.
Decision | Reality | |
---|---|---|
\(H_0\) is true | \(H_0\) is false | |
Reject \(H_0\), (conclude \(H_a\)) | Correct decision | |
Fail to reject \(H_0\) | Correct decision |
So what happens when we do not make the correct decision?
When doing hypothesis testing, two types of mistakes may be made and we call them Type I error and Type II error. If we reject the null hypothesis when it is true, then we made a type I error. If the null hypothesis is false and we failed to reject it, we made another error called a Type II error.
Decision | Reality | |
---|---|---|
\(H_0\) is true | \(H_0\) is false | |
Reject \(H_0\), (conclude \(H_a\)) | Type I error | Correct decision |
Fail to reject \(H_0\) | Correct decision | Type II error |
The “reality”, or truth, about the null hypothesis is unknown and therefore we do not know if we have made the correct decision or if we committed an error. We can, however, define the likelihood of these events.
\(\alpha\) and \(\beta\) are probabilities of committing an error so we want these values to be low. However, we cannot decrease both. As \(\alpha\) decreases, \(\beta\) increases.
A man, Mr. Orangejuice, goes to trial and is tried for the murder of his ex-wife. He is either guilty or not guilty. We found before that...
Interpret Type I error, \(\alpha \), Type II error, \(\beta \).
As you can see here, the Type I error (putting an innocent man in jail) is the more serious error. Ethically, it is more serious to put an innocent man in jail than to let a guilty man go free. So to minimize the probability of a type I error we would choose a smaller significance level.
An inspector has to choose between certifying a building as safe or saying that the building is not safe. There are two hypotheses:
Set up the null and alternative hypotheses. Interpret Type I and Type II error.
\( H_0\colon\) Building is not safe vs \(H_a\colon \) Building is safe
Decision | Reality | |
---|---|---|
\(H_0\) is true | \(H_0\) is false | |
Reject \(H_0\), (conclude \(H_a\)) | Reject "building is not safe" when it is not safe (Type I Error) | Correct decision |
Fail to reject \(H_0\) | Correct decision | Failing to reject 'building not is safe' when it is safe (Type II Error) |
Power and \(\beta \) are complements of each other. Therefore, they have an inverse relationship, i.e. as one increases, the other decreases.
Our websites may use cookies to personalize and enhance your experience. By continuing without changing your cookie settings, you agree to this collection. For more information, please see our University Websites Privacy Notice .
Neag School of Education
Null and alternative hypotheses.
Converting research questions to hypothesis is a simple task. Take the questions and make it a positive statement that says a relationship exists (correlation studies) or a difference exists between the groups (experiment study) and you have the alternative hypothesis. Write the statement such that a relationship does not exist or a difference does not exist and you have the null hypothesis. You can reverse the process if you have a hypothesis and wish to write a research question.
When you are comparing two groups, the groups are the independent variable. When you are testing whether something affects something else, the cause is the independent variable. The independent variable is the one you manipulate.
Teachers given higher pay will have more positive attitudes toward children than teachers given lower pay. The first step is to ask yourself “Are there two or more groups being compared?” The answer is “Yes.” What are the groups? Teachers who are given higher pay and teachers who are given lower pay. The independent variable is teacher pay. The dependent variable (the outcome) is attitude towards school.
You could also approach is another way. “Is something causing something else?” The answer is “Yes.” What is causing what? Teacher pay is causing attitude towards school. Therefore, teacher pay is the independent variable (cause) and attitude towards school is the dependent variable (outcome).
By tradition, we try to disprove (reject) the null hypothesis. We can never prove a null hypothesis, because it is impossible to prove something does not exist. We can disprove something does not exist by finding an example of it. Therefore, in research we try to disprove the null hypothesis. When we do find that a relationship (or difference) exists then we reject the null and accept the alternative. If we do not find that a relationship (or difference) exists, we fail to reject the null hypothesis (and go with it). We never say we accept the null hypothesis because it is never possible to prove something does not exist. That is why we say that we failed to reject the null hypothesis, rather than we accepted it.
Del Siegle, Ph.D. Neag School of Education – University of Connecticut [email protected] www.delsiegle.com
What is the null hypothesis, how to state the null hypothesis, null hypothesis overview.
The word “null” in this context means that it’s a commonly accepted fact that researchers work to nullify . It doesn’t mean that the statement is null (i.e. amounts to nothing) itself! (Perhaps the term should be called the “nullifiable hypothesis” as that might cause less confusion).
The short answer is, as a scientist, you are required to ; It’s part of the scientific process. Science uses a battery of processes to prove or disprove theories, making sure than any new hypothesis has no flaws. Including both a null and an alternate hypothesis is one safeguard to ensure your research isn’t flawed. Not including the null hypothesis in your research is considered very bad practice by the scientific community. If you set out to prove an alternate hypothesis without considering it, you are likely setting yourself up for failure. At a minimum, your experiment will likely not be taken seriously.
Several scientists, including Copernicus , set out to disprove the null hypothesis. This eventually led to the rejection of the null and the acceptance of the alternate. Most people accepted it — the ones that didn’t created the Flat Earth Society !. What would have happened if Copernicus had not disproved the it and merely proved the alternate? No one would have listened to him. In order to change people’s thinking, he first had to prove that their thinking was wrong .
You’ll be asked to convert a word problem into a hypothesis statement in statistics that will include a null hypothesis and an alternate hypothesis . Breaking your problem into a few small steps makes these problems much easier to handle.
Step 2: Convert the hypothesis to math . Remember that the average is sometimes written as μ.
H 1 : μ > 8.2
Broken down into (somewhat) English, that’s H 1 (The hypothesis): μ (the average) > (is greater than) 8.2
Step 3: State what will happen if the hypothesis doesn’t come true. If the recovery time isn’t greater than 8.2 weeks, there are only two possibilities, that the recovery time is equal to 8.2 weeks or less than 8.2 weeks.
H 0 : μ ≤ 8.2
Broken down again into English, that’s H 0 (The null hypothesis): μ (the average) ≤ (is less than or equal to) 8.2
But what if the researcher doesn’t have any idea what will happen.
Example Problem: A researcher is studying the effects of radical exercise program on knee surgery patients. There is a good chance the therapy will improve recovery time, but there’s also the possibility it will make it worse. Average recovery times for knee surgery patients is 8.2 weeks.
Step 1: State what will happen if the experiment doesn’t make any difference. That’s the null hypothesis–that nothing will happen. In this experiment, if nothing happens, then the recovery time will stay at 8.2 weeks.
H 0 : μ = 8.2
Broken down into English, that’s H 0 (The null hypothesis): μ (the average) = (is equal to) 8.2
Step 2: Figure out the alternate hypothesis . The alternate hypothesis is the opposite of the null hypothesis. In other words, what happens if our experiment makes a difference?
H 1 : μ ≠ 8.2
In English again, that’s H 1 (The alternate hypothesis): μ (the average) ≠ (is not equal to) 8.2
That’s How to State the Null Hypothesis!
Check out our Youtube channel for more stats tips!
Gonick, L. (1993). The Cartoon Guide to Statistics . HarperPerennial. Kotz, S.; et al., eds. (2006), Encyclopedia of Statistical Sciences , Wiley.
An official website of the United States government
The .gov means it’s official. Federal government websites often end in .gov or .mil. Before sharing sensitive information, make sure you’re on a federal government site.
The site is secure. The https:// ensures that you are connecting to the official website and that any information you provide is encrypted and transmitted securely.
The PMC website is updating on October 15, 2024. Learn More or Try it out now .
Edward barroga.
1 Department of General Education, Graduate School of Nursing Science, St. Luke’s International University, Tokyo, Japan.
2 Department of Biological Sciences, Messiah University, Mechanicsburg, PA, USA.
The development of research questions and the subsequent hypotheses are prerequisites to defining the main research purpose and specific objectives of a study. Consequently, these objectives determine the study design and research outcome. The development of research questions is a process based on knowledge of current trends, cutting-edge studies, and technological advances in the research field. Excellent research questions are focused and require a comprehensive literature search and in-depth understanding of the problem being investigated. Initially, research questions may be written as descriptive questions which could be developed into inferential questions. These questions must be specific and concise to provide a clear foundation for developing hypotheses. Hypotheses are more formal predictions about the research outcomes. These specify the possible results that may or may not be expected regarding the relationship between groups. Thus, research questions and hypotheses clarify the main purpose and specific objectives of the study, which in turn dictate the design of the study, its direction, and outcome. Studies developed from good research questions and hypotheses will have trustworthy outcomes with wide-ranging social and health implications.
Scientific research is usually initiated by posing evidenced-based research questions which are then explicitly restated as hypotheses. 1 , 2 The hypotheses provide directions to guide the study, solutions, explanations, and expected results. 3 , 4 Both research questions and hypotheses are essentially formulated based on conventional theories and real-world processes, which allow the inception of novel studies and the ethical testing of ideas. 5 , 6
It is crucial to have knowledge of both quantitative and qualitative research 2 as both types of research involve writing research questions and hypotheses. 7 However, these crucial elements of research are sometimes overlooked; if not overlooked, then framed without the forethought and meticulous attention it needs. Planning and careful consideration are needed when developing quantitative or qualitative research, particularly when conceptualizing research questions and hypotheses. 4
There is a continuing need to support researchers in the creation of innovative research questions and hypotheses, as well as for journal articles that carefully review these elements. 1 When research questions and hypotheses are not carefully thought of, unethical studies and poor outcomes usually ensue. Carefully formulated research questions and hypotheses define well-founded objectives, which in turn determine the appropriate design, course, and outcome of the study. This article then aims to discuss in detail the various aspects of crafting research questions and hypotheses, with the goal of guiding researchers as they develop their own. Examples from the authors and peer-reviewed scientific articles in the healthcare field are provided to illustrate key points.
A research question is what a study aims to answer after data analysis and interpretation. The answer is written in length in the discussion section of the paper. Thus, the research question gives a preview of the different parts and variables of the study meant to address the problem posed in the research question. 1 An excellent research question clarifies the research writing while facilitating understanding of the research topic, objective, scope, and limitations of the study. 5
On the other hand, a research hypothesis is an educated statement of an expected outcome. This statement is based on background research and current knowledge. 8 , 9 The research hypothesis makes a specific prediction about a new phenomenon 10 or a formal statement on the expected relationship between an independent variable and a dependent variable. 3 , 11 It provides a tentative answer to the research question to be tested or explored. 4
Hypotheses employ reasoning to predict a theory-based outcome. 10 These can also be developed from theories by focusing on components of theories that have not yet been observed. 10 The validity of hypotheses is often based on the testability of the prediction made in a reproducible experiment. 8
Conversely, hypotheses can also be rephrased as research questions. Several hypotheses based on existing theories and knowledge may be needed to answer a research question. Developing ethical research questions and hypotheses creates a research design that has logical relationships among variables. These relationships serve as a solid foundation for the conduct of the study. 4 , 11 Haphazardly constructed research questions can result in poorly formulated hypotheses and improper study designs, leading to unreliable results. Thus, the formulations of relevant research questions and verifiable hypotheses are crucial when beginning research. 12
Excellent research questions are specific and focused. These integrate collective data and observations to confirm or refute the subsequent hypotheses. Well-constructed hypotheses are based on previous reports and verify the research context. These are realistic, in-depth, sufficiently complex, and reproducible. More importantly, these hypotheses can be addressed and tested. 13
There are several characteristics of well-developed hypotheses. Good hypotheses are 1) empirically testable 7 , 10 , 11 , 13 ; 2) backed by preliminary evidence 9 ; 3) testable by ethical research 7 , 9 ; 4) based on original ideas 9 ; 5) have evidenced-based logical reasoning 10 ; and 6) can be predicted. 11 Good hypotheses can infer ethical and positive implications, indicating the presence of a relationship or effect relevant to the research theme. 7 , 11 These are initially developed from a general theory and branch into specific hypotheses by deductive reasoning. In the absence of a theory to base the hypotheses, inductive reasoning based on specific observations or findings form more general hypotheses. 10
Research questions and hypotheses are developed according to the type of research, which can be broadly classified into quantitative and qualitative research. We provide a summary of the types of research questions and hypotheses under quantitative and qualitative research categories in Table 1 .
Quantitative research questions | Quantitative research hypotheses |
---|---|
Descriptive research questions | Simple hypothesis |
Comparative research questions | Complex hypothesis |
Relationship research questions | Directional hypothesis |
Non-directional hypothesis | |
Associative hypothesis | |
Causal hypothesis | |
Null hypothesis | |
Alternative hypothesis | |
Working hypothesis | |
Statistical hypothesis | |
Logical hypothesis | |
Hypothesis-testing | |
Qualitative research questions | Qualitative research hypotheses |
Contextual research questions | Hypothesis-generating |
Descriptive research questions | |
Evaluation research questions | |
Explanatory research questions | |
Exploratory research questions | |
Generative research questions | |
Ideological research questions | |
Ethnographic research questions | |
Phenomenological research questions | |
Grounded theory questions | |
Qualitative case study questions |
In quantitative research, research questions inquire about the relationships among variables being investigated and are usually framed at the start of the study. These are precise and typically linked to the subject population, dependent and independent variables, and research design. 1 Research questions may also attempt to describe the behavior of a population in relation to one or more variables, or describe the characteristics of variables to be measured ( descriptive research questions ). 1 , 5 , 14 These questions may also aim to discover differences between groups within the context of an outcome variable ( comparative research questions ), 1 , 5 , 14 or elucidate trends and interactions among variables ( relationship research questions ). 1 , 5 We provide examples of descriptive, comparative, and relationship research questions in quantitative research in Table 2 .
Quantitative research questions | |
---|---|
Descriptive research question | |
- Measures responses of subjects to variables | |
- Presents variables to measure, analyze, or assess | |
What is the proportion of resident doctors in the hospital who have mastered ultrasonography (response of subjects to a variable) as a diagnostic technique in their clinical training? | |
Comparative research question | |
- Clarifies difference between one group with outcome variable and another group without outcome variable | |
Is there a difference in the reduction of lung metastasis in osteosarcoma patients who received the vitamin D adjunctive therapy (group with outcome variable) compared with osteosarcoma patients who did not receive the vitamin D adjunctive therapy (group without outcome variable)? | |
- Compares the effects of variables | |
How does the vitamin D analogue 22-Oxacalcitriol (variable 1) mimic the antiproliferative activity of 1,25-Dihydroxyvitamin D (variable 2) in osteosarcoma cells? | |
Relationship research question | |
- Defines trends, association, relationships, or interactions between dependent variable and independent variable | |
Is there a relationship between the number of medical student suicide (dependent variable) and the level of medical student stress (independent variable) in Japan during the first wave of the COVID-19 pandemic? |
In quantitative research, hypotheses predict the expected relationships among variables. 15 Relationships among variables that can be predicted include 1) between a single dependent variable and a single independent variable ( simple hypothesis ) or 2) between two or more independent and dependent variables ( complex hypothesis ). 4 , 11 Hypotheses may also specify the expected direction to be followed and imply an intellectual commitment to a particular outcome ( directional hypothesis ) 4 . On the other hand, hypotheses may not predict the exact direction and are used in the absence of a theory, or when findings contradict previous studies ( non-directional hypothesis ). 4 In addition, hypotheses can 1) define interdependency between variables ( associative hypothesis ), 4 2) propose an effect on the dependent variable from manipulation of the independent variable ( causal hypothesis ), 4 3) state a negative relationship between two variables ( null hypothesis ), 4 , 11 , 15 4) replace the working hypothesis if rejected ( alternative hypothesis ), 15 explain the relationship of phenomena to possibly generate a theory ( working hypothesis ), 11 5) involve quantifiable variables that can be tested statistically ( statistical hypothesis ), 11 6) or express a relationship whose interlinks can be verified logically ( logical hypothesis ). 11 We provide examples of simple, complex, directional, non-directional, associative, causal, null, alternative, working, statistical, and logical hypotheses in quantitative research, as well as the definition of quantitative hypothesis-testing research in Table 3 .
Quantitative research hypotheses | |
---|---|
Simple hypothesis | |
- Predicts relationship between single dependent variable and single independent variable | |
If the dose of the new medication (single independent variable) is high, blood pressure (single dependent variable) is lowered. | |
Complex hypothesis | |
- Foretells relationship between two or more independent and dependent variables | |
The higher the use of anticancer drugs, radiation therapy, and adjunctive agents (3 independent variables), the higher would be the survival rate (1 dependent variable). | |
Directional hypothesis | |
- Identifies study direction based on theory towards particular outcome to clarify relationship between variables | |
Privately funded research projects will have a larger international scope (study direction) than publicly funded research projects. | |
Non-directional hypothesis | |
- Nature of relationship between two variables or exact study direction is not identified | |
- Does not involve a theory | |
Women and men are different in terms of helpfulness. (Exact study direction is not identified) | |
Associative hypothesis | |
- Describes variable interdependency | |
- Change in one variable causes change in another variable | |
A larger number of people vaccinated against COVID-19 in the region (change in independent variable) will reduce the region’s incidence of COVID-19 infection (change in dependent variable). | |
Causal hypothesis | |
- An effect on dependent variable is predicted from manipulation of independent variable | |
A change into a high-fiber diet (independent variable) will reduce the blood sugar level (dependent variable) of the patient. | |
Null hypothesis | |
- A negative statement indicating no relationship or difference between 2 variables | |
There is no significant difference in the severity of pulmonary metastases between the new drug (variable 1) and the current drug (variable 2). | |
Alternative hypothesis | |
- Following a null hypothesis, an alternative hypothesis predicts a relationship between 2 study variables | |
The new drug (variable 1) is better on average in reducing the level of pain from pulmonary metastasis than the current drug (variable 2). | |
Working hypothesis | |
- A hypothesis that is initially accepted for further research to produce a feasible theory | |
Dairy cows fed with concentrates of different formulations will produce different amounts of milk. | |
Statistical hypothesis | |
- Assumption about the value of population parameter or relationship among several population characteristics | |
- Validity tested by a statistical experiment or analysis | |
The mean recovery rate from COVID-19 infection (value of population parameter) is not significantly different between population 1 and population 2. | |
There is a positive correlation between the level of stress at the workplace and the number of suicides (population characteristics) among working people in Japan. | |
Logical hypothesis | |
- Offers or proposes an explanation with limited or no extensive evidence | |
If healthcare workers provide more educational programs about contraception methods, the number of adolescent pregnancies will be less. | |
Hypothesis-testing (Quantitative hypothesis-testing research) | |
- Quantitative research uses deductive reasoning. | |
- This involves the formation of a hypothesis, collection of data in the investigation of the problem, analysis and use of the data from the investigation, and drawing of conclusions to validate or nullify the hypotheses. |
Unlike research questions in quantitative research, research questions in qualitative research are usually continuously reviewed and reformulated. The central question and associated subquestions are stated more than the hypotheses. 15 The central question broadly explores a complex set of factors surrounding the central phenomenon, aiming to present the varied perspectives of participants. 15
There are varied goals for which qualitative research questions are developed. These questions can function in several ways, such as to 1) identify and describe existing conditions ( contextual research question s); 2) describe a phenomenon ( descriptive research questions ); 3) assess the effectiveness of existing methods, protocols, theories, or procedures ( evaluation research questions ); 4) examine a phenomenon or analyze the reasons or relationships between subjects or phenomena ( explanatory research questions ); or 5) focus on unknown aspects of a particular topic ( exploratory research questions ). 5 In addition, some qualitative research questions provide new ideas for the development of theories and actions ( generative research questions ) or advance specific ideologies of a position ( ideological research questions ). 1 Other qualitative research questions may build on a body of existing literature and become working guidelines ( ethnographic research questions ). Research questions may also be broadly stated without specific reference to the existing literature or a typology of questions ( phenomenological research questions ), may be directed towards generating a theory of some process ( grounded theory questions ), or may address a description of the case and the emerging themes ( qualitative case study questions ). 15 We provide examples of contextual, descriptive, evaluation, explanatory, exploratory, generative, ideological, ethnographic, phenomenological, grounded theory, and qualitative case study research questions in qualitative research in Table 4 , and the definition of qualitative hypothesis-generating research in Table 5 .
Qualitative research questions | |
---|---|
Contextual research question | |
- Ask the nature of what already exists | |
- Individuals or groups function to further clarify and understand the natural context of real-world problems | |
What are the experiences of nurses working night shifts in healthcare during the COVID-19 pandemic? (natural context of real-world problems) | |
Descriptive research question | |
- Aims to describe a phenomenon | |
What are the different forms of disrespect and abuse (phenomenon) experienced by Tanzanian women when giving birth in healthcare facilities? | |
Evaluation research question | |
- Examines the effectiveness of existing practice or accepted frameworks | |
How effective are decision aids (effectiveness of existing practice) in helping decide whether to give birth at home or in a healthcare facility? | |
Explanatory research question | |
- Clarifies a previously studied phenomenon and explains why it occurs | |
Why is there an increase in teenage pregnancy (phenomenon) in Tanzania? | |
Exploratory research question | |
- Explores areas that have not been fully investigated to have a deeper understanding of the research problem | |
What factors affect the mental health of medical students (areas that have not yet been fully investigated) during the COVID-19 pandemic? | |
Generative research question | |
- Develops an in-depth understanding of people’s behavior by asking ‘how would’ or ‘what if’ to identify problems and find solutions | |
How would the extensive research experience of the behavior of new staff impact the success of the novel drug initiative? | |
Ideological research question | |
- Aims to advance specific ideas or ideologies of a position | |
Are Japanese nurses who volunteer in remote African hospitals able to promote humanized care of patients (specific ideas or ideologies) in the areas of safe patient environment, respect of patient privacy, and provision of accurate information related to health and care? | |
Ethnographic research question | |
- Clarifies peoples’ nature, activities, their interactions, and the outcomes of their actions in specific settings | |
What are the demographic characteristics, rehabilitative treatments, community interactions, and disease outcomes (nature, activities, their interactions, and the outcomes) of people in China who are suffering from pneumoconiosis? | |
Phenomenological research question | |
- Knows more about the phenomena that have impacted an individual | |
What are the lived experiences of parents who have been living with and caring for children with a diagnosis of autism? (phenomena that have impacted an individual) | |
Grounded theory question | |
- Focuses on social processes asking about what happens and how people interact, or uncovering social relationships and behaviors of groups | |
What are the problems that pregnant adolescents face in terms of social and cultural norms (social processes), and how can these be addressed? | |
Qualitative case study question | |
- Assesses a phenomenon using different sources of data to answer “why” and “how” questions | |
- Considers how the phenomenon is influenced by its contextual situation. | |
How does quitting work and assuming the role of a full-time mother (phenomenon assessed) change the lives of women in Japan? |
Qualitative research hypotheses | |
---|---|
Hypothesis-generating (Qualitative hypothesis-generating research) | |
- Qualitative research uses inductive reasoning. | |
- This involves data collection from study participants or the literature regarding a phenomenon of interest, using the collected data to develop a formal hypothesis, and using the formal hypothesis as a framework for testing the hypothesis. | |
- Qualitative exploratory studies explore areas deeper, clarifying subjective experience and allowing formulation of a formal hypothesis potentially testable in a future quantitative approach. |
Qualitative studies usually pose at least one central research question and several subquestions starting with How or What . These research questions use exploratory verbs such as explore or describe . These also focus on one central phenomenon of interest, and may mention the participants and research site. 15
Hypotheses in qualitative research are stated in the form of a clear statement concerning the problem to be investigated. Unlike in quantitative research where hypotheses are usually developed to be tested, qualitative research can lead to both hypothesis-testing and hypothesis-generating outcomes. 2 When studies require both quantitative and qualitative research questions, this suggests an integrative process between both research methods wherein a single mixed-methods research question can be developed. 1
Research questions followed by hypotheses should be developed before the start of the study. 1 , 12 , 14 It is crucial to develop feasible research questions on a topic that is interesting to both the researcher and the scientific community. This can be achieved by a meticulous review of previous and current studies to establish a novel topic. Specific areas are subsequently focused on to generate ethical research questions. The relevance of the research questions is evaluated in terms of clarity of the resulting data, specificity of the methodology, objectivity of the outcome, depth of the research, and impact of the study. 1 , 5 These aspects constitute the FINER criteria (i.e., Feasible, Interesting, Novel, Ethical, and Relevant). 1 Clarity and effectiveness are achieved if research questions meet the FINER criteria. In addition to the FINER criteria, Ratan et al. described focus, complexity, novelty, feasibility, and measurability for evaluating the effectiveness of research questions. 14
The PICOT and PEO frameworks are also used when developing research questions. 1 The following elements are addressed in these frameworks, PICOT: P-population/patients/problem, I-intervention or indicator being studied, C-comparison group, O-outcome of interest, and T-timeframe of the study; PEO: P-population being studied, E-exposure to preexisting conditions, and O-outcome of interest. 1 Research questions are also considered good if these meet the “FINERMAPS” framework: Feasible, Interesting, Novel, Ethical, Relevant, Manageable, Appropriate, Potential value/publishable, and Systematic. 14
As we indicated earlier, research questions and hypotheses that are not carefully formulated result in unethical studies or poor outcomes. To illustrate this, we provide some examples of ambiguous research question and hypotheses that result in unclear and weak research objectives in quantitative research ( Table 6 ) 16 and qualitative research ( Table 7 ) 17 , and how to transform these ambiguous research question(s) and hypothesis(es) into clear and good statements.
Variables | Unclear and weak statement (Statement 1) | Clear and good statement (Statement 2) | Points to avoid |
---|---|---|---|
Research question | Which is more effective between smoke moxibustion and smokeless moxibustion? | “Moreover, regarding smoke moxibustion versus smokeless moxibustion, it remains unclear which is more effective, safe, and acceptable to pregnant women, and whether there is any difference in the amount of heat generated.” | 1) Vague and unfocused questions |
2) Closed questions simply answerable by yes or no | |||
3) Questions requiring a simple choice | |||
Hypothesis | The smoke moxibustion group will have higher cephalic presentation. | “Hypothesis 1. The smoke moxibustion stick group (SM group) and smokeless moxibustion stick group (-SLM group) will have higher rates of cephalic presentation after treatment than the control group. | 1) Unverifiable hypotheses |
Hypothesis 2. The SM group and SLM group will have higher rates of cephalic presentation at birth than the control group. | 2) Incompletely stated groups of comparison | ||
Hypothesis 3. There will be no significant differences in the well-being of the mother and child among the three groups in terms of the following outcomes: premature birth, premature rupture of membranes (PROM) at < 37 weeks, Apgar score < 7 at 5 min, umbilical cord blood pH < 7.1, admission to neonatal intensive care unit (NICU), and intrauterine fetal death.” | 3) Insufficiently described variables or outcomes | ||
Research objective | To determine which is more effective between smoke moxibustion and smokeless moxibustion. | “The specific aims of this pilot study were (a) to compare the effects of smoke moxibustion and smokeless moxibustion treatments with the control group as a possible supplement to ECV for converting breech presentation to cephalic presentation and increasing adherence to the newly obtained cephalic position, and (b) to assess the effects of these treatments on the well-being of the mother and child.” | 1) Poor understanding of the research question and hypotheses |
2) Insufficient description of population, variables, or study outcomes |
a These statements were composed for comparison and illustrative purposes only.
b These statements are direct quotes from Higashihara and Horiuchi. 16
Variables | Unclear and weak statement (Statement 1) | Clear and good statement (Statement 2) | Points to avoid |
---|---|---|---|
Research question | Does disrespect and abuse (D&A) occur in childbirth in Tanzania? | How does disrespect and abuse (D&A) occur and what are the types of physical and psychological abuses observed in midwives’ actual care during facility-based childbirth in urban Tanzania? | 1) Ambiguous or oversimplistic questions |
2) Questions unverifiable by data collection and analysis | |||
Hypothesis | Disrespect and abuse (D&A) occur in childbirth in Tanzania. | Hypothesis 1: Several types of physical and psychological abuse by midwives in actual care occur during facility-based childbirth in urban Tanzania. | 1) Statements simply expressing facts |
Hypothesis 2: Weak nursing and midwifery management contribute to the D&A of women during facility-based childbirth in urban Tanzania. | 2) Insufficiently described concepts or variables | ||
Research objective | To describe disrespect and abuse (D&A) in childbirth in Tanzania. | “This study aimed to describe from actual observations the respectful and disrespectful care received by women from midwives during their labor period in two hospitals in urban Tanzania.” | 1) Statements unrelated to the research question and hypotheses |
2) Unattainable or unexplorable objectives |
a This statement is a direct quote from Shimoda et al. 17
The other statements were composed for comparison and illustrative purposes only.
To construct effective research questions and hypotheses, it is very important to 1) clarify the background and 2) identify the research problem at the outset of the research, within a specific timeframe. 9 Then, 3) review or conduct preliminary research to collect all available knowledge about the possible research questions by studying theories and previous studies. 18 Afterwards, 4) construct research questions to investigate the research problem. Identify variables to be accessed from the research questions 4 and make operational definitions of constructs from the research problem and questions. Thereafter, 5) construct specific deductive or inductive predictions in the form of hypotheses. 4 Finally, 6) state the study aims . This general flow for constructing effective research questions and hypotheses prior to conducting research is shown in Fig. 1 .
Research questions are used more frequently in qualitative research than objectives or hypotheses. 3 These questions seek to discover, understand, explore or describe experiences by asking “What” or “How.” The questions are open-ended to elicit a description rather than to relate variables or compare groups. The questions are continually reviewed, reformulated, and changed during the qualitative study. 3 Research questions are also used more frequently in survey projects than hypotheses in experiments in quantitative research to compare variables and their relationships.
Hypotheses are constructed based on the variables identified and as an if-then statement, following the template, ‘If a specific action is taken, then a certain outcome is expected.’ At this stage, some ideas regarding expectations from the research to be conducted must be drawn. 18 Then, the variables to be manipulated (independent) and influenced (dependent) are defined. 4 Thereafter, the hypothesis is stated and refined, and reproducible data tailored to the hypothesis are identified, collected, and analyzed. 4 The hypotheses must be testable and specific, 18 and should describe the variables and their relationships, the specific group being studied, and the predicted research outcome. 18 Hypotheses construction involves a testable proposition to be deduced from theory, and independent and dependent variables to be separated and measured separately. 3 Therefore, good hypotheses must be based on good research questions constructed at the start of a study or trial. 12
In summary, research questions are constructed after establishing the background of the study. Hypotheses are then developed based on the research questions. Thus, it is crucial to have excellent research questions to generate superior hypotheses. In turn, these would determine the research objectives and the design of the study, and ultimately, the outcome of the research. 12 Algorithms for building research questions and hypotheses are shown in Fig. 2 for quantitative research and in Fig. 3 for qualitative research.
Research questions and hypotheses are crucial components to any type of research, whether quantitative or qualitative. These questions should be developed at the very beginning of the study. Excellent research questions lead to superior hypotheses, which, like a compass, set the direction of research, and can often determine the successful conduct of the study. Many research studies have floundered because the development of research questions and subsequent hypotheses was not given the thought and meticulous attention needed. The development of research questions and hypotheses is an iterative process based on extensive knowledge of the literature and insightful grasp of the knowledge gap. Focused, concise, and specific research questions provide a strong foundation for constructing hypotheses which serve as formal predictions about the research outcomes. Research questions and hypotheses are crucial elements of research that should not be overlooked. They should be carefully thought of and constructed when planning research. This avoids unethical studies and poor outcomes by defining well-founded objectives that determine the design, course, and outcome of the study.
Disclosure: The authors have no potential conflicts of interest to disclose.
Author Contributions:
Hypothesis testing involves the careful construction of two statements: the null hypothesis and the alternative hypothesis. These hypotheses can look very similar but are actually different.
How do we know which hypothesis is the null and which one is the alternative? We will see that there are a few ways to tell the difference.
The null hypothesis reflects that there will be no observed effect in our experiment. In a mathematical formulation of the null hypothesis, there will typically be an equal sign. This hypothesis is denoted by H 0 .
The null hypothesis is what we attempt to find evidence against in our hypothesis test. We hope to obtain a small enough p-value that it is lower than our level of significance alpha and we are justified in rejecting the null hypothesis. If our p-value is greater than alpha, then we fail to reject the null hypothesis.
If the null hypothesis is not rejected, then we must be careful to say what this means. The thinking on this is similar to a legal verdict. Just because a person has been declared "not guilty", it does not mean that he is innocent. In the same way, just because we failed to reject a null hypothesis it does not mean that the statement is true.
For example, we may want to investigate the claim that despite what convention has told us, the mean adult body temperature is not the accepted value of 98.6 degrees Fahrenheit . The null hypothesis for an experiment to investigate this is “The mean adult body temperature for healthy individuals is 98.6 degrees Fahrenheit.” If we fail to reject the null hypothesis, then our working hypothesis remains that the average adult who is healthy has a temperature of 98.6 degrees. We do not prove that this is true.
If we are studying a new treatment, the null hypothesis is that our treatment will not change our subjects in any meaningful way. In other words, the treatment will not produce any effect in our subjects.
The alternative or experimental hypothesis reflects that there will be an observed effect for our experiment. In a mathematical formulation of the alternative hypothesis, there will typically be an inequality, or not equal to symbol. This hypothesis is denoted by either H a or by H 1 .
The alternative hypothesis is what we are attempting to demonstrate in an indirect way by the use of our hypothesis test. If the null hypothesis is rejected, then we accept the alternative hypothesis. If the null hypothesis is not rejected, then we do not accept the alternative hypothesis. Going back to the above example of mean human body temperature, the alternative hypothesis is “The average adult human body temperature is not 98.6 degrees Fahrenheit.”
If we are studying a new treatment, then the alternative hypothesis is that our treatment does, in fact, change our subjects in a meaningful and measurable way.
The following set of negations may help when you are forming your null and alternative hypotheses. Most technical papers rely on just the first formulation, even though you may see some of the others in a statistics textbook.
Stack Exchange network consists of 183 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Q&A for work
Connect and share knowledge within a single location that is structured and easy to search.
I'm practicing with the hypothesis test and I find myself in trouble with the decision about how to set a null and an alternative hypothesis. My main issue is to determine, in every situation, a "general rule" on how I can decide correctly which is the null and which is the alternative hypothesis.. can someone help me?
Here is an example: As an established scholar, you are requested to evaluate if Customer Relationship Management affects the financial performance of firms. The main issue will be solved by means of a test of hypothesis. Two hypothesis will be tested one against the other: CRM is related to performance, CRM is not related.
The rule for the proper formulation of a hypothesis test is that the alternative or research hypothesis is the statement that, if true, is strongly supported by the evidence furnished by the data.
The null hypothesis is generally the complement of the alternative hypothesis. Frequently, it is (or contains) the assumption that you are making about how the data are distributed in order to calculate the test statistic.
Here are a few examples to help you understand how these are properly chosen.
Suppose I am an epidemiologist in public health, and I'm investigating whether the incidence of smoking among a certain ethnic group is greater than the population as a whole, and therefore there is a need to target anti-smoking campaigns for this sub-population through greater community outreach and education. From previous studies that have been published in the literature, I find that the incidence among the general population is $p_0$. I can then go about collecting sample data (that's actually the hard part!) to test $$H_0 : p = p_0 \quad \mathrm{vs.} \quad H_a : p > p_0.$$ This is a one-sided binomial proportion test. $H_a$ is the statement that, if it were true, would need to be strongly supported by the data we collected. It is the statement that carries the burden of proof . This is because any conclusion we draw from the test is conditional upon assuming that the null is true: either $H_a$ is accepted, or the test is inconclusive and there is insufficient evidence from the data to suggest $H_a$ is true. The choice of $H_0$ reflects the underlying assumption that there is no difference in the smoking rates of the sub-population compared to the whole.
Now suppose I am a researcher investigating a new drug that I believe to be equally effective to an existing standard of treatment, but with fewer side effects and therefore a more desirable safety profile. I would like to demonstrate the equal efficacy by conducting a bioequivalence test. If $\mu_0$ is the mean existing standard treatment effect, then my hypothesis might look like this: $$H_0 : |\mu - \mu_0| \ge \Delta \quad \mathrm{vs.} \quad H_a : |\mu - \mu_0| < \Delta,$$ for some choice of margin $\Delta$ that I consider to be clinically significant. For example, a clinician might say that two treatments are sufficiently bioequivalent if there is less than a $\Delta = 10\%$ difference in treatment effect. Note again that $H_a$ is the statement that carries the burden of proof: the data we collect must strongly support it, in order for us to accept it; otherwise, it could still be true but we don't have the evidence to support the claim .
Now suppose I am doing an analysis for a small business owner who sells three products $A$, $B$, $C$. They suspect that there is a statistically significant preference for these three products. Then my hypothesis is $$H_0 : \mu_A = \mu_B = \mu_C \quad \mathrm{vs.} \quad H_a : \exists i \ne j \text{ such that } \mu_i \ne \mu_j.$$ Really, all that $H_a$ is saying is that there are two means that are not equal to each other, which would then suggest that some difference in preference exists.
The null hypothesis is nearly always "something didn't happen" or "there is no effect" or "there is no relationship" or something similar. But it need not be this.
In your case, the null would be "there is no relationship between CRM and performance"
The usual method is to test the null at some significance level (most often, 0.05). Whether this is a good method is another matter, but it is what is commonly done.
In science proofs, you can never prove anything, you can only demonstrate that your model describes the data better than another model. You want your alternate hypothesis to come from the new model under test, and the null hypothesis to be from a different model.
The null hypothesis should come from a model which others would choose to use when challenging your scientific claims! The most common pattern for a scientific claim is "I think that X is a factor in process Y. If everyone already believes X is a factor in the process, then there is nothing to prove, and everyone can just go out and talk about it over drinks. Scientific arguments with null hypothesis are interesting because, if someone takes the opposing view, "X is not a factor in process Y, then there is a disagreement. This is where science does its thing.
If you believe "X is a factor in process Y" enough to run an experiment, you should generally know what you're looking to see in the results. So now your phrase becomes "X is a factor in process Y, producing visible outcome Z."
This is where you pick your null hypothesis. If someone believes X is not a factor, and your experiment does indeed show Z, then they need an explanation for Z. With your choice of null hypothesis, you are effectively challenging their explanation . The dead simplest explanation is always "Z was caused by random chance because science is based on statistics." Accordingly, most null hypothesis are in the form of "The outcome should be predicted using the previously accepted model plus some random chance to account for statistics.
Both hypothesis should be phrased in terms of the visible outcome, NOT the model you intend to prove. [note] You never start with an alternate hypothesis of "I believe X is a factor." You phrase it "I expect to see this result when I observe Z." The null hypothesis will be phrased similarly, "The status quo predicts that we will see this different result when I observe Z." There is always a statistical phrasing in there such as "I expect to observe a normal distribution on Z when I do this experiment over and over." Once you observe results that defend your alternate hypothesis and reject the null hypothsis, you are THEN in a position to make claims about the validity of your model.
[note] This bolded statement is my opinion, but I feel confident enough in its wording choice to post it. The hypotheses draw a strong line between the intuitive portion of the science, and the data and analysis of the science. If your phrasing is too close to the model, it becomes hard to separate the model from the data, and makes it harder for the next scientist to use your data
In the case of our simple model with process Y and visible outcome Z, the existing belief is that Z will fit a distribution that everyone is already comfortable with, such as "the randomness expected by your particular laboratory equipment setup" or "the purity of the reagents used in the experiment." When you "reject the null hypothesis" what you are saying is most literally, "I have run this experiment, and it is so tremendously unlikely that random chance generated the observed behavior, that everybody should start considering that maybe there's more to this than meets the eye."
The alternative hypothesis is what you offer to the world to replace the null hypothesis . It is one thing to go do experiments to poke at holes in other's models, but that doesn't promote science nearly as well as poking holes in other's models and then replacing them with new models that do a better job.
With the null and alternate hypothesis, you are trying to challenge the current conventional thinking of the day. Choose the hypotheses so that they effectively declare "Here is a result everybody would expect (null hypothesis). However, I actually went out and did the experiment and gathered data, and it is VERY unlikely that the null hypothesis is true. Here is the result I expected (the alternate hypothesis). Nobody expected this hypothesis to be true but me, but when I gathered the data and did the statistics, it is very likely that my model does a better job of describing reality than the existing model . Accordingly, I reject the null hypothesis, accept my hypothesis, and challenge my fellow scientists to work from this new data."
And the fellow scientists are free to:
The last outcome causes strife and bickering, but is ABSOLUTELY part of the scientific process. By using the scientific method to publish your results, you accept that others are free to use the scientific method to contradict your results. They will do so, and publish their results.
At this point, the scientific community will make a political decision: who has to go out and spend the money to test their model, and whose model do we accept. TYPICALLY, because you published the model and the data first, and they are refuting your data, the onus is on them to run the experiments which proves why their model is better than yours. But this is now WELL beyond the hypothesis that caused the strife in the first place, so I leave you to experience them in your lifetime!
Written by:
Aashi Verma has dedicated herself to covering the forefront of enterprise and cloud technologies. As an Passionate researcher, learner, and writer, Aashi Verma interests extend beyond technology to include a deep appreciation for the outdoors, music, literature, and a commitment to environmental and social sustainability.
Summary: Explore the difference between Null and Alternate Hypotheses in hypothesis testing. The Null Hypothesis assumes no effect, while the Alternate Hypothesis suggests a significant impact. Accurate formulation of these hypotheses is crucial for reliable research outcomes.
Hypothesis testing is a fundamental concept in statistical analysis used to determine if enough evidence exists to support a specific claim or hypothesis. Understanding the difference between the Null and Alternate Hypothesis is crucial for accurate data interpretation and decision-making.
The Null Hypothesis (H₀) assumes no effect or difference, while the Alternate Hypothesis (H₁) suggests a potential impact or difference. This article aims to clarify these concepts, highlight their importance in research, and provide practical examples to enhance your comprehension of hypothesis testing.
Read Blog: Let’s Understand the Difference Between Data and Information .
A hypothesis is a testable statement or prediction about the relationship between variables. It serves as the foundation for research and data analysis by providing a clear, focused question that guides the investigation. A hypothesis suggests an expected outcome or a potential explanation that researchers can test through experimentation or observation.
Hypotheses play a crucial role in research and data analysis . They help frame the research question and set the direction for the study. By proposing a hypothesis, researchers can design experiments or analyses to collect data and test the validity of their predictions. This process allows them to determine whether the evidence supports or refutes their hypothesis.
Hypotheses drive the scientific method, enabling researchers to conclude, make informed decisions, and contribute to the body of knowledge in their field. They direct the research process and provide a basis for statistical testing, helping to ensure that findings are based on empirical evidence rather than assumptions or guesswork.
The null hypothesis (H₀) is a fundamental concept in statistical hypothesis testing . It represents a default assumption that no effect, relationship, or difference exists between groups or variables.
The null hypothesis provides a benchmark against which researchers can test their alternative hypothesis (H₁ or Ha). It asserts that any observed differences or effects in the data are due to chance rather than a significant effect or relationship.
Characteristics of a Null Hypothesis:
Examples of Null Hypotheses:
Researchers use statistical tests to evaluate the null hypothesis. If the evidence strongly contradicts H₀, they may reject it in favour of the alternate hypothesis. However, accepting H₀ does not prove it true; it merely indicates insufficient evidence against it.
Also Read More About:
Statistical Tools for Data-Driven Research .
Exploring 5 Statistical Data Analysis Techniques with Real-World Examples .
The alternate hypothesis (H₁ or Ha) is a crucial component of hypothesis testing that represents the researcher’s prediction or the effect they aim to prove.
Unlike the null hypothesis, which suggests no effect or relationship, the alternate hypothesis proposes a significant effect or relationship between variables. Its primary purpose is to challenge the null hypothesis and provide a basis for further investigation.
Characteristics of an Alternate Hypothesis:
Examples of Alternate Hypotheses:
Understanding the distinctions between the null and alternate hypotheses is crucial for effective hypothesis testing. Both serve specific roles in statistical analysis and research, but their purposes and definitions differ significantly.
The null hypothesis (H₀) represents a statement of no effect or difference. It posits that any observed differences or effects in the data are due to random chance rather than a real underlying cause. Essentially, the null hypothesis acts as a default assumption that no significant effect or relationship exists.
In contrast, the alternate hypothesis (H₁ or Ha) challenges the null hypothesis by suggesting that there is a significant effect or relationship. It asserts that any observed differences are not due to chance but result from a specific factor or intervention. The alternate hypothesis is what researchers aim to support through their analysis.
In hypothesis testing, the null hypothesis serves as the baseline against which the alternate hypothesis is tested. Researchers use statistical tests to evaluate whether there is enough evidence to reject the null hypothesis in favour of the alternate hypothesis. If the evidence is strong enough, they reject the null hypothesis and accept the alternate hypothesis.
For example, in a clinical trial testing a new drug, the null hypothesis might state that the drug does not affect patients’ recovery times. The alternate hypothesis would propose that the drug does have an effect.
Statistical tests analyse the data to determine if sufficient evidence exists to reject the null hypothesis and support the claim that the drug has a significant impact.
Consider a study on the effectiveness of a new teaching method. The null hypothesis might claim that the new method has no impact on student performance compared to traditional methods.
The alternate hypothesis would suggest that the new method does improve student performance. Researchers test these hypotheses through statistical analysis to determine if the observed improvements are statistically significant.
Understanding these differences helps researchers design better experiments and interpret results accurately, ensuring that conclusions drawn from data are based on sound evidence.
Explore More: An Introduction to Statistical Inference .
Formulating null and alternate hypotheses is a crucial step in hypothesis testing. It sets the stage for statistical analysis by defining the assumptions you will test against. Properly formulating these hypotheses ensures that your research is grounded in clear and testable statements.
The null hypothesis (H₀) represents a statement of no effect or difference. It assumes that any observed effect in your data is due to chance rather than a specific cause. To formulate a null hypothesis:
The alternate hypothesis is the statement that you aim to provide evidence for, suggesting that there is an effect or a difference. It contrasts with the null hypothesis and often indicates the presence of a relationship or effect. To formulate an alternate hypothesis:
Avoiding common pitfalls in hypothesis formulation is crucial for maintaining the integrity and validity of your research. Understanding and steering clear of these errors helps ensure that your hypotheses are robust and your subsequent analysis is credible.
Following these guidelines and avoiding common pitfalls can create robust null and alternate hypotheses that lay a strong foundation for effective hypothesis testing. This careful formulation will help ensure your research is accurate, reliable, and meaningful.
Further Discover:
Exploring The Top Key Statistical Concepts .
Different Types of Statistical Sampling in Data Analytics .
Understanding the difference between Null and Alternate Hypotheses is essential for accurate hypothesis testing. The Null Hypothesis assumes no significant effect, while the Alternate Hypothesis proposes a meaningful difference. Formulating these hypotheses correctly helps ensure robust research and valid conclusions, supporting sound data interpretation and decision-making.
What is the difference between null and alternate hypothesis .
The Null Hypothesis (H₀) assumes no effect or difference, while the Alternate Hypothesis (H₁) suggests a significant effect or relationship. Researchers test evidence to reject H₀ in favour of H₁.
The Null Hypothesis (H₀) is a default assumption of no effect. It provides a baseline for testing and allows researchers to determine if observed differences are statistically significant.
An Alternate Hypothesis (H₁) proposes a significant effect or difference. It should directly oppose the Null Hypothesis and be specific, clear, and testable to guide statistical analysis effectively.
Reviewed by:
You may also like.
Defining the hypothesis, the role of a hypothesis in the scientific method, types of hypotheses, hypothesis formulation, hypotheses and variables.
In sociology, as in other scientific disciplines, the hypothesis serves as a crucial building block for research. It is a central element that directs the inquiry and provides a framework for testing the relationships between social phenomena. This article will explore what a hypothesis is, how it is formulated, and its role within the broader scientific method. By understanding the hypothesis, students of sociology can grasp how sociologists construct and test theories about the social world.
A hypothesis is a specific, testable statement about the relationship between two or more variables. It acts as a proposed explanation or prediction based on limited evidence, which researchers then test through empirical investigation. In essence, it is a statement that can be supported or refuted by data gathered from observation, experimentation, or other forms of systematic inquiry. The hypothesis typically takes the form of an “if-then” statement: if one variable changes, then another will change in response.
In sociological research, a hypothesis helps to focus the investigation by offering a clear proposition that can be tested. For instance, a sociologist might hypothesize that an increase in education levels leads to a decrease in crime rates. This hypothesis gives the researcher a direction, guiding them to collect data on education and crime, and analyze the relationship between the two variables. By doing so, the hypothesis serves as a tool for making sense of complex social phenomena.
The hypothesis is a key component of the scientific method, which is the systematic process by which sociologists and other scientists investigate the world. The scientific method begins with an observation of the world, followed by the formulation of a question or problem. Based on prior knowledge, theory, or preliminary observations, researchers then develop a hypothesis, which predicts an outcome or proposes a relationship between variables.
Once a hypothesis is established, researchers gather data to test it. If the data supports the hypothesis, it may be used to build a broader theory or to further refine the understanding of the social phenomenon in question. If the data contradicts the hypothesis, researchers may revise their hypothesis or abandon it altogether, depending on the strength of the evidence. In either case, the hypothesis helps to organize the research process, ensuring that it remains focused and methodologically sound.
In sociology, this method is particularly important because the social world is highly complex. Researchers must navigate a vast range of variables—age, gender, class, race, education, and countless others—that interact in unpredictable ways. A well-constructed hypothesis allows sociologists to narrow their focus to a manageable set of variables, making the investigation more precise and efficient.
Sociologists use different types of hypotheses, depending on the nature of their research question and the methods they plan to use. Broadly speaking, hypotheses can be classified into two main types: null hypotheses and alternative (or research) hypotheses.
The null hypothesis, denoted as H0, states that there is no relationship between the variables being studied. It is a default assumption that any observed differences or relationships are due to random chance rather than a real underlying cause. In research, the null hypothesis serves as a point of comparison. Researchers collect data to see if the results allow them to reject the null hypothesis in favor of an alternative explanation.
For example, a sociologist studying the relationship between income and political participation might propose a null hypothesis that income has no effect on political participation. The goal of the research would then be to determine whether this null hypothesis can be rejected based on the data. If the data shows a significant correlation between income and political participation, the null hypothesis would be rejected.
The alternative hypothesis, denoted as H1 or Ha, proposes that there is a significant relationship between the variables. This is the hypothesis that researchers aim to support with their data. In contrast to the null hypothesis, the alternative hypothesis predicts a specific direction or effect. For example, a researcher might hypothesize that higher levels of education lead to greater political engagement. In this case, the alternative hypothesis is proposing a positive correlation between the two variables.
The alternative hypothesis is the one that guides the research design, as it directs the researcher toward gathering evidence that will either support or refute the predicted relationship. The research process is structured around testing this hypothesis and determining whether the evidence is strong enough to reject the null hypothesis.
The process of formulating a hypothesis is both an art and a science. It requires a deep understanding of the social phenomena under investigation, as well as a clear sense of what is possible to observe and measure. Hypothesis formulation is closely linked to the theoretical framework that guides the research. Sociologists draw on existing theories to generate hypotheses, ensuring that their predictions are grounded in established knowledge.
To formulate a good hypothesis, a researcher must identify the key variables and determine how they are expected to relate to one another. Variables are the factors or characteristics that are being measured in a study. In sociology, these variables often include social attributes such as class, race, gender, age, education, and income, as well as behavioral variables like voting, criminal activity, or social participation.
For example, a sociologist studying the effects of social media on self-esteem might propose the following hypothesis: “Increased time spent on social media leads to lower levels of self-esteem among adolescents.” Here, the independent variable is the time spent on social media, and the dependent variable is the level of self-esteem. The hypothesis predicts a negative relationship between the two variables: as time spent on social media increases, self-esteem decreases.
A strong hypothesis has several key characteristics. It should be clear and specific, meaning that it unambiguously states the relationship between the variables. It should also be testable, meaning that it can be supported or refuted through empirical investigation. Finally, it should be grounded in theory, meaning that it is based on existing knowledge about the social phenomenon in question.
You must be a member to access this content.
View Membership Levels
Mr Edwards has a PhD in sociology and 10 years of experience in sociological knowledge
Learn about the concept of data in sociology and its importance in understanding social phenomena. Explore the types of data...
Learn about analytic induction, a qualitative research method used in sociology to develop theories based on empirical evidence. Discover the...
Get the latest sociology.
How would you rate the content on Easy Sociology?
24 hour trending.
Functionalism: an introduction, understanding the concept of ‘community’ in sociology, understanding gemeinschaft and gesellschaft, gender roles: an introduction.
Easy Sociology makes sociology as easy as possible. Our aim is to make sociology accessible for everybody. © 2023 Easy Sociology
© 2023 Easy Sociology
What are null and alternative hypotheses.
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.
As the degrees of freedom increase, Student’s t distribution becomes less leptokurtic , meaning that the probability of extreme values decreases. The distribution becomes more and more similar to a standard normal distribution .
The three categories of kurtosis are:
Probability distributions belong to two broad categories: discrete probability distributions and continuous probability distributions . Within each category, there are many types of probability distributions.
Probability is the relative frequency over an infinite number of trials.
For example, the probability of a coin landing on heads is .5, meaning that if you flip the coin an infinite number of times, it will land on heads half the time.
Since doing something an infinite number of times is impossible, relative frequency is often used as an estimate of probability. If you flip a coin 1000 times and get 507 heads, the relative frequency, .507, is a good estimate of the probability.
Categorical variables can be described by a frequency distribution. Quantitative variables can also be described by a frequency distribution, but first they need to be grouped into interval classes .
A histogram is an effective way to tell if a frequency distribution appears to have a normal distribution .
Plot a histogram and look at the shape of the bars. If the bars roughly follow a symmetrical bell or hill shape, like the example below, then the distribution is approximately normally distributed.
You can use the CHISQ.INV.RT() function to find a chi-square critical value in Excel.
For example, to calculate the chi-square critical value for a test with df = 22 and α = .05, click any blank cell and type:
=CHISQ.INV.RT(0.05,22)
You can use the qchisq() function to find a chi-square critical value in R.
For example, to calculate the chi-square critical value for a test with df = 22 and α = .05:
qchisq(p = .05, df = 22, lower.tail = FALSE)
You can use the chisq.test() function to perform a chi-square test of independence in R. Give the contingency table as a matrix for the “x” argument. For example:
m = matrix(data = c(89, 84, 86, 9, 8, 24), nrow = 3, ncol = 2)
chisq.test(x = m)
You can use the CHISQ.TEST() function to perform a chi-square test of independence in Excel. It takes two arguments, CHISQ.TEST(observed_range, expected_range), and returns the p value.
Chi-square goodness of fit tests are often used in genetics. One common application is to check if two genes are linked (i.e., if the assortment is independent). When genes are linked, the allele inherited for one gene affects the allele inherited for another gene.
Suppose that you want to know if the genes for pea texture (R = round, r = wrinkled) and color (Y = yellow, y = green) are linked. You perform a dihybrid cross between two heterozygous ( RY / ry ) pea plants. The hypotheses you’re testing with your experiment are:
You observe 100 peas:
To calculate the expected values, you can make a Punnett square. If the two genes are unlinked, the probability of each genotypic combination is equal.
RRYY | RrYy | RRYy | RrYY | |
RrYy | rryy | Rryy | rrYy | |
RRYy | Rryy | RRyy | RrYy | |
RrYY | rrYy | RrYy | rrYY |
The expected phenotypic ratios are therefore 9 round and yellow: 3 round and green: 3 wrinkled and yellow: 1 wrinkled and green.
From this, you can calculate the expected phenotypic frequencies for 100 peas:
Round and yellow | 78 | 100 * (9/16) = 56.25 |
Round and green | 6 | 100 * (3/16) = 18.75 |
Wrinkled and yellow | 4 | 100 * (3/16) = 18.75 |
Wrinkled and green | 12 | 100 * (1/16) = 6.21 |
− | − | ||||
Round and yellow | 78 | 56.25 | 21.75 | 473.06 | 8.41 |
Round and green | 6 | 18.75 | −12.75 | 162.56 | 8.67 |
Wrinkled and yellow | 4 | 18.75 | −14.75 | 217.56 | 11.6 |
Wrinkled and green | 12 | 6.21 | 5.79 | 33.52 | 5.4 |
Χ 2 = 8.41 + 8.67 + 11.6 + 5.4 = 34.08
Since there are four groups (round and yellow, round and green, wrinkled and yellow, wrinkled and green), there are three degrees of freedom .
For a test of significance at α = .05 and df = 3, the Χ 2 critical value is 7.82.
Χ 2 = 34.08
Critical value = 7.82
The Χ 2 value is greater than the critical value .
The Χ 2 value is greater than the critical value, so we reject the null hypothesis that the population of offspring have an equal probability of inheriting all possible genotypic combinations. There is a significant difference between the observed and expected genotypic frequencies ( p < .05).
The data supports the alternative hypothesis that the offspring do not have an equal probability of inheriting all possible genotypic combinations, which suggests that the genes are linked
You can use the chisq.test() function to perform a chi-square goodness of fit test in R. Give the observed values in the “x” argument, give the expected values in the “p” argument, and set “rescale.p” to true. For example:
chisq.test(x = c(22,30,23), p = c(25,25,25), rescale.p = TRUE)
You can use the CHISQ.TEST() function to perform a chi-square goodness of fit test in Excel. It takes two arguments, CHISQ.TEST(observed_range, expected_range), and returns the p value .
Both correlations and chi-square tests can test for relationships between two variables. However, a correlation is used when you have two quantitative variables and a chi-square test of independence is used when you have two categorical variables.
Both chi-square tests and t tests can test for differences between two groups. However, a t test is used when you have a dependent quantitative variable and an independent categorical variable (with two groups). A chi-square test of independence is used when you have two categorical variables.
The two main chi-square tests are the chi-square goodness of fit test and the chi-square test of independence .
A chi-square distribution is a continuous probability distribution . The shape of a chi-square distribution depends on its degrees of freedom , k . The mean of a chi-square distribution is equal to its degrees of freedom ( k ) and the variance is 2 k . The range is 0 to ∞.
As the degrees of freedom ( k ) increases, the chi-square distribution goes from a downward curve to a hump shape. As the degrees of freedom increases further, the hump goes from being strongly right-skewed to being approximately normal.
To find the quartiles of a probability distribution, you can use the distribution’s quantile function.
You can use the quantile() function to find quartiles in R. If your data is called “data”, then “quantile(data, prob=c(.25,.5,.75), type=1)” will return the three quartiles.
You can use the QUARTILE() function to find quartiles in Excel. If your data is in column A, then click any blank cell and type “=QUARTILE(A:A,1)” for the first quartile, “=QUARTILE(A:A,2)” for the second quartile, and “=QUARTILE(A:A,3)” for the third quartile.
You can use the PEARSON() function to calculate the Pearson correlation coefficient in Excel. If your variables are in columns A and B, then click any blank cell and type “PEARSON(A:A,B:B)”.
There is no function to directly test the significance of the correlation.
You can use the cor() function to calculate the Pearson correlation coefficient in R. To test the significance of the correlation, you can use the cor.test() function.
You should use the Pearson correlation coefficient when (1) the relationship is linear and (2) both variables are quantitative and (3) normally distributed and (4) have no outliers.
The Pearson correlation coefficient ( r ) is the most common way of measuring a linear correlation. It is a number between –1 and 1 that measures the strength and direction of the relationship between two variables.
This table summarizes the most important differences between normal distributions and Poisson distributions :
Characteristic | Normal | Poisson |
---|---|---|
Continuous | ||
Mean (µ) and standard deviation (σ) | Lambda (λ) | |
Shape | Bell-shaped | Depends on λ |
Symmetrical | Asymmetrical (right-skewed). As λ increases, the asymmetry decreases. | |
Range | −∞ to ∞ | 0 to ∞ |
When the mean of a Poisson distribution is large (>10), it can be approximated by a normal distribution.
In the Poisson distribution formula, lambda (λ) is the mean number of events within a given interval of time or space. For example, λ = 0.748 floods per year.
The e in the Poisson distribution formula stands for the number 2.718. This number is called Euler’s constant. You can simply substitute e with 2.718 when you’re calculating a Poisson probability. Euler’s constant is a very useful number and is especially important in calculus.
The three types of skewness are:
Skewness and kurtosis are both important measures of a distribution’s shape.
A research hypothesis is your proposed answer to your research question. The research hypothesis usually includes an explanation (“ x affects y because …”).
A statistical hypothesis, on the other hand, is a mathematical statement about a population parameter. Statistical hypotheses always come in pairs: the null and alternative hypotheses . In a well-designed study , the statistical hypotheses correspond logically to the research hypothesis.
The alternative hypothesis is often abbreviated as H a or H 1 . When the alternative hypothesis is written using mathematical symbols, it always includes an inequality symbol (usually ≠, but sometimes < or >).
The null hypothesis is often abbreviated as H 0 . When the null hypothesis is written using mathematical symbols, it always includes an equality symbol (usually =, but sometimes ≥ or ≤).
The t distribution was first described by statistician William Sealy Gosset under the pseudonym “Student.”
To calculate a confidence interval of a mean using the critical value of t , follow these four steps:
To test a hypothesis using the critical value of t , follow these four steps:
You can use the T.INV() function to find the critical value of t for one-tailed tests in Excel, and you can use the T.INV.2T() function for two-tailed tests.
You can use the qt() function to find the critical value of t in R. The function gives the critical value of t for the one-tailed test. If you want the critical value of t for a two-tailed test, divide the significance level by two.
You can use the RSQ() function to calculate R² in Excel. If your dependent variable is in column A and your independent variable is in column B, then click any blank cell and type “RSQ(A:A,B:B)”.
You can use the summary() function to view the R² of a linear model in R. You will see the “R-squared” near the bottom of the output.
There are two formulas you can use to calculate the coefficient of determination (R²) of a simple linear regression .
The coefficient of determination (R²) is a number between 0 and 1 that measures how well a statistical model predicts an outcome. You can interpret the R² as the proportion of variation in the dependent variable that is predicted by the statistical model.
There are three main types of missing data .
Missing completely at random (MCAR) data are randomly distributed across the variable and unrelated to other variables .
Missing at random (MAR) data are not randomly distributed but they are accounted for by other observed variables.
Missing not at random (MNAR) data systematically differ from the observed values.
To tidy up your missing data , your options usually include accepting, removing, or recreating the missing data.
Missing data are important because, depending on the type, they can sometimes bias your results. This means your results may not be generalizable outside of your study because your data come from an unrepresentative sample .
Missing data , or missing values, occur when you don’t have data stored for certain variables or participants.
In any dataset, there’s usually some missing data. In quantitative research , missing values appear as blank cells in your spreadsheet.
There are two steps to calculating the geometric mean :
Before calculating the geometric mean, note that:
The arithmetic mean is the most commonly used type of mean and is often referred to simply as “the mean.” While the arithmetic mean is based on adding and dividing values, the geometric mean multiplies and finds the root of values.
Even though the geometric mean is a less common measure of central tendency , it’s more accurate than the arithmetic mean for percentage change and positively skewed data. The geometric mean is often reported for financial indices and population growth rates.
The geometric mean is an average that multiplies all values and finds a root of the number. For a dataset with n numbers, you find the n th root of their product.
Outliers are extreme values that differ from most values in the dataset. You find outliers at the extreme ends of your dataset.
It’s best to remove outliers only when you have a sound reason for doing so.
Some outliers represent natural variations in the population , and they should be left as is in your dataset. These are called true outliers.
Other outliers are problematic and should be removed because they represent measurement errors , data entry or processing errors, or poor sampling.
You can choose from four main ways to detect outliers :
Outliers can have a big impact on your statistical analyses and skew the results of any hypothesis test if they are inaccurate.
These extreme values can impact your statistical power as well, making it hard to detect a true effect if there is one.
No, the steepness or slope of the line isn’t related to the correlation coefficient value. The correlation coefficient only tells you how closely your data fit on a line, so two datasets with the same correlation coefficient can have very different slopes.
To find the slope of the line, you’ll need to perform a regression analysis .
Correlation coefficients always range between -1 and 1.
The sign of the coefficient tells you the direction of the relationship: a positive value means the variables change together in the same direction, while a negative value means they change together in opposite directions.
The absolute value of a number is equal to the number without its sign. The absolute value of a correlation coefficient tells you the magnitude of the correlation: the greater the absolute value, the stronger the correlation.
These are the assumptions your data must meet if you want to use Pearson’s r :
A correlation coefficient is a single number that describes the strength and direction of the relationship between your variables.
Different types of correlation coefficients might be appropriate for your data based on their levels of measurement and distributions . The Pearson product-moment correlation coefficient (Pearson’s r ) is commonly used to assess a linear relationship between two quantitative variables.
There are various ways to improve power:
A power analysis is a calculation that helps you determine a minimum sample size for your study. It’s made up of four main components. If you know or have estimates for any three of these, you can calculate the fourth component.
Statistical analysis is the main method for analyzing quantitative research data . It uses probabilities and models to test predictions about a population from sample data.
The risk of making a Type II error is inversely related to the statistical power of a test. Power is the extent to which a test can correctly detect a real effect when there is one.
To (indirectly) reduce the risk of a Type II error, you can increase the sample size or the significance level to increase statistical power.
The risk of making a Type I error is the significance level (or alpha) that you choose. That’s a value that you set at the beginning of your study to assess the statistical probability of obtaining your results ( p value ).
The significance level is usually set at 0.05 or 5%. This means that your results only have a 5% chance of occurring, or less, if the null hypothesis is actually true.
To reduce the Type I error probability, you can set a lower significance level.
In statistics, a Type I error means rejecting the null hypothesis when it’s actually true, while a Type II error means failing to reject the null hypothesis when it’s actually false.
In statistics, power refers to the likelihood of a hypothesis test detecting a true effect if there is one. A statistically powerful test is more likely to reject a false negative (a Type II error).
If you don’t ensure enough power in your study, you may not be able to detect a statistically significant result even when it has practical significance. Your study might not have the ability to answer your research question.
While statistical significance shows that an effect exists in a study, practical significance shows that the effect is large enough to be meaningful in the real world.
Statistical significance is denoted by p -values whereas practical significance is represented by effect sizes .
There are dozens of measures of effect sizes . The most common effect sizes are Cohen’s d and Pearson’s r . Cohen’s d measures the size of the difference between two groups while Pearson’s r measures the strength of the relationship between two variables .
Effect size tells you how meaningful the relationship between variables or the difference between groups is.
A large effect size means that a research finding has practical significance, while a small effect size indicates limited practical applications.
Using descriptive and inferential statistics , you can make two types of estimates about the population : point estimates and interval estimates.
Both types of estimates are important for gathering a clear idea of where a parameter is likely to lie.
Standard error and standard deviation are both measures of variability . The standard deviation reflects variability within a sample, while the standard error estimates the variability across samples of a population.
The standard error of the mean , or simply standard error , indicates how different the population mean is likely to be from a sample mean. It tells you how much the sample mean would vary if you were to repeat a study using new samples from within a single population.
To figure out whether a given number is a parameter or a statistic , ask yourself the following:
If the answer is yes to both questions, the number is likely to be a parameter. For small populations, data can be collected from the whole population and summarized in parameters.
If the answer is no to either of the questions, then the number is more likely to be a statistic.
The arithmetic mean is the most commonly used mean. It’s often simply called the mean or the average. But there are some other types of means you can calculate depending on your research purposes:
You can find the mean , or average, of a data set in two simple steps:
This method is the same whether you are dealing with sample or population data or positive or negative numbers.
The median is the most informative measure of central tendency for skewed distributions or distributions with outliers. For example, the median is often used as a measure of central tendency for income distributions, which are generally highly skewed.
Because the median only uses one or two values, it’s unaffected by extreme outliers or non-symmetric distributions of scores. In contrast, the mean and mode can vary in skewed distributions.
To find the median , first order your data. Then calculate the middle position based on n , the number of values in your data set.
A data set can often have no mode, one mode or more than one mode – it all depends on how many different values repeat most frequently.
Your data can be:
To find the mode :
Then you simply need to identify the most frequently occurring value.
The interquartile range is the best measure of variability for skewed distributions or data sets with outliers. Because it’s based on values that come from the middle half of the distribution, it’s unlikely to be influenced by outliers .
The two most common methods for calculating interquartile range are the exclusive and inclusive methods.
The exclusive method excludes the median when identifying Q1 and Q3, while the inclusive method includes the median as a value in the data set in identifying the quartiles.
For each of these methods, you’ll need different procedures for finding the median, Q1 and Q3 depending on whether your sample size is even- or odd-numbered. The exclusive method works best for even-numbered sample sizes, while the inclusive method is often used with odd-numbered sample sizes.
While the range gives you the spread of the whole data set, the interquartile range gives you the spread of the middle half of a data set.
Homoscedasticity, or homogeneity of variances, is an assumption of equal or similar variances in different groups being compared.
This is an important assumption of parametric statistical tests because they are sensitive to any dissimilarities. Uneven variances in samples result in biased and skewed test results.
Statistical tests such as variance tests or the analysis of variance (ANOVA) use sample variance to assess group differences of populations. They use the variances of the samples to assess whether the populations they come from significantly differ from each other.
Variance is the average squared deviations from the mean, while standard deviation is the square root of this number. Both measures reflect variability in a distribution, but their units differ:
Although the units of variance are harder to intuitively understand, variance is important in statistical tests .
The empirical rule, or the 68-95-99.7 rule, tells you where most of the values lie in a normal distribution :
The empirical rule is a quick way to get an overview of your data and check for any outliers or extreme values that don’t follow this pattern.
In a normal distribution , data are symmetrically distributed with no skew. Most values cluster around a central region, with values tapering off as they go further away from the center.
The measures of central tendency (mean, mode, and median) are exactly the same in a normal distribution.
The standard deviation is the average amount of variability in your data set. It tells you, on average, how far each score lies from the mean .
In normal distributions, a high standard deviation means that values are generally far from the mean, while a low standard deviation indicates that values are clustered close to the mean.
No. Because the range formula subtracts the lowest number from the highest number, the range is always zero or a positive number.
In statistics, the range is the spread of your data from the lowest to the highest value in the distribution. It is the simplest measure of variability .
While central tendency tells you where most of your data points lie, variability summarizes how far apart your points from each other.
Data sets can have the same central tendency but different levels of variability or vice versa . Together, they give you a complete picture of your data.
Variability is most commonly measured with the following descriptive statistics :
Variability tells you how far apart points lie from each other and from the center of a distribution or a data set.
Variability is also referred to as spread, scatter or dispersion.
While interval and ratio data can both be categorized, ranked, and have equal spacing between adjacent values, only ratio scales have a true zero.
For example, temperature in Celsius or Fahrenheit is at an interval scale because zero is not the lowest possible temperature. In the Kelvin scale, a ratio scale, zero represents a total lack of thermal energy.
A critical value is the value of the test statistic which defines the upper and lower bounds of a confidence interval , or which defines the threshold of statistical significance in a statistical test. It describes how far from the mean of the distribution you have to go to cover a certain amount of the total variation in the data (i.e. 90%, 95%, 99%).
If you are constructing a 95% confidence interval and are using a threshold of statistical significance of p = 0.05, then your critical value will be identical in both cases.
The t -distribution gives more probability to observations in the tails of the distribution than the standard normal distribution (a.k.a. the z -distribution).
In this way, the t -distribution is more conservative than the standard normal distribution: to reach the same level of confidence or statistical significance , you will need to include a wider range of the data.
A t -score (a.k.a. a t -value) is equivalent to the number of standard deviations away from the mean of the t -distribution .
The t -score is the test statistic used in t -tests and regression tests. It can also be used to describe how far from the mean an observation is when the data follow a t -distribution.
The t -distribution is a way of describing a set of observations where most observations fall close to the mean , and the rest of the observations make up the tails on either side. It is a type of normal distribution used for smaller sample sizes, where the variance in the data is unknown.
The t -distribution forms a bell curve when plotted on a graph. It can be described mathematically using the mean and the standard deviation .
In statistics, ordinal and nominal variables are both considered categorical variables .
Even though ordinal data can sometimes be numerical, not all mathematical operations can be performed on them.
Ordinal data has two characteristics:
However, unlike with interval data, the distances between the categories are uneven or unknown.
Nominal and ordinal are two of the four levels of measurement . Nominal level data can only be classified, while ordinal level data can be classified and ordered.
Nominal data is data that can be labelled or classified into mutually exclusive categories within a variable. These categories cannot be ordered in a meaningful way.
For example, for the nominal variable of preferred mode of transportation, you may have the categories of car, bus, train, tram or bicycle.
If your confidence interval for a difference between groups includes zero, that means that if you run your experiment again you have a good chance of finding no difference between groups.
If your confidence interval for a correlation or regression includes zero, that means that if you run your experiment again there is a good chance of finding no correlation in your data.
In both of these cases, you will also find a high p -value when you run your statistical test, meaning that your results could have occurred under the null hypothesis of no relationship between variables or no difference between groups.
If you want to calculate a confidence interval around the mean of data that is not normally distributed , you have two choices:
The standard normal distribution , also called the z -distribution, is a special normal distribution where the mean is 0 and the standard deviation is 1.
Any normal distribution can be converted into the standard normal distribution by turning the individual values into z -scores. In a z -distribution, z -scores tell you how many standard deviations away from the mean each value lies.
The z -score and t -score (aka z -value and t -value) show how many standard deviations away from the mean of the distribution you are, assuming your data follow a z -distribution or a t -distribution .
These scores are used in statistical tests to show how far from the mean of the predicted distribution your statistical estimate is. If your test produces a z -score of 2.5, this means that your estimate is 2.5 standard deviations from the predicted mean.
The predicted mean and distribution of your estimate are generated by the null hypothesis of the statistical test you are using. The more standard deviations away from the predicted mean your estimate is, the less likely it is that the estimate could have occurred under the null hypothesis .
To calculate the confidence interval , you need to know:
Then you can plug these components into the confidence interval formula that corresponds to your data. The formula depends on the type of estimate (e.g. a mean or a proportion) and on the distribution of your data.
The confidence level is the percentage of times you expect to get close to the same estimate if you run your experiment again or resample the population in the same way.
The confidence interval consists of the upper and lower bounds of the estimate you expect to find at a given level of confidence.
For example, if you are estimating a 95% confidence interval around the mean proportion of female babies born every year based on a random sample of babies, you might find an upper bound of 0.56 and a lower bound of 0.48. These are the upper and lower bounds of the confidence interval. The confidence level is 95%.
The mean is the most frequently used measure of central tendency because it uses all values in the data set to give you an average.
For data from skewed distributions, the median is better than the mean because it isn’t influenced by extremely large values.
The mode is the only measure you can use for nominal or categorical data that can’t be ordered.
The measures of central tendency you can use depends on the level of measurement of your data.
Measures of central tendency help you find the middle, or the average, of a data set.
The 3 most common measures of central tendency are the mean, median and mode.
Some variables have fixed levels. For example, gender and ethnicity are always nominal level data because they cannot be ranked.
However, for other variables, you can choose the level of measurement . For example, income is a variable that can be recorded on an ordinal or a ratio scale:
If you have a choice, the ratio level is always preferable because you can analyze data in more ways. The higher the level of measurement, the more precise your data is.
The level at which you measure a variable determines how you can analyze your data.
Depending on the level of measurement , you can perform different descriptive statistics to get an overall summary of your data and inferential statistics to see if your results support or refute your hypothesis .
Levels of measurement tell you how precisely variables are recorded. There are 4 levels of measurement, which can be ranked from low to high:
No. The p -value only tells you how likely the data you have observed is to have occurred under the null hypothesis .
If the p -value is below your threshold of significance (typically p < 0.05), then you can reject the null hypothesis, but this does not necessarily mean that your alternative hypothesis is true.
The alpha value, or the threshold for statistical significance , is arbitrary – which value you use depends on your field of study.
In most cases, researchers use an alpha of 0.05, which means that there is a less than 5% chance that the data being tested could have occurred under the null hypothesis.
P -values are usually automatically calculated by the program you use to perform your statistical test. They can also be estimated using p -value tables for the relevant test statistic .
P -values are calculated from the null distribution of the test statistic. They tell you how often a test statistic is expected to occur under the null hypothesis of the statistical test, based on where it falls in the null distribution.
If the test statistic is far from the mean of the null distribution, then the p -value will be small, showing that the test statistic is not likely to have occurred under the null hypothesis.
A p -value , or probability value, is a number describing how likely it is that your data would have occurred under the null hypothesis of your statistical test .
The test statistic you use will be determined by the statistical test.
You can choose the right statistical test by looking at what type of data you have collected and what type of relationship you want to test.
The test statistic will change based on the number of observations in your data, how variable your observations are, and how strong the underlying patterns in the data are.
For example, if one data set has higher variability while another has lower variability, the first data set will produce a test statistic closer to the null hypothesis , even if the true correlation between two variables is the same in either data set.
The formula for the test statistic depends on the statistical test being used.
Generally, the test statistic is calculated as the pattern in your data (i.e. the correlation between variables or difference between groups) divided by the variance in the data (i.e. the standard deviation ).
The 3 main types of descriptive statistics concern the frequency distribution, central tendency, and variability of a dataset.
Descriptive statistics summarize the characteristics of a data set. Inferential statistics allow you to test a hypothesis or assess whether your data is generalizable to the broader population.
In statistics, model selection is a process researchers use to compare the relative value of different statistical models and determine which one is the best fit for the observed data.
The Akaike information criterion is one of the most common methods of model selection. AIC weights the ability of the model to predict the observed data against the number of parameters the model requires to reach that level of precision.
AIC model selection can help researchers find a model that explains the observed variation in their data while avoiding overfitting.
In statistics, a model is the collection of one or more independent variables and their predicted interactions that researchers use to try to explain variation in their dependent variable.
You can test a model using a statistical test . To compare how well different models fit your data, you can use Akaike’s information criterion for model selection.
The Akaike information criterion is calculated from the maximum log-likelihood of the model and the number of parameters (K) used to reach that likelihood. The AIC function is 2K – 2(log-likelihood) .
Lower AIC values indicate a better-fit model, and a model with a delta-AIC (the difference between the two AIC values being compared) of more than -2 is considered significantly better than the model it is being compared to.
The Akaike information criterion is a mathematical test used to evaluate how well a model fits the data it is meant to describe. It penalizes models which use more independent variables (parameters) as a way to avoid over-fitting.
AIC is most often used to compare the relative goodness-of-fit among different models under consideration and to then choose the model that best fits the data.
A factorial ANOVA is any ANOVA that uses more than one categorical independent variable . A two-way ANOVA is a type of factorial ANOVA.
Some examples of factorial ANOVAs include:
In ANOVA, the null hypothesis is that there is no difference among group means. If any group differs significantly from the overall group mean, then the ANOVA will report a statistically significant result.
Significant differences among group means are calculated using the F statistic, which is the ratio of the mean sum of squares (the variance explained by the independent variable) to the mean square error (the variance left over).
If the F statistic is higher than the critical value (the value of F that corresponds with your alpha value, usually 0.05), then the difference among groups is deemed statistically significant.
The only difference between one-way and two-way ANOVA is the number of independent variables . A one-way ANOVA has one independent variable, while a two-way ANOVA has two.
All ANOVAs are designed to test for differences among three or more groups. If you are only testing for a difference between two groups, use a t-test instead.
Multiple linear regression is a regression model that estimates the relationship between a quantitative dependent variable and two or more independent variables using a straight line.
Linear regression most often uses mean-square error (MSE) to calculate the error of the model. MSE is calculated by:
Linear regression fits a line to the data by finding the regression coefficient that results in the smallest MSE.
Simple linear regression is a regression model that estimates the relationship between one independent variable and one dependent variable using a straight line. Both variables should be quantitative.
For example, the relationship between temperature and the expansion of mercury in a thermometer can be modeled using a straight line: as temperature increases, the mercury expands. This linear relationship is so certain that we can use mercury thermometers to measure temperature.
A regression model is a statistical model that estimates the relationship between one dependent variable and one or more independent variables using a line (or a plane in the case of two or more independent variables).
A regression model can be used when the dependent variable is quantitative, except in the case of logistic regression, where the dependent variable is binary.
A t-test should not be used to measure differences among more than two groups, because the error structure for a t-test will underestimate the actual error when many groups are being compared.
If you want to compare the means of several groups at once, it’s best to use another statistical test such as ANOVA or a post-hoc test.
A one-sample t-test is used to compare a single population to a standard value (for example, to determine whether the average lifespan of a specific town is different from the country average).
A paired t-test is used to compare a single population before and after some experimental intervention or at two different points in time (for example, measuring student performance on a test before and after being taught the material).
A t-test measures the difference in group means divided by the pooled standard error of the two group means.
In this way, it calculates a number (the t-value) illustrating the magnitude of the difference between the two group means being compared, and estimates the likelihood that this difference exists purely by chance (p-value).
Your choice of t-test depends on whether you are studying one group or two groups, and whether you care about the direction of the difference in group means.
If you are studying one group, use a paired t-test to compare the group mean over time or after an intervention, or use a one-sample t-test to compare the group mean to a standard value. If you are studying two groups, use a two-sample t-test .
If you want to know only whether a difference exists, use a two-tailed test . If you want to know if one group mean is greater or less than the other, use a left-tailed or right-tailed one-tailed test .
A t-test is a statistical test that compares the means of two samples . It is used in hypothesis testing , with a null hypothesis that the difference in group means is zero and an alternate hypothesis that the difference in group means is different from zero.
Statistical significance is a term used by researchers to state that it is unlikely their observations could have occurred under the null hypothesis of a statistical test . Significance is usually denoted by a p -value , or probability value.
Statistical significance is arbitrary – it depends on the threshold, or alpha value, chosen by the researcher. The most common threshold is p < 0.05, which means that the data is likely to occur less than 5% of the time under the null hypothesis .
When the p -value falls below the chosen alpha value, then we say the result of the test is statistically significant.
A test statistic is a number calculated by a statistical test . It describes how far your observed data is from the null hypothesis of no relationship between variables or no difference among sample groups.
The test statistic tells you how different two or more groups are from the overall population mean , or how different a linear slope is from the slope predicted by a null hypothesis . Different test statistics are used in different statistical tests.
Statistical tests commonly assume that:
If your data does not meet these assumptions you might still be able to use a nonparametric statistical test , which have fewer requirements but also make weaker inferences.
Want to contact us directly? No problem. We are always here for you.
Our team helps students graduate by offering:
Scribbr specializes in editing study-related documents . We proofread:
Scribbr’s Plagiarism Checker is powered by elements of Turnitin’s Similarity Checker , namely the plagiarism detection software and the Internet Archive and Premium Scholarly Publications content databases .
The add-on AI detector is powered by Scribbr’s proprietary software.
The Scribbr Citation Generator is developed using the open-source Citation Style Language (CSL) project and Frank Bennett’s citeproc-js . It’s the same technology used by dozens of other popular citation tools, including Mendeley and Zotero.
You can find all the citation styles and locales used in the Scribbr Citation Generator in our publicly accessible repository on Github .
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.
H a : The alternative hypothesis: It is a claim about the population that is contradictory to H 0 and what we conclude when we reject H 0 . This is usually what the researcher is trying to prove.
Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data.
After you have determined which hypothesis the sample supports, you make a decision. There are two options for a decision. They are "reject H 0 " if the sample information favors the alternative hypothesis or "do not reject H 0 " or "decline to reject H 0 " if the sample information is insufficient to reject the null hypothesis.
Mathematical Symbols Used in H 0 and H a :
equal (=) | not equal (≠) greater than (>) less than (<) |
greater than or equal to (≥) | less than (<) |
less than or equal to (≤) | more 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.
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.
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.
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.
Bring to class a newspaper, some news magazines, and some Internet articles . In groups, find articles from which your group can write null and alternative hypotheses. Discuss your hypotheses with the rest of the class.
This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.
Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.
Access for free at https://openstax.org/books/introductory-statistics-2e/pages/1-introduction
© Jul 18, 2024 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.
As you were browsing something about your browser made us think you were a bot. There are a few reasons this might happen:
To regain access, please make sure that cookies and JavaScript are enabled before reloading the page.
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.
Yarilet Perez is an experienced multimedia journalist and fact-checker with a Master of Science in Journalism. She has worked in multiple cities covering breaking news, politics, education, and more. Her expertise is in personal finance and investing, and real estate.
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 their 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.”
Correction—July 23, 2024: This article was corrected to state accurate examples of null hypothesis in the Null Hypothesis Examples section.
COMMENTS
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 population.". The null and alternative are always claims about the population. That's because the goal of hypothesis testing is to make inferences about a population based on a sample.
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 (H 0) answers "No, there's no effect in the population.". On the other hand, the alternative hypothesis (H A) answers "Yes, there ...
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.
Table of contents. 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.
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.
Research Question: Does the data suggest that the population mean dosage of this brand is different than 50 mg? Response Variable: dosage of the active ingredient found by a chemical assay. State Null and Alternative Hypotheses. Null Hypothesis: On the average, the dosage sold under this brand is 50 mg (population mean dosage = 50 mg).
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.
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.
These kinds of null hypotheses are the subject of Chapters 8 through 12. The Null hypothesis (HO) (H O) is a statement about the comparisons, e.g., between a sample statistic and the population, or between two treatment groups. The former is referred to as a one-tailed test whereas the latter is called a two-tailed test.
6. Write a null hypothesis. If your research involves statistical hypothesis testing, you will also have to write a null hypothesis. The null hypothesis is the default position that there is no association between the variables. The null hypothesis is written as H 0, while the alternative hypothesis is H 1 or H a.
The null hypothesis is a presumption of status quo or no change. Alternative Hypothesis (H a) - This is also known as the claim. This hypothesis should state what you expect the data to show, based on your research on the topic. This is your answer to your research question. Examples: Null Hypothesis: H 0: There is no difference in the salary ...
The actual test begins by considering two hypotheses. They are called the null hypothesis and the alternative hypothesis. These hypotheses contain opposing viewpoints. H0 H 0: The null hypothesis: It is a statement of no difference between a sample mean or proportion and a population mean or proportion. In other words, the difference equals 0.
The first step in hypothesis testing is to set up two competing hypotheses. The hypotheses are the most important aspect. If the hypotheses are incorrect, your conclusion will also be incorrect. The two hypotheses are named the null hypothesis and the alternative hypothesis. The null hypothesis is typically denoted as H 0.
Null and Alternative Hypotheses. Converting research questions to hypothesis is a simple task. Take the questions and make it a positive statement that says a relationship exists (correlation studies) or a difference exists between the groups (experiment study) and you have the alternative hypothesis. Write the statement such that a ...
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.
- Following a null hypothesis, an alternative hypothesis predicts a relationship between 2 study variables: The new drug (variable 1) is better on average in reducing the level of pain from pulmonary metastasis than the current drug (variable 2). ... Examples of ambiguous research question and hypothesis that result in unclear and weak research ...
The null and alternative hypotheses are both statements about the population that you are studying. The null hypothesis is often stated as the assumption that there is no change, no difference between two groups, or no relationship between two variables. The alternative hypothesis, on the other hand, is the statement that there is a change, difference, or relationship.
Most technical papers rely on just the first formulation, even though you may see some of the others in a statistics textbook. Null hypothesis: " x is equal to y.". Alternative hypothesis " x is not equal to y.". Null hypothesis: " x is at least y.". Alternative hypothesis " x is less than y.". Null hypothesis: " x is at most ...
14. The rule for the proper formulation of a hypothesis test is that the alternative or research hypothesis is the statement that, if true, is strongly supported by the evidence furnished by the data. The null hypothesis is generally the complement of the alternative hypothesis. Frequently, it is (or contains) the assumption that you are making ...
The null hypothesis should not reflect personal beliefs but a neutral stance on the research question. Guidelines for Creating Alternate Hypotheses The alternate hypothesis is the statement that you aim to provide evidence for, suggesting that there is an effect or a difference.
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 alternative hypothesis is the one that guides the research design, as it directs the researcher toward gathering evidence that will either support or refute the predicted relationship. The research process is structured around testing this hypothesis and determining whether the evidence is strong enough to reject the null hypothesis.
A research hypothesis is your proposed answer to your research question. The research hypothesis usually includes an explanation ("x affects y because …"). A statistical hypothesis, on the other hand, is a mathematical statement about a population parameter. Statistical hypotheses always come in pairs: the null and alternative hypotheses.
The actual test begins by considering two hypotheses.They are called the null hypothesis and the alternative hypothesis.These hypotheses contain opposing viewpoints. 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 Research Hypothesis. A research hypothesis is a mathematical way of stating a research question. A research hypothesis names the groups (we'll start with a sample and a population), what was measured, and which we think will have a higher mean. The last one gives the research hypothesis a direction. In other words, a research hypothesis ...
In step 2, we assume the null hypothesis is true and simulate a sampling distribution to visualize what we might expect to see under this claim. Try It! Assuming the null hypothesis is true Let's reframe the research question into two possible answers that will form the basis of the competing claims about the population parameter.
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.