how to prove a hypothesis is true

Step-by-step guide to hypothesis testing in statistics

Hypothesis testing in statistics helps us use data to make informed decisions. It starts with an assumption or guess about a group or population—something we believe might be true. We then collect sample data to check if there is enough evidence to support or reject that guess. This method is useful in many fields, like science, business, and healthcare, where decisions need to be based on facts.

Learning how to do hypothesis testing in statistics step-by-step can help you better understand data and make smarter choices, even when things are uncertain. This guide will take you through each step, from creating your hypothesis to making sense of the results, so you can see how it works in practical situations.

What is Hypothesis Testing?

Table of Contents

Hypothesis testing is a method for determining whether data supports a certain idea or assumption about a larger group. It starts by making a guess, like an average or a proportion, and then uses a small sample of data to see if that guess seems true or not.

For example, if a company wants to know if its new product is more popular than its old one, it can use hypothesis testing. They start with a statement like “The new product is not more popular than the old one” (this is the null hypothesis) and compare it with “The new product is more popular” (this is the alternative hypothesis). Then, they look at customer feedback to see if there’s enough evidence to reject the first statement and support the second one.

Simply put, hypothesis testing is a way to use data to help make decisions and understand what the data is really telling us, even when we don’t have all the answers.

Importance Of Hypothesis Testing In Decision-Making And Data Analysis

Hypothesis testing is important because it helps us make smart choices and understand data better. Here’s why it’s useful:

• Reduces Guesswork : It helps us see if our guesses or ideas are likely correct, even when we don’t have all the details.
• Uses Real Data : Instead of just guessing, it checks if our ideas match up with real data, which makes our decisions more reliable.
• Avoids Errors : It helps us avoid mistakes by carefully checking if our ideas are right so we don’t make costly errors.
• Shows What to Do Next : It tells us if our ideas work or not, helping us decide whether to keep, change, or drop something. For example, a company might test a new ad and decide what to do based on the results.
• Confirms Research Findings : It makes sure that research results are accurate and not just random chance so that we can trust the findings.

Here’s a simple guide to understanding hypothesis testing, with an example:

1. Set Up Your Hypotheses

Explanation: Start by defining two statements:

• Null Hypothesis (H0): This is the idea that there is no change or effect. It’s what you assume is true.
• Alternative Hypothesis (H1): This is what you want to test. It suggests there is a change or effect.

Example: Suppose a company says their new batteries last an average of 500 hours. To check this:

• Null Hypothesis (H0): The average battery life is 500 hours.
• Alternative Hypothesis (H1): The average battery life is not 500 hours.

2. Choose the Test

Explanation: Pick a statistical test that fits your data and your hypotheses. Different tests are used for various kinds of data.

Example: Since you’re comparing the average battery life, you use a one-sample t-test .

3. Set the Significance Level

Explanation: Decide how much risk you’re willing to take if you make a wrong decision. This is called the significance level, often set at 0.05 or 5%.

Example: You choose a significance level of 0.05, meaning you’re okay with a 5% chance of being wrong.

4. Gather and Analyze Data

Explanation: Collect your data and perform the test. Calculate the test statistic to see how far your sample result is from what you assumed.

Example: You test 30 batteries and find they last an average of 485 hours. You then calculate how this average compares to the claimed 500 hours using the t-test.

5. Find the p-Value

Explanation: The p-value tells you the probability of getting a result as extreme as yours if the null hypothesis is true.

Example: You find a p-value of 0.0001. This means there’s a very small chance (0.01%) of getting an average battery life of 485 hours or less if the true average is 500 hours.

6. Make Your Decision

Explanation: Compare the p-value to your significance level. If the p-value is smaller, you reject the null hypothesis. If it’s larger, you do not reject it.

Example: Since 0.0001 is much less than 0.05, you reject the null hypothesis. This means the data suggests the average battery life is different from 500 hours.

7. Report Your Findings

Explanation: Summarize what the results mean. State whether you rejected the null hypothesis and what that implies.

Example: You conclude that the average battery life is likely different from 500 hours. This suggests the company’s claim might not be accurate.

Hypothesis testing is a way to use data to check if your guesses or assumptions are likely true. By following these steps—setting up your hypotheses, choosing the right test, deciding on a significance level, analyzing your data, finding the p-value, making a decision, and reporting results—you can determine if your data supports or challenges your initial idea.

Understanding Hypothesis Testing: A Simple Explanation

Hypothesis testing is a way to use data to make decisions. Here’s a straightforward guide:

1. What is the Null and Alternative Hypotheses?

• Null Hypothesis (H0): This is your starting assumption. It says that nothing has changed or that there is no effect. It’s what you assume to be true until your data shows otherwise. Example: If a company says their batteries last 500 hours, the null hypothesis is: “The average battery life is 500 hours.” This means you think the claim is correct unless you find evidence to prove otherwise.
• Alternative Hypothesis (H1): This is what you want to find out. It suggests that there is an effect or a difference. It’s what you are testing to see if it might be true. Example: To test the company’s claim, you might say: “The average battery life is not 500 hours.” This means you think the average battery life might be different from what the company says.

2. One-Tailed vs. Two-Tailed Tests

• One-Tailed Test: This test checks for an effect in only one direction. You use it when you’re only interested in finding out if something is either more or less than a specific value. Example: If you think the battery lasts longer than 500 hours, you would use a one-tailed test to see if the battery life is significantly more than 500 hours.
• Two-Tailed Test: This test checks for an effect in both directions. Use this when you want to see if something is different from a specific value, whether it’s more or less. Example: If you want to see if the battery life is different from 500 hours, whether it’s more or less, you would use a two-tailed test. This checks for any significant difference, regardless of the direction.

3. Common Misunderstandings

• Clarification: Hypothesis testing doesn’t prove that the null hypothesis is true. It just helps you decide if you should reject it. If there isn’t enough evidence against it, you don’t reject it, but that doesn’t mean it’s definitely true.
• Clarification: A small p-value shows that your data is unlikely if the null hypothesis is true. It suggests that the alternative hypothesis might be right, but it doesn’t prove the null hypothesis is false.
• Clarification: The significance level (alpha) is a set threshold, like 0.05, that helps you decide how much risk you’re willing to take for making a wrong decision. It should be chosen carefully, not randomly.
• Clarification: Hypothesis testing helps you make decisions based on data, but it doesn’t guarantee your results are correct. The quality of your data and the right choice of test affect how reliable your results are.

Benefits and Limitations of Hypothesis Testing

• Clear Decisions: Hypothesis testing helps you make clear decisions based on data. It shows whether the evidence supports or goes against your initial idea.
• Objective Analysis: It relies on data rather than personal opinions, so your decisions are based on facts rather than feelings.
• Concrete Numbers: You get specific numbers, like p-values, to understand how strong the evidence is against your idea.
• Control Risk: You can set a risk level (alpha level) to manage the chance of making an error, which helps avoid incorrect conclusions.
• Widely Used: It can be used in many areas, from science and business to social studies and engineering, making it a versatile tool.

Limitations

• Sample Size Matters: The results can be affected by the size of the sample. Small samples might give unreliable results, while large samples might find differences that aren’t meaningful in real life.
• Risk of Misinterpretation: A small p-value means the results are unlikely if the null hypothesis is true, but it doesn’t show how important the effect is.
• Needs Assumptions: Hypothesis testing requires certain conditions, like data being normally distributed . If these aren’t met, the results might not be accurate.
• Simple Decisions: It often results in a basic yes or no decision without giving detailed information about the size or impact of the effect.
• Can Be Misused: Sometimes, people misuse hypothesis testing, tweaking data to get a desired result or focusing only on whether the result is statistically significant.
• No Absolute Proof: Hypothesis testing doesn’t prove that your hypothesis is true. It only helps you decide if there’s enough evidence to reject the null hypothesis, so the conclusions are based on likelihood, not certainty.

Final Thoughts 

Hypothesis testing helps you make decisions based on data. It involves setting up your initial idea, picking a significance level, doing the test, and looking at the results. By following these steps, you can make sure your conclusions are based on solid information, not just guesses.

This approach lets you see if the evidence supports or contradicts your initial idea, helping you make better decisions. But remember that hypothesis testing isn’t perfect. Things like sample size and assumptions can affect the results, so it’s important to be aware of these limitations.

In simple terms, using a step-by-step guide for hypothesis testing is a great way to better understand your data. Follow the steps carefully and keep in mind the method’s limits.

What is the difference between one-tailed and two-tailed tests?

 A one-tailed test assesses the probability of the observed data in one direction (either greater than or less than a certain value). In contrast, a two-tailed test looks at both directions (greater than and less than) to detect any significant deviation from the null hypothesis.

How do you choose the appropriate test for hypothesis testing?

The choice of test depends on the type of data you have and the hypotheses you are testing. Common tests include t-tests, chi-square tests, and ANOVA. You get more details about ANOVA, you may read Complete Details on What is ANOVA in Statistics ?  It’s important to match the test to the data characteristics and the research question.

What is the role of sample size in hypothesis testing?  

Sample size affects the reliability of hypothesis testing. Larger samples provide more reliable estimates and can detect smaller effects, while smaller samples may lead to less accurate results and reduced power.

Can hypothesis testing prove that a hypothesis is true?  

Hypothesis testing cannot prove that a hypothesis is true. It can only provide evidence to support or reject the null hypothesis. A result can indicate whether the data is consistent with the null hypothesis or not, but it does not prove the alternative hypothesis with certainty.

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What is a scientific hypothesis?

It's the initial building block in the scientific method.

Hypothesis basics

What makes a hypothesis testable.

• Types of hypotheses
• Hypothesis versus theory

Additional resources

Bibliography.

A scientific hypothesis is a tentative, testable explanation for a phenomenon in the natural world. It's the initial building block in the scientific method . Many describe it as an "educated guess" based on prior knowledge and observation. While this is true, a hypothesis is more informed than a guess. While an "educated guess" suggests a random prediction based on a person's expertise, developing a hypothesis requires active observation and background research. 

The basic idea of a hypothesis is that there is no predetermined outcome. For a solution to be termed a scientific hypothesis, it has to be an idea that can be supported or refuted through carefully crafted experimentation or observation. This concept, called falsifiability and testability, was advanced in the mid-20th century by Austrian-British philosopher Karl Popper in his famous book "The Logic of Scientific Discovery" (Routledge, 1959).

A key function of a hypothesis is to derive predictions about the results of future experiments and then perform those experiments to see whether they support the predictions.

A hypothesis is usually written in the form of an if-then statement, which gives a possibility (if) and explains what may happen because of the possibility (then). The statement could also include "may," according to California State University, Bakersfield .

Here are some examples of hypothesis statements:

• If garlic repels fleas, then a dog that is given garlic every day will not get fleas.
• If sugar causes cavities, then people who eat a lot of candy may be more prone to cavities.
• If ultraviolet light can damage the eyes, then maybe this light can cause blindness.

A useful hypothesis should be testable and falsifiable. That means that it should be possible to prove it wrong. A theory that can't be proved wrong is nonscientific, according to Karl Popper's 1963 book " Conjectures and Refutations ."

An example of an untestable statement is, "Dogs are better than cats." That's because the definition of "better" is vague and subjective. However, an untestable statement can be reworded to make it testable. For example, the previous statement could be changed to this: "Owning a dog is associated with higher levels of physical fitness than owning a cat." With this statement, the researcher can take measures of physical fitness from dog and cat owners and compare the two.

Types of scientific hypotheses

In an experiment, researchers generally state their hypotheses in two ways. The null hypothesis predicts that there will be no relationship between the variables tested, or no difference between the experimental groups. The alternative hypothesis predicts the opposite: that there will be a difference between the experimental groups. This is usually the hypothesis scientists are most interested in, according to the University of Miami .

For example, a null hypothesis might state, "There will be no difference in the rate of muscle growth between people who take a protein supplement and people who don't." The alternative hypothesis would state, "There will be a difference in the rate of muscle growth between people who take a protein supplement and people who don't."

If the results of the experiment show a relationship between the variables, then the null hypothesis has been rejected in favor of the alternative hypothesis, according to the book " Research Methods in Psychology " (​​BCcampus, 2015). 

There are other ways to describe an alternative hypothesis. The alternative hypothesis above does not specify a direction of the effect, only that there will be a difference between the two groups. That type of prediction is called a two-tailed hypothesis. If a hypothesis specifies a certain direction — for example, that people who take a protein supplement will gain more muscle than people who don't — it is called a one-tailed hypothesis, according to William M. K. Trochim , a professor of Policy Analysis and Management at Cornell University.

Sometimes, errors take place during an experiment. These errors can happen in one of two ways. A type I error is when the null hypothesis is rejected when it is true. This is also known as a false positive. A type II error occurs when the null hypothesis is not rejected when it is false. This is also known as a false negative, according to the University of California, Berkeley . 

A hypothesis can be rejected or modified, but it can never be proved correct 100% of the time. For example, a scientist can form a hypothesis stating that if a certain type of tomato has a gene for red pigment, that type of tomato will be red. During research, the scientist then finds that each tomato of this type is red. Though the findings confirm the hypothesis, there may be a tomato of that type somewhere in the world that isn't red. Thus, the hypothesis is true, but it may not be true 100% of the time.

Scientific theory vs. scientific hypothesis

The best hypotheses are simple. They deal with a relatively narrow set of phenomena. But theories are broader; they generally combine multiple hypotheses into a general explanation for a wide range of phenomena, according to the University of California, Berkeley . For example, a hypothesis might state, "If animals adapt to suit their environments, then birds that live on islands with lots of seeds to eat will have differently shaped beaks than birds that live on islands with lots of insects to eat." After testing many hypotheses like these, Charles Darwin formulated an overarching theory: the theory of evolution by natural selection.

"Theories are the ways that we make sense of what we observe in the natural world," Tanner said. "Theories are structures of ideas that explain and interpret facts." 

• Read more about writing a hypothesis, from the American Medical Writers Association.
• Find out why a hypothesis isn't always necessary in science, from The American Biology Teacher.
• Learn about null and alternative hypotheses, from Prof. Essa on YouTube .

Encyclopedia Britannica. Scientific Hypothesis. Jan. 13, 2022. https://www.britannica.com/science/scientific-hypothesis

Karl Popper, "The Logic of Scientific Discovery," Routledge, 1959.

California State University, Bakersfield, "Formatting a testable hypothesis." https://www.csub.edu/~ddodenhoff/Bio100/Bio100sp04/formattingahypothesis.htm  

Karl Popper, "Conjectures and Refutations," Routledge, 1963.

Price, P., Jhangiani, R., & Chiang, I., "Research Methods of Psychology — 2nd Canadian Edition," BCcampus, 2015.‌

University of Miami, "The Scientific Method" http://www.bio.miami.edu/dana/161/evolution/161app1_scimethod.pdf  

William M.K. Trochim, "Research Methods Knowledge Base," https://conjointly.com/kb/hypotheses-explained/  

University of California, Berkeley, "Multiple Hypothesis Testing and False Discovery Rate" https://www.stat.berkeley.edu/~hhuang/STAT141/Lecture-FDR.pdf  

University of California, Berkeley, "Science at multiple levels" https://undsci.berkeley.edu/article/0_0_0/howscienceworks_19

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scientific hypothesis

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• National Center for Biotechnology Information - PubMed Central - On the scope of scientific hypotheses
• LiveScience - What is a scientific hypothesis?
• The Royal Society - Open Science - On the scope of scientific hypotheses

scientific hypothesis , an idea that proposes a tentative explanation about a phenomenon or a narrow set of phenomena observed in the natural world. The two primary features of a scientific hypothesis are falsifiability and testability, which are reflected in an “If…then” statement summarizing the idea and in the ability to be supported or refuted through observation and experimentation. The notion of the scientific hypothesis as both falsifiable and testable was advanced in the mid-20th century by Austrian-born British philosopher Karl Popper .

The formulation and testing of a hypothesis is part of the scientific method , the approach scientists use when attempting to understand and test ideas about natural phenomena. The generation of a hypothesis frequently is described as a creative process and is based on existing scientific knowledge, intuition , or experience. Therefore, although scientific hypotheses commonly are described as educated guesses, they actually are more informed than a guess. In addition, scientists generally strive to develop simple hypotheses, since these are easier to test relative to hypotheses that involve many different variables and potential outcomes. Such complex hypotheses may be developed as scientific models ( see scientific modeling ).

Depending on the results of scientific evaluation, a hypothesis typically is either rejected as false or accepted as true. However, because a hypothesis inherently is falsifiable, even hypotheses supported by scientific evidence and accepted as true are susceptible to rejection later, when new evidence has become available. In some instances, rather than rejecting a hypothesis because it has been falsified by new evidence, scientists simply adapt the existing idea to accommodate the new information. In this sense a hypothesis is never incorrect but only incomplete.

The investigation of scientific hypotheses is an important component in the development of scientific theory . Hence, hypotheses differ fundamentally from theories; whereas the former is a specific tentative explanation and serves as the main tool by which scientists gather data, the latter is a broad general explanation that incorporates data from many different scientific investigations undertaken to explore hypotheses.

Countless hypotheses have been developed and tested throughout the history of science . Several examples include the idea that living organisms develop from nonliving matter, which formed the basis of spontaneous generation , a hypothesis that ultimately was disproved (first in 1668, with the experiments of Italian physician Francesco Redi , and later in 1859, with the experiments of French chemist and microbiologist Louis Pasteur ); the concept proposed in the late 19th century that microorganisms cause certain diseases (now known as germ theory ); and the notion that oceanic crust forms along submarine mountain zones and spreads laterally away from them ( seafloor spreading hypothesis ).

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Methodology

• How to Write a Strong Hypothesis | Steps & Examples

How to Write a Strong Hypothesis | Steps & Examples

Published on May 6, 2022 by Shona McCombes . Revised on November 20, 2023.

A hypothesis is a statement that can be tested by scientific research. If you want to test a relationship between two or more variables, you need to write hypotheses before you start your experiment or data collection .

Example: Hypothesis

Daily apple consumption leads to fewer doctor’s visits.

Table of contents

What is a hypothesis, developing a hypothesis (with example), hypothesis examples, other interesting articles, frequently asked questions about writing hypotheses.

A hypothesis states your predictions about what your research will find. It is a tentative answer to your research question that has not yet been tested. For some research projects, you might have to write several hypotheses that address different aspects of your research question.

A hypothesis is not just a guess – it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

Variables in hypotheses

Hypotheses propose a relationship between two or more types of variables .

• An independent variable is something the researcher changes or controls.
• A dependent variable is something the researcher observes and measures.

If there are any control variables , extraneous variables , or confounding variables , be sure to jot those down as you go to minimize the chances that research bias  will affect your results.

In this example, the independent variable is exposure to the sun – the assumed cause . The dependent variable is the level of happiness – the assumed effect .

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Step 1. ask a question.

Writing a hypothesis begins with a research question that you want to answer. The question should be focused, specific, and researchable within the constraints of your project.

Step 2. Do some preliminary research

Your initial answer to the question should be based on what is already known about the topic. Look for theories and previous studies to help you form educated assumptions about what your research will find.

At this stage, you might construct a conceptual framework to ensure that you’re embarking on a relevant topic . This can also help you identify which variables you will study and what you think the relationships are between them. Sometimes, you’ll have to operationalize more complex constructs.

Step 3. Formulate your hypothesis

Now you should have some idea of what you expect to find. Write your initial answer to the question in a clear, concise sentence.

4. Refine your hypothesis

You need to make sure your hypothesis is specific and testable. There are various ways of phrasing a hypothesis, but all the terms you use should have clear definitions, and the hypothesis should contain:

• The relevant variables
• The specific group being studied
• The predicted outcome of the experiment or analysis

5. Phrase your hypothesis in three ways

To identify the variables, you can write a simple prediction in  if…then form. The first part of the sentence states the independent variable and the second part states the dependent variable.

In academic research, hypotheses are more commonly phrased in terms of correlations or effects, where you directly state the predicted relationship between variables.

If you are comparing two groups, the hypothesis can state what difference you expect to find between them.

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 .

• H 0 : The number of lectures attended by first-year students has no effect on their final exam scores.
• H 1 : The number of lectures attended by first-year students has a positive effect on their final exam scores.

If you want to know more about the research process , methodology , research bias , or statistics , make sure to check out some of our other articles with explanations and examples.

• Sampling methods
• Simple random sampling
• Stratified sampling
• Cluster sampling
• Likert scales
• Reproducibility

 Statistics

• Null hypothesis
• Statistical power
• Probability distribution
• Effect size
• Poisson distribution

Research bias

• Optimism bias
• Cognitive bias
• Implicit bias
• Hawthorne effect
• Anchoring bias
• Explicit bias

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A hypothesis is not just a guess — it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

Null and alternative hypotheses are used in statistical hypothesis testing . The null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship.

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

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Home » What is a Hypothesis – Types, Examples and Writing Guide

What is a Hypothesis – Types, Examples and Writing Guide

Table of Contents

Definition:

Hypothesis is an educated guess or proposed explanation for a phenomenon, based on some initial observations or data. It is a tentative statement that can be tested and potentially proven or disproven through further investigation and experimentation.

Hypothesis is often used in scientific research to guide the design of experiments and the collection and analysis of data. It is an essential element of the scientific method, as it allows researchers to make predictions about the outcome of their experiments and to test those predictions to determine their accuracy.

Types of Hypothesis

Types of Hypothesis are as follows:

Research Hypothesis

A research hypothesis is a statement that predicts a relationship between variables. It is usually formulated as a specific statement that can be tested through research, and it is often used in scientific research to guide the design of experiments.

Null Hypothesis

The null hypothesis is a statement that assumes there is no significant difference or relationship between variables. It is often used as a starting point for testing the research hypothesis, and if the results of the study reject the null hypothesis, it suggests that there is a significant difference or relationship between variables.

Alternative Hypothesis

An alternative hypothesis is a statement that assumes there is a significant difference or relationship between variables. It is often used as an alternative to the null hypothesis and is tested against the null hypothesis to determine which statement is more accurate.

Directional Hypothesis

A directional hypothesis is a statement that predicts the direction of the relationship between variables. For example, a researcher might predict that increasing the amount of exercise will result in a decrease in body weight.

Non-directional Hypothesis

A non-directional hypothesis is a statement that predicts the relationship between variables but does not specify the direction. For example, a researcher might predict that there is a relationship between the amount of exercise and body weight, but they do not specify whether increasing or decreasing exercise will affect body weight.

Statistical Hypothesis

A statistical hypothesis is a statement that assumes a particular statistical model or distribution for the data. It is often used in statistical analysis to test the significance of a particular result.

Composite Hypothesis

A composite hypothesis is a statement that assumes more than one condition or outcome. It can be divided into several sub-hypotheses, each of which represents a different possible outcome.

Empirical Hypothesis

An empirical hypothesis is a statement that is based on observed phenomena or data. It is often used in scientific research to develop theories or models that explain the observed phenomena.

Simple Hypothesis

A simple hypothesis is a statement that assumes only one outcome or condition. It is often used in scientific research to test a single variable or factor.

Complex Hypothesis

A complex hypothesis is a statement that assumes multiple outcomes or conditions. It is often used in scientific research to test the effects of multiple variables or factors on a particular outcome.

Applications of Hypothesis

Hypotheses are used in various fields to guide research and make predictions about the outcomes of experiments or observations. Here are some examples of how hypotheses are applied in different fields:

• Science : In scientific research, hypotheses are used to test the validity of theories and models that explain natural phenomena. For example, a hypothesis might be formulated to test the effects of a particular variable on a natural system, such as the effects of climate change on an ecosystem.
• Medicine : In medical research, hypotheses are used to test the effectiveness of treatments and therapies for specific conditions. For example, a hypothesis might be formulated to test the effects of a new drug on a particular disease.
• Psychology : In psychology, hypotheses are used to test theories and models of human behavior and cognition. For example, a hypothesis might be formulated to test the effects of a particular stimulus on the brain or behavior.
• Sociology : In sociology, hypotheses are used to test theories and models of social phenomena, such as the effects of social structures or institutions on human behavior. For example, a hypothesis might be formulated to test the effects of income inequality on crime rates.
• Business : In business research, hypotheses are used to test the validity of theories and models that explain business phenomena, such as consumer behavior or market trends. For example, a hypothesis might be formulated to test the effects of a new marketing campaign on consumer buying behavior.
• Engineering : In engineering, hypotheses are used to test the effectiveness of new technologies or designs. For example, a hypothesis might be formulated to test the efficiency of a new solar panel design.

How to write a Hypothesis

Here are the steps to follow when writing a hypothesis:

Identify the Research Question

The first step is to identify the research question that you want to answer through your study. This question should be clear, specific, and focused. It should be something that can be investigated empirically and that has some relevance or significance in the field.

Conduct a Literature Review

Before writing your hypothesis, it’s essential to conduct a thorough literature review to understand what is already known about the topic. This will help you to identify the research gap and formulate a hypothesis that builds on existing knowledge.

Determine the Variables

The next step is to identify the variables involved in the research question. A variable is any characteristic or factor that can vary or change. There are two types of variables: independent and dependent. The independent variable is the one that is manipulated or changed by the researcher, while the dependent variable is the one that is measured or observed as a result of the independent variable.

Formulate the Hypothesis

Based on the research question and the variables involved, you can now formulate your hypothesis. A hypothesis should be a clear and concise statement that predicts the relationship between the variables. It should be testable through empirical research and based on existing theory or evidence.

Write the Null Hypothesis

The null hypothesis is the opposite of the alternative hypothesis, which is the hypothesis that you are testing. The null hypothesis states that there is no significant difference or relationship between the variables. It is important to write the null hypothesis because it allows you to compare your results with what would be expected by chance.

Refine the Hypothesis

After formulating the hypothesis, it’s important to refine it and make it more precise. This may involve clarifying the variables, specifying the direction of the relationship, or making the hypothesis more testable.

Examples of Hypothesis

Here are a few examples of hypotheses in different fields:

• Psychology : “Increased exposure to violent video games leads to increased aggressive behavior in adolescents.”
• Biology : “Higher levels of carbon dioxide in the atmosphere will lead to increased plant growth.”
• Sociology : “Individuals who grow up in households with higher socioeconomic status will have higher levels of education and income as adults.”
• Education : “Implementing a new teaching method will result in higher student achievement scores.”
• Marketing : “Customers who receive a personalized email will be more likely to make a purchase than those who receive a generic email.”
• Physics : “An increase in temperature will cause an increase in the volume of a gas, assuming all other variables remain constant.”
• Medicine : “Consuming a diet high in saturated fats will increase the risk of developing heart disease.”

Purpose of Hypothesis

The purpose of a hypothesis is to provide a testable explanation for an observed phenomenon or a prediction of a future outcome based on existing knowledge or theories. A hypothesis is an essential part of the scientific method and helps to guide the research process by providing a clear focus for investigation. It enables scientists to design experiments or studies to gather evidence and data that can support or refute the proposed explanation or prediction.

The formulation of a hypothesis is based on existing knowledge, observations, and theories, and it should be specific, testable, and falsifiable. A specific hypothesis helps to define the research question, which is important in the research process as it guides the selection of an appropriate research design and methodology. Testability of the hypothesis means that it can be proven or disproven through empirical data collection and analysis. Falsifiability means that the hypothesis should be formulated in such a way that it can be proven wrong if it is incorrect.

In addition to guiding the research process, the testing of hypotheses can lead to new discoveries and advancements in scientific knowledge. When a hypothesis is supported by the data, it can be used to develop new theories or models to explain the observed phenomenon. When a hypothesis is not supported by the data, it can help to refine existing theories or prompt the development of new hypotheses to explain the phenomenon.

When to use Hypothesis

Here are some common situations in which hypotheses are used:

• In scientific research , hypotheses are used to guide the design of experiments and to help researchers make predictions about the outcomes of those experiments.
• In social science research , hypotheses are used to test theories about human behavior, social relationships, and other phenomena.
• I n business , hypotheses can be used to guide decisions about marketing, product development, and other areas. For example, a hypothesis might be that a new product will sell well in a particular market, and this hypothesis can be tested through market research.

Characteristics of Hypothesis

Here are some common characteristics of a hypothesis:

• Testable : A hypothesis must be able to be tested through observation or experimentation. This means that it must be possible to collect data that will either support or refute the hypothesis.
• Falsifiable : A hypothesis must be able to be proven false if it is not supported by the data. If a hypothesis cannot be falsified, then it is not a scientific hypothesis.
• Clear and concise : A hypothesis should be stated in a clear and concise manner so that it can be easily understood and tested.
• Based on existing knowledge : A hypothesis should be based on existing knowledge and research in the field. It should not be based on personal beliefs or opinions.
• Specific : A hypothesis should be specific in terms of the variables being tested and the predicted outcome. This will help to ensure that the research is focused and well-designed.
• Tentative: A hypothesis is a tentative statement or assumption that requires further testing and evidence to be confirmed or refuted. It is not a final conclusion or assertion.
• Relevant : A hypothesis should be relevant to the research question or problem being studied. It should address a gap in knowledge or provide a new perspective on the issue.

Advantages of Hypothesis

Hypotheses have several advantages in scientific research and experimentation:

• Guides research: A hypothesis provides a clear and specific direction for research. It helps to focus the research question, select appropriate methods and variables, and interpret the results.
• Predictive powe r: A hypothesis makes predictions about the outcome of research, which can be tested through experimentation. This allows researchers to evaluate the validity of the hypothesis and make new discoveries.
• Facilitates communication: A hypothesis provides a common language and framework for scientists to communicate with one another about their research. This helps to facilitate the exchange of ideas and promotes collaboration.
• Efficient use of resources: A hypothesis helps researchers to use their time, resources, and funding efficiently by directing them towards specific research questions and methods that are most likely to yield results.
• Provides a basis for further research: A hypothesis that is supported by data provides a basis for further research and exploration. It can lead to new hypotheses, theories, and discoveries.
• Increases objectivity: A hypothesis can help to increase objectivity in research by providing a clear and specific framework for testing and interpreting results. This can reduce bias and increase the reliability of research findings.

Limitations of Hypothesis

Some Limitations of the Hypothesis are as follows:

• Limited to observable phenomena: Hypotheses are limited to observable phenomena and cannot account for unobservable or intangible factors. This means that some research questions may not be amenable to hypothesis testing.
• May be inaccurate or incomplete: Hypotheses are based on existing knowledge and research, which may be incomplete or inaccurate. This can lead to flawed hypotheses and erroneous conclusions.
• May be biased: Hypotheses may be biased by the researcher’s own beliefs, values, or assumptions. This can lead to selective interpretation of data and a lack of objectivity in research.
• Cannot prove causation: A hypothesis can only show a correlation between variables, but it cannot prove causation. This requires further experimentation and analysis.
• Limited to specific contexts: Hypotheses are limited to specific contexts and may not be generalizable to other situations or populations. This means that results may not be applicable in other contexts or may require further testing.
• May be affected by chance : Hypotheses may be affected by chance or random variation, which can obscure or distort the true relationship between variables.

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S.3 hypothesis testing.

In reviewing hypothesis tests, we start first with the general idea. Then, we keep returning to the basic procedures of hypothesis testing, each time adding a little more detail.

The general idea of hypothesis testing involves:

• Making an initial assumption.
• Collecting evidence (data).
• Based on the available evidence (data), deciding whether to reject or not reject the initial assumption.

Every hypothesis test — regardless of the population parameter involved — requires the above three steps.

Example S.3.1

Is normal body temperature really 98.6 degrees f section  .

Consider the population of many, many adults. A researcher hypothesized that the average adult body temperature is lower than the often-advertised 98.6 degrees F. That is, the researcher wants an answer to the question: "Is the average adult body temperature 98.6 degrees? Or is it lower?" To answer his research question, the researcher starts by assuming that the average adult body temperature was 98.6 degrees F.

Then, the researcher went out and tried to find evidence that refutes his initial assumption. In doing so, he selects a random sample of 130 adults. The average body temperature of the 130 sampled adults is 98.25 degrees.

Then, the researcher uses the data he collected to make a decision about his initial assumption. It is either likely or unlikely that the researcher would collect the evidence he did given his initial assumption that the average adult body temperature is 98.6 degrees:

• If it is likely , then the researcher does not reject his initial assumption that the average adult body temperature is 98.6 degrees. There is not enough evidence to do otherwise.
• either the researcher's initial assumption is correct and he experienced a very unusual event;
• or the researcher's initial assumption is incorrect.

In statistics, we generally don't make claims that require us to believe that a very unusual event happened. That is, in the practice of statistics, if the evidence (data) we collected is unlikely in light of the initial assumption, then we reject our initial assumption.

Example S.3.2

Criminal trial analogy section  .

One place where you can consistently see the general idea of hypothesis testing in action is in criminal trials held in the United States. Our criminal justice system assumes "the defendant is innocent until proven guilty." That is, our initial assumption is that the defendant is innocent.

In the practice of statistics, we make our initial assumption when we state our two competing hypotheses -- the null hypothesis ( H 0 ) and the alternative hypothesis ( H A ). Here, our hypotheses are:

• H 0 : Defendant is not guilty (innocent)
• H A : Defendant is guilty

In statistics, we always assume the null hypothesis is true . That is, the null hypothesis is always our initial assumption.

The prosecution team then collects evidence — such as finger prints, blood spots, hair samples, carpet fibers, shoe prints, ransom notes, and handwriting samples — with the hopes of finding "sufficient evidence" to make the assumption of innocence refutable.

In statistics, the data are the evidence.

The jury then makes a decision based on the available evidence:

• If the jury finds sufficient evidence — beyond a reasonable doubt — to make the assumption of innocence refutable, the jury rejects the null hypothesis and deems the defendant guilty. We behave as if the defendant is guilty.
• If there is insufficient evidence, then the jury does not reject the null hypothesis . We behave as if the defendant is innocent.

In statistics, we always make one of two decisions. We either "reject the null hypothesis" or we "fail to reject the null hypothesis."

Errors in Hypothesis Testing Section  

Did you notice the use of the phrase "behave as if" in the previous discussion? We "behave as if" the defendant is guilty; we do not "prove" that the defendant is guilty. And, we "behave as if" the defendant is innocent; we do not "prove" that the defendant is innocent.

This is a very important distinction! We make our decision based on evidence not on 100% guaranteed proof. Again:

• If we reject the null hypothesis, we do not prove that the alternative hypothesis is true.
• If we do not reject the null hypothesis, we do not prove that the null hypothesis is true.

We merely state that there is enough evidence to behave one way or the other. This is always true in statistics! Because of this, whatever the decision, there is always a chance that we made an error .

Let's review the two types of errors that can be made in criminal trials:

Table S.3.2 shows how this corresponds to the two types of errors in hypothesis testing.

Note that, in statistics, we call the two types of errors by two different  names -- one is called a "Type I error," and the other is called  a "Type II error." Here are the formal definitions of the two types of errors:

There is always a chance of making one of these errors. But, a good scientific study will minimize the chance of doing so!

Making the Decision Section  

Recall that it is either likely or unlikely that we would observe the evidence we did given our initial assumption. If it is likely , we do not reject the null hypothesis. If it is unlikely , then we reject the null hypothesis in favor of the alternative hypothesis. Effectively, then, making the decision reduces to determining "likely" or "unlikely."

In statistics, there are two ways to determine whether the evidence is likely or unlikely given the initial assumption:

• We could take the " critical value approach " (favored in many of the older textbooks).
• Or, we could take the " P -value approach " (what is used most often in research, journal articles, and statistical software).

In the next two sections, we review the procedures behind each of these two approaches. To make our review concrete, let's imagine that μ is the average grade point average of all American students who major in mathematics. We first review the critical value approach for conducting each of the following three hypothesis tests about the population mean $\mu$:

In Practice

• We would want to conduct the first hypothesis test if we were interested in concluding that the average grade point average of the group is more than 3.
• We would want to conduct the second hypothesis test if we were interested in concluding that the average grade point average of the group is less than 3.
• And, we would want to conduct the third hypothesis test if we were only interested in concluding that the average grade point average of the group differs from 3 (without caring whether it is more or less than 3).

Upon completing the review of the critical value approach, we review the P -value approach for conducting each of the above three hypothesis tests about the population mean \(\mu\). The procedures that we review here for both approaches easily extend to hypothesis tests about any other population parameter.

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• v.23(Suppl 3); 2019 Sep

An Introduction to Statistics: Understanding Hypothesis Testing and Statistical Errors

Priya ranganathan.

1 Department of Anesthesiology, Critical Care and Pain, Tata Memorial Hospital, Mumbai, Maharashtra, India

2 Department of Surgical Oncology, Tata Memorial Centre, Mumbai, Maharashtra, India

The second article in this series on biostatistics covers the concepts of sample, population, research hypotheses and statistical errors.

How to cite this article

Ranganathan P, Pramesh CS. An Introduction to Statistics: Understanding Hypothesis Testing and Statistical Errors. Indian J Crit Care Med 2019;23(Suppl 3):S230–S231.

Two papers quoted in this issue of the Indian Journal of Critical Care Medicine report. The results of studies aim to prove that a new intervention is better than (superior to) an existing treatment. In the ABLE study, the investigators wanted to show that transfusion of fresh red blood cells would be superior to standard-issue red cells in reducing 90-day mortality in ICU patients. 1 The PROPPR study was designed to prove that transfusion of a lower ratio of plasma and platelets to red cells would be superior to a higher ratio in decreasing 24-hour and 30-day mortality in critically ill patients. 2 These studies are known as superiority studies (as opposed to noninferiority or equivalence studies which will be discussed in a subsequent article).

SAMPLE VERSUS POPULATION

A sample represents a group of participants selected from the entire population. Since studies cannot be carried out on entire populations, researchers choose samples, which are representative of the population. This is similar to walking into a grocery store and examining a few grains of rice or wheat before purchasing an entire bag; we assume that the few grains that we select (the sample) are representative of the entire sack of grains (the population).

The results of the study are then extrapolated to generate inferences about the population. We do this using a process known as hypothesis testing. This means that the results of the study may not always be identical to the results we would expect to find in the population; i.e., there is the possibility that the study results may be erroneous.

HYPOTHESIS TESTING

A clinical trial begins with an assumption or belief, and then proceeds to either prove or disprove this assumption. In statistical terms, this belief or assumption is known as a hypothesis. Counterintuitively, what the researcher believes in (or is trying to prove) is called the “alternate” hypothesis, and the opposite is called the “null” hypothesis; every study has a null hypothesis and an alternate hypothesis. For superiority studies, the alternate hypothesis states that one treatment (usually the new or experimental treatment) is superior to the other; the null hypothesis states that there is no difference between the treatments (the treatments are equal). For example, in the ABLE study, we start by stating the null hypothesis—there is no difference in mortality between groups receiving fresh RBCs and standard-issue RBCs. We then state the alternate hypothesis—There is a difference between groups receiving fresh RBCs and standard-issue RBCs. It is important to note that we have stated that the groups are different, without specifying which group will be better than the other. This is known as a two-tailed hypothesis and it allows us to test for superiority on either side (using a two-sided test). This is because, when we start a study, we are not 100% certain that the new treatment can only be better than the standard treatment—it could be worse, and if it is so, the study should pick it up as well. One tailed hypothesis and one-sided statistical testing is done for non-inferiority studies, which will be discussed in a subsequent paper in this series.

STATISTICAL ERRORS

There are two possibilities to consider when interpreting the results of a superiority study. The first possibility is that there is truly no difference between the treatments but the study finds that they are different. This is called a Type-1 error or false-positive error or alpha error. This means falsely rejecting the null hypothesis.

The second possibility is that there is a difference between the treatments and the study does not pick up this difference. This is called a Type 2 error or false-negative error or beta error. This means falsely accepting the null hypothesis.

The power of the study is the ability to detect a difference between groups and is the converse of the beta error; i.e., power = 1-beta error. Alpha and beta errors are finalized when the protocol is written and form the basis for sample size calculation for the study. In an ideal world, we would not like any error in the results of our study; however, we would need to do the study in the entire population (infinite sample size) to be able to get a 0% alpha and beta error. These two errors enable us to do studies with realistic sample sizes, with the compromise that there is a small possibility that the results may not always reflect the truth. The basis for this will be discussed in a subsequent paper in this series dealing with sample size calculation.

Conventionally, type 1 or alpha error is set at 5%. This means, that at the end of the study, if there is a difference between groups, we want to be 95% certain that this is a true difference and allow only a 5% probability that this difference has occurred by chance (false positive). Type 2 or beta error is usually set between 10% and 20%; therefore, the power of the study is 90% or 80%. This means that if there is a difference between groups, we want to be 80% (or 90%) certain that the study will detect that difference. For example, in the ABLE study, sample size was calculated with a type 1 error of 5% (two-sided) and power of 90% (type 2 error of 10%) (1).

Table 1 gives a summary of the two types of statistical errors with an example

Statistical errors

In the next article in this series, we will look at the meaning and interpretation of ‘ p ’ value and confidence intervals for hypothesis testing.

Source of support: Nil

Conflict of interest: None

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How to Write a Great Hypothesis

Hypothesis Definition, Format, Examples, and Tips

Verywell / Alex Dos Diaz

• The Scientific Method

Hypothesis Format

Falsifiability of a hypothesis.

• Operationalization

Hypothesis Types

Hypotheses examples.

• Collecting Data

A hypothesis is a tentative statement about the relationship between two or more variables. It is a specific, testable prediction about what you expect to happen in a study. It is a preliminary answer to your question that helps guide the research process.

Consider a study designed to examine the relationship between sleep deprivation and test performance. The hypothesis might be: "This study is designed to assess the hypothesis that sleep-deprived people will perform worse on a test than individuals who are not sleep-deprived."

At a Glance

A hypothesis is crucial to scientific research because it offers a clear direction for what the researchers are looking to find. This allows them to design experiments to test their predictions and add to our scientific knowledge about the world. This article explores how a hypothesis is used in psychology research, how to write a good hypothesis, and the different types of hypotheses you might use.

The Hypothesis in the Scientific Method

In the scientific method , whether it involves research in psychology, biology, or some other area, a hypothesis represents what the researchers think will happen in an experiment. The scientific method involves the following steps:

• Forming a question
• Performing background research
• Creating a hypothesis
• Designing an experiment
• Collecting data
• Analyzing the results
• Drawing conclusions
• Communicating the results

The hypothesis is a prediction, but it involves more than a guess. Most of the time, the hypothesis begins with a question which is then explored through background research. At this point, researchers then begin to develop a testable hypothesis.

Unless you are creating an exploratory study, your hypothesis should always explain what you  expect  to happen.

In a study exploring the effects of a particular drug, the hypothesis might be that researchers expect the drug to have some type of effect on the symptoms of a specific illness. In psychology, the hypothesis might focus on how a certain aspect of the environment might influence a particular behavior.

Remember, a hypothesis does not have to be correct. While the hypothesis predicts what the researchers expect to see, the goal of the research is to determine whether this guess is right or wrong. When conducting an experiment, researchers might explore numerous factors to determine which ones might contribute to the ultimate outcome.

In many cases, researchers may find that the results of an experiment  do not  support the original hypothesis. When writing up these results, the researchers might suggest other options that should be explored in future studies.

In many cases, researchers might draw a hypothesis from a specific theory or build on previous research. For example, prior research has shown that stress can impact the immune system. So a researcher might hypothesize: "People with high-stress levels will be more likely to contract a common cold after being exposed to the virus than people who have low-stress levels."

In other instances, researchers might look at commonly held beliefs or folk wisdom. "Birds of a feather flock together" is one example of folk adage that a psychologist might try to investigate. The researcher might pose a specific hypothesis that "People tend to select romantic partners who are similar to them in interests and educational level."

Elements of a Good Hypothesis

So how do you write a good hypothesis? When trying to come up with a hypothesis for your research or experiments, ask yourself the following questions:

• Is your hypothesis based on your research on a topic?
• Can your hypothesis be tested?
• Does your hypothesis include independent and dependent variables?

Before you come up with a specific hypothesis, spend some time doing background research. Once you have completed a literature review, start thinking about potential questions you still have. Pay attention to the discussion section in the  journal articles you read . Many authors will suggest questions that still need to be explored.

How to Formulate a Good Hypothesis

To form a hypothesis, you should take these steps:

• Collect as many observations about a topic or problem as you can.
• Evaluate these observations and look for possible causes of the problem.
• Create a list of possible explanations that you might want to explore.
• After you have developed some possible hypotheses, think of ways that you could confirm or disprove each hypothesis through experimentation. This is known as falsifiability.

In the scientific method ,  falsifiability is an important part of any valid hypothesis. In order to test a claim scientifically, it must be possible that the claim could be proven false.

Students sometimes confuse the idea of falsifiability with the idea that it means that something is false, which is not the case. What falsifiability means is that  if  something was false, then it is possible to demonstrate that it is false.

One of the hallmarks of pseudoscience is that it makes claims that cannot be refuted or proven false.

The Importance of Operational Definitions

A variable is a factor or element that can be changed and manipulated in ways that are observable and measurable. However, the researcher must also define how the variable will be manipulated and measured in the study.

Operational definitions are specific definitions for all relevant factors in a study. This process helps make vague or ambiguous concepts detailed and measurable.

For example, a researcher might operationally define the variable " test anxiety " as the results of a self-report measure of anxiety experienced during an exam. A "study habits" variable might be defined by the amount of studying that actually occurs as measured by time.

These precise descriptions are important because many things can be measured in various ways. Clearly defining these variables and how they are measured helps ensure that other researchers can replicate your results.

Replicability

One of the basic principles of any type of scientific research is that the results must be replicable.

Replication means repeating an experiment in the same way to produce the same results. By clearly detailing the specifics of how the variables were measured and manipulated, other researchers can better understand the results and repeat the study if needed.

Some variables are more difficult than others to define. For example, how would you operationally define a variable such as aggression ? For obvious ethical reasons, researchers cannot create a situation in which a person behaves aggressively toward others.

To measure this variable, the researcher must devise a measurement that assesses aggressive behavior without harming others. The researcher might utilize a simulated task to measure aggressiveness in this situation.

Hypothesis Checklist

• Does your hypothesis focus on something that you can actually test?
• Does your hypothesis include both an independent and dependent variable?
• Can you manipulate the variables?
• Can your hypothesis be tested without violating ethical standards?

The hypothesis you use will depend on what you are investigating and hoping to find. Some of the main types of hypotheses that you might use include:

• Simple hypothesis : This type of hypothesis suggests there is a relationship between one independent variable and one dependent variable.
• Complex hypothesis : This type suggests a relationship between three or more variables, such as two independent and dependent variables.
• Null hypothesis : This hypothesis suggests no relationship exists between two or more variables.
• Alternative hypothesis : This hypothesis states the opposite of the null hypothesis.
• Statistical hypothesis : This hypothesis uses statistical analysis to evaluate a representative population sample and then generalizes the findings to the larger group.
• Logical hypothesis : This hypothesis assumes a relationship between variables without collecting data or evidence.

A hypothesis often follows a basic format of "If {this happens} then {this will happen}." One way to structure your hypothesis is to describe what will happen to the  dependent variable  if you change the  independent variable .

The basic format might be: "If {these changes are made to a certain independent variable}, then we will observe {a change in a specific dependent variable}."

A few examples of simple hypotheses:

• "Students who eat breakfast will perform better on a math exam than students who do not eat breakfast."
• "Students who experience test anxiety before an English exam will get lower scores than students who do not experience test anxiety."​
• "Motorists who talk on the phone while driving will be more likely to make errors on a driving course than those who do not talk on the phone."
• "Children who receive a new reading intervention will have higher reading scores than students who do not receive the intervention."

Examples of a complex hypothesis include:

• "People with high-sugar diets and sedentary activity levels are more likely to develop depression."
• "Younger people who are regularly exposed to green, outdoor areas have better subjective well-being than older adults who have limited exposure to green spaces."

Examples of a null hypothesis include:

• "There is no difference in anxiety levels between people who take St. John's wort supplements and those who do not."
• "There is no difference in scores on a memory recall task between children and adults."
• "There is no difference in aggression levels between children who play first-person shooter games and those who do not."

Examples of an alternative hypothesis:

• "People who take St. John's wort supplements will have less anxiety than those who do not."
• "Adults will perform better on a memory task than children."
• "Children who play first-person shooter games will show higher levels of aggression than children who do not." 

Collecting Data on Your Hypothesis

Once a researcher has formed a testable hypothesis, the next step is to select a research design and start collecting data. The research method depends largely on exactly what they are studying. There are two basic types of research methods: descriptive research and experimental research.

Descriptive Research Methods

Descriptive research such as  case studies ,  naturalistic observations , and surveys are often used when  conducting an experiment is difficult or impossible. These methods are best used to describe different aspects of a behavior or psychological phenomenon.

Once a researcher has collected data using descriptive methods, a  correlational study  can examine how the variables are related. This research method might be used to investigate a hypothesis that is difficult to test experimentally.

Experimental Research Methods

Experimental methods  are used to demonstrate causal relationships between variables. In an experiment, the researcher systematically manipulates a variable of interest (known as the independent variable) and measures the effect on another variable (known as the dependent variable).

Unlike correlational studies, which can only be used to determine if there is a relationship between two variables, experimental methods can be used to determine the actual nature of the relationship—whether changes in one variable actually  cause  another to change.

The hypothesis is a critical part of any scientific exploration. It represents what researchers expect to find in a study or experiment. In situations where the hypothesis is unsupported by the research, the research still has value. Such research helps us better understand how different aspects of the natural world relate to one another. It also helps us develop new hypotheses that can then be tested in the future.

Thompson WH, Skau S. On the scope of scientific hypotheses .  R Soc Open Sci . 2023;10(8):230607. doi:10.1098/rsos.230607

Taran S, Adhikari NKJ, Fan E. Falsifiability in medicine: what clinicians can learn from Karl Popper [published correction appears in Intensive Care Med. 2021 Jun 17;:].  Intensive Care Med . 2021;47(9):1054-1056. doi:10.1007/s00134-021-06432-z

Eyler AA. Research Methods for Public Health . 1st ed. Springer Publishing Company; 2020. doi:10.1891/9780826182067.0004

Nosek BA, Errington TM. What is replication ?  PLoS Biol . 2020;18(3):e3000691. doi:10.1371/journal.pbio.3000691

Aggarwal R, Ranganathan P. Study designs: Part 2 - Descriptive studies .  Perspect Clin Res . 2019;10(1):34-36. doi:10.4103/picr.PICR_154_18

Nevid J. Psychology: Concepts and Applications. Wadworth, 2013.

By Kendra Cherry, MSEd Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

What Is a Testable Hypothesis?

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• Ph.D., Biomedical Sciences, University of Tennessee at Knoxville
• B.A., Physics and Mathematics, Hastings College

A hypothesis is a tentative answer to a scientific question. A testable hypothesis is a  hypothesis that can be proved or disproved as a result of testing, data collection, or experience. Only testable hypotheses can be used to conceive and perform an experiment using the scientific method .

Requirements for a Testable Hypothesis

In order to be considered testable, two criteria must be met:

• It must be possible to prove that the hypothesis is true.
• It must be possible to prove that the hypothesis is false.
• It must be possible to reproduce the results of the hypothesis.

Examples of a Testable Hypothesis

All the following hypotheses are testable. It's important, however, to note that while it's possible to say that the hypothesis is correct, much more research would be required to answer the question " why is this hypothesis correct?" 

• Students who attend class have higher grades than students who skip class.  This is testable because it is possible to compare the grades of students who do and do not skip class and then analyze the resulting data. Another person could conduct the same research and come up with the same results.
• People exposed to high levels of ultraviolet light have a higher incidence of cancer than the norm.  This is testable because it is possible to find a group of people who have been exposed to high levels of ultraviolet light and compare their cancer rates to the average.
• If you put people in a dark room, then they will be unable to tell when an infrared light turns on.  This hypothesis is testable because it is possible to put a group of people into a dark room, turn on an infrared light, and ask the people in the room whether or not an infrared light has been turned on.

Examples of a Hypothesis Not Written in a Testable Form

• It doesn't matter whether or not you skip class.  This hypothesis can't be tested because it doesn't make any actual claim regarding the outcome of skipping class. "It doesn't matter" doesn't have any specific meaning, so it can't be tested.
• Ultraviolet light could cause cancer.  The word "could" makes a hypothesis extremely difficult to test because it is very vague. There "could," for example, be UFOs watching us at every moment, even though it's impossible to prove that they are there!
• Goldfish make better pets than guinea pigs.  This is not a hypothesis; it's a matter of opinion. There is no agreed-upon definition of what a "better" pet is, so while it is possible to argue the point, there is no way to prove it.

How to Test a Hypothesis

Last Updated: July 4, 2023 Fact Checked

This article was co-authored by Meredith Juncker, PhD . Meredith Juncker is a PhD candidate in Biochemistry and Molecular Biology at Louisiana State University Health Sciences Center. Her studies are focused on proteins and neurodegenerative diseases. There are 8 references cited in this article, which can be found at the bottom of the page. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 41,012 times.

Testing a hypothesis is an important part of the scientific method. It allows you to evaluate the validity of an educated guess. In a typical process, you’ll form a hypothesis based on evidence that you’ve gathered, and then test that hypothesis through experiments. As you gather more and more data, you’ll be able to see if your original hypothesis was correct. If there were flaws in your first guess, you can revise your hypothesis to better match what you’ve learned from your data.

Asking a Question and Researching

• For example, a question could be something like, “Which brand of stain remover will remove stains from fabrics most effectively?” [2] X Research source

• For the stain-remover experiment, you could dirty 4 types of fabric (e.g. cotton, linen, wool, polyester) each with 4 different types of stains (e.g. red wine, grass, mud and dirt, grease), and then test the top four or five brands of stain remover (e.g. Mr. Clean, Tide, Shout, Clorox) to see which removes the largest number of stains.

• In the case of the stain-remover experiment, you’d need to purchase a bottle of each of the major stain-remover brands and dirty a variety of fabrics with a variety of stains.
• Then, test out each type of detergent on each of the stained fabrics. (If you live at your parents’ house, you’ll need to get permission to use the laundry room for most of a day.)

Making and Challenging Your Hypothesis

• For instance, if you’ve run some loads of laundry (maybe testing which brand of stain remover works best on removing different stains from linen), you can use your results to take a stab at a hypothesis.
• A good working hypothesis would look like: “If fabrics are stained with common household items, Tide stain remover will remove the stains most effectively.”

• In our example, since you only tested 1 type of fabric (linen), you’ll need to repeat the laundry test with the other 3 fabrics (cotton, wool, polyester) and note which stain remover most effectively eliminates the stains.

• While it can be tempting to only accept data that supports your hypothesis, it’s not scientific or ethical.
• You must accept all the data and watch for whatever patterns appear, even if it proves your hypothesis to be likely false.
• Note that significant results don’t mean your hypothesis is proven, but rather that based on the data you collected, the differences you observed are likely not due to chance.

Revising Your Hypothesis

• For example, if you began the experiment thinking that Tide would have the most effective stain remover, but you’ve noticed that Tide does a poor job of removing stains from red wine and mud, you likely need to change your working assumptions.

• So, if Tide turns out to be ineffective at removing certain types of stains, your early working hypothesis will have been incorrect.

• A final, tested hypothesis would look like: “Shout is the most effective stain remover for removing a variety of household stains from a variety of common fabrics.”

Expert Q&A

• Deductive (or “top-down”) reasoning won’t help you much in testing out a scientific hypothesis. Your hypothesis needs to be grounded in the tests you’ve run and the data that you’ve gathered. Thanks Helpful 0 Not Helpful 0
• Depending on the type of hypothesis you’re testing, you may also need a control group. For example, if you’re testing how effective a drug is, you need to have a group on a placebo. Thanks Helpful 0 Not Helpful 0
• Remember that a null hypothesis (when the control and tested variable are the same) is different from an alternative hypothesis (when the control and tested variable are different). Thanks Helpful 0 Not Helpful 0

You Might Also Like

• ↑ https://grammar.yourdictionary.com/for-students-and-parents/how-create-hypothesis.html
• ↑ https://www.education.com/science-fair/article/best-way-stains-out-of-clothes/
• ↑ https://www.grammarly.com/blog/how-to-write-a-hypothesis/
• ↑ https://www.sciencebuddies.org/science-fair-projects/science-fair/steps-of-the-scientific-method
• ↑ https://online.stat.psu.edu/stat502_fa21/lesson/1/1.1
• ↑ https://pressbooks.pub/scientificinquiryinsocialwork/chapter/6-3-inductive-and-deductive-reasoning/
• ↑ https://online.stat.psu.edu/stat100/lesson/10/10.1
• ↑ https://www.khanacademy.org/science/biology/intro-to-biology/science-of-biology/a/the-science-of-biology

About this article

If you need to test a hypothesis, first come up with a question you’d like to answer, then develop an experiment to answer your question. As you set up your experiment, create a statement about what you think is happening. This is your hypothesis. As you perform the test, compare your data to your hypothesis. By the end of the experiment, you should be able to conclude whether your hypothesis was true or not. Read on to learn more from our Science co-author about how to set up an experiment! Did this summary help you? Yes No

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What Is a Hypothesis and How Do I Write One?

General Education

Think about something strange and unexplainable in your life. Maybe you get a headache right before it rains, or maybe you think your favorite sports team wins when you wear a certain color. If you wanted to see whether these are just coincidences or scientific fact, you would form a hypothesis, then create an experiment to see whether that hypothesis is true or not.

But what is a hypothesis, anyway? If you’re not sure about what a hypothesis is--or how to test for one!--you’re in the right place. This article will teach you everything you need to know about hypotheses, including: 

• Defining the term “hypothesis” 
• Providing hypothesis examples 
• Giving you tips for how to write your own hypothesis

So let’s get started!

What Is a Hypothesis?

Merriam Webster defines a hypothesis as “an assumption or concession made for the sake of argument.” In other words, a hypothesis is an educated guess . Scientists make a reasonable assumption--or a hypothesis--then design an experiment to test whether it’s true or not. Keep in mind that in science, a hypothesis should be testable. You have to be able to design an experiment that tests your hypothesis in order for it to be valid. 

As you could assume from that statement, it’s easy to make a bad hypothesis. But when you’re holding an experiment, it’s even more important that your guesses be good...after all, you’re spending time (and maybe money!) to figure out more about your observation. That’s why we refer to a hypothesis as an educated guess--good hypotheses are based on existing data and research to make them as sound as possible.

Hypotheses are one part of what’s called the scientific method .  Every (good) experiment or study is based in the scientific method. The scientific method gives order and structure to experiments and ensures that interference from scientists or outside influences does not skew the results. It’s important that you understand the concepts of the scientific method before holding your own experiment. Though it may vary among scientists, the scientific method is generally made up of six steps (in order):

• Observation
• Asking questions
• Forming a hypothesis
• Analyze the data
• Communicate your results

You’ll notice that the hypothesis comes pretty early on when conducting an experiment. That’s because experiments work best when they’re trying to answer one specific question. And you can’t conduct an experiment until you know what you’re trying to prove!

Independent and Dependent Variables 

After doing your research, you’re ready for another important step in forming your hypothesis: identifying variables. Variables are basically any factor that could influence the outcome of your experiment . Variables have to be measurable and related to the topic being studied.

There are two types of variables:  independent variables and dependent variables. I ndependent variables remain constant . For example, age is an independent variable; it will stay the same, and researchers can look at different ages to see if it has an effect on the dependent variable. 

Speaking of dependent variables... dependent variables are subject to the influence of the independent variable , meaning that they are not constant. Let’s say you want to test whether a person’s age affects how much sleep they need. In that case, the independent variable is age (like we mentioned above), and the dependent variable is how much sleep a person gets. 

Variables will be crucial in writing your hypothesis. You need to be able to identify which variable is which, as both the independent and dependent variables will be written into your hypothesis. For instance, in a study about exercise, the independent variable might be the speed at which the respondents walk for thirty minutes, and the dependent variable would be their heart rate. In your study and in your hypothesis, you’re trying to understand the relationship between the two variables.

Elements of a Good Hypothesis

The best hypotheses start by asking the right questions . For instance, if you’ve observed that the grass is greener when it rains twice a week, you could ask what kind of grass it is, what elevation it’s at, and if the grass across the street responds to rain in the same way. Any of these questions could become the backbone of experiments to test why the grass gets greener when it rains fairly frequently.

As you’re asking more questions about your first observation, make sure you’re also making more observations . If it doesn’t rain for two weeks and the grass still looks green, that’s an important observation that could influence your hypothesis. You'll continue observing all throughout your experiment, but until the hypothesis is finalized, every observation should be noted.

Finally, you should consult secondary research before writing your hypothesis . Secondary research is comprised of results found and published by other people. You can usually find this information online or at your library. Additionally, m ake sure the research you find is credible and related to your topic. If you’re studying the correlation between rain and grass growth, it would help you to research rain patterns over the past twenty years for your county, published by a local agricultural association. You should also research the types of grass common in your area, the type of grass in your lawn, and whether anyone else has conducted experiments about your hypothesis. Also be sure you’re checking the quality of your research . Research done by a middle school student about what minerals can be found in rainwater would be less useful than an article published by a local university.

Writing Your Hypothesis

Once you’ve considered all of the factors above, you’re ready to start writing your hypothesis. Hypotheses usually take a certain form when they’re written out in a research report.

When you boil down your hypothesis statement, you are writing down your best guess and not the question at hand . This means that your statement should be written as if it is fact already, even though you are simply testing it.

The reason for this is that, after you have completed your study, you'll either accept or reject your if-then or your null hypothesis. All hypothesis testing examples should be measurable and able to be confirmed or denied. You cannot confirm a question, only a statement! 

In fact, you come up with hypothesis examples all the time! For instance, when you guess on the outcome of a basketball game, you don’t say, “Will the Miami Heat beat the Boston Celtics?” but instead, “I think the Miami Heat will beat the Boston Celtics.” You state it as if it is already true, even if it turns out you’re wrong. You do the same thing when writing your hypothesis.

Additionally, keep in mind that hypotheses can range from very specific to very broad.  These hypotheses can be specific, but if your hypothesis testing examples involve a broad range of causes and effects, your hypothesis can also be broad.  

The Two Types of Hypotheses

Now that you understand what goes into a hypothesis, it’s time to look more closely at the two most common types of hypothesis: the if-then hypothesis and the null hypothesis.

#1: If-Then Hypotheses

First of all, if-then hypotheses typically follow this formula:

If ____ happens, then ____ will happen.

The goal of this type of hypothesis is to test the causal relationship between the independent and dependent variable. It’s fairly simple, and each hypothesis can vary in how detailed it can be. We create if-then hypotheses all the time with our daily predictions. Here are some examples of hypotheses that use an if-then structure from daily life: 

• If I get enough sleep, I’ll be able to get more work done tomorrow.
• If the bus is on time, I can make it to my friend’s birthday party. 
• If I study every night this week, I’ll get a better grade on my exam. 

In each of these situations, you’re making a guess on how an independent variable (sleep, time, or studying) will affect a dependent variable (the amount of work you can do, making it to a party on time, or getting better grades). 

You may still be asking, “What is an example of a hypothesis used in scientific research?” Take one of the hypothesis examples from a real-world study on whether using technology before bed affects children’s sleep patterns. The hypothesis read s:

“We hypothesized that increased hours of tablet- and phone-based screen time at bedtime would be inversely correlated with sleep quality and child attention.”

It might not look like it, but this is an if-then statement. The researchers basically said, “If children have more screen usage at bedtime, then their quality of sleep and attention will be worse.” The sleep quality and attention are the dependent variables and the screen usage is the independent variable. (Usually, the independent variable comes after the “if” and the dependent variable comes after the “then,” as it is the independent variable that affects the dependent variable.) This is an excellent example of how flexible hypothesis statements can be, as long as the general idea of “if-then” and the independent and dependent variables are present.

#2: Null Hypotheses

Your if-then hypothesis is not the only one needed to complete a successful experiment, however. You also need a null hypothesis to test it against. In its most basic form, the null hypothesis is the opposite of your if-then hypothesis . When you write your null hypothesis, you are writing a hypothesis that suggests that your guess is not true, and that the independent and dependent variables have no relationship .

One null hypothesis for the cell phone and sleep study from the last section might say: 

“If children have more screen usage at bedtime, their quality of sleep and attention will not be worse.” 

In this case, this is a null hypothesis because it’s asking the opposite of the original thesis! 

Conversely, if your if-then hypothesis suggests that your two variables have no relationship, then your null hypothesis would suggest that there is one. So, pretend that there is a study that is asking the question, “Does the amount of followers on Instagram influence how long people spend on the app?” The independent variable is the amount of followers, and the dependent variable is the time spent. But if you, as the researcher, don’t think there is a relationship between the number of followers and time spent, you might write an if-then hypothesis that reads:

“If people have many followers on Instagram, they will not spend more time on the app than people who have less.”

In this case, the if-then suggests there isn’t a relationship between the variables. In that case, one of the null hypothesis examples might say:

“If people have many followers on Instagram, they will spend more time on the app than people who have less.”

You then test both the if-then and the null hypothesis to gauge if there is a relationship between the variables, and if so, how much of a relationship. 

4 Tips to Write the Best Hypothesis

If you’re going to take the time to hold an experiment, whether in school or by yourself, you’re also going to want to take the time to make sure your hypothesis is a good one. The best hypotheses have four major elements in common: plausibility, defined concepts, observability, and general explanation.

#1: Plausibility

At first glance, this quality of a hypothesis might seem obvious. When your hypothesis is plausible, that means it’s possible given what we know about science and general common sense. However, improbable hypotheses are more common than you might think. 

Imagine you’re studying weight gain and television watching habits. If you hypothesize that people who watch more than  twenty hours of television a week will gain two hundred pounds or more over the course of a year, this might be improbable (though it’s potentially possible). Consequently, c ommon sense can tell us the results of the study before the study even begins.

Improbable hypotheses generally go against  science, as well. Take this hypothesis example: 

“If a person smokes one cigarette a day, then they will have lungs just as healthy as the average person’s.” 

This hypothesis is obviously untrue, as studies have shown again and again that cigarettes negatively affect lung health. You must be careful that your hypotheses do not reflect your own personal opinion more than they do scientifically-supported findings. This plausibility points to the necessity of research before the hypothesis is written to make sure that your hypothesis has not already been disproven.

#2: Defined Concepts

The more advanced you are in your studies, the more likely that the terms you’re using in your hypothesis are specific to a limited set of knowledge. One of the hypothesis testing examples might include the readability of printed text in newspapers, where you might use words like “kerning” and “x-height.” Unless your readers have a background in graphic design, it’s likely that they won’t know what you mean by these terms. Thus, it’s important to either write what they mean in the hypothesis itself or in the report before the hypothesis.

Here’s what we mean. Which of the following sentences makes more sense to the common person?

If the kerning is greater than average, more words will be read per minute.

If the space between letters is greater than average, more words will be read per minute.

For people reading your report that are not experts in typography, simply adding a few more words will be helpful in clarifying exactly what the experiment is all about. It’s always a good idea to make your research and findings as accessible as possible. 

Good hypotheses ensure that you can observe the results. 

#3: Observability

In order to measure the truth or falsity of your hypothesis, you must be able to see your variables and the way they interact. For instance, if your hypothesis is that the flight patterns of satellites affect the strength of certain television signals, yet you don’t have a telescope to view the satellites or a television to monitor the signal strength, you cannot properly observe your hypothesis and thus cannot continue your study.

Some variables may seem easy to observe, but if you do not have a system of measurement in place, you cannot observe your hypothesis properly. Here’s an example: if you’re experimenting on the effect of healthy food on overall happiness, but you don’t have a way to monitor and measure what “overall happiness” means, your results will not reflect the truth. Monitoring how often someone smiles for a whole day is not reasonably observable, but having the participants state how happy they feel on a scale of one to ten is more observable. 

In writing your hypothesis, always keep in mind how you'll execute the experiment.

#4: Generalizability 

Perhaps you’d like to study what color your best friend wears the most often by observing and documenting the colors she wears each day of the week. This might be fun information for her and you to know, but beyond you two, there aren’t many people who could benefit from this experiment. When you start an experiment, you should note how generalizable your findings may be if they are confirmed. Generalizability is basically how common a particular phenomenon is to other people’s everyday life.

Let’s say you’re asking a question about the health benefits of eating an apple for one day only, you need to realize that the experiment may be too specific to be helpful. It does not help to explain a phenomenon that many people experience. If you find yourself with too specific of a hypothesis, go back to asking the big question: what is it that you want to know, and what do you think will happen between your two variables?

Hypothesis Testing Examples

We know it can be hard to write a good hypothesis unless you’ve seen some good hypothesis examples. We’ve included four hypothesis examples based on some made-up experiments. Use these as templates or launch pads for coming up with your own hypotheses.

Experiment #1: Students Studying Outside (Writing a Hypothesis)

You are a student at PrepScholar University. When you walk around campus, you notice that, when the temperature is above 60 degrees, more students study in the quad. You want to know when your fellow students are more likely to study outside. With this information, how do you make the best hypothesis possible?

You must remember to make additional observations and do secondary research before writing your hypothesis. In doing so, you notice that no one studies outside when it’s 75 degrees and raining, so this should be included in your experiment. Also, studies done on the topic beforehand suggested that students are more likely to study in temperatures less than 85 degrees. With this in mind, you feel confident that you can identify your variables and write your hypotheses:

If-then: “If the temperature in Fahrenheit is less than 60 degrees, significantly fewer students will study outside.”

Null: “If the temperature in Fahrenheit is less than 60 degrees, the same number of students will study outside as when it is more than 60 degrees.”

These hypotheses are plausible, as the temperatures are reasonably within the bounds of what is possible. The number of people in the quad is also easily observable. It is also not a phenomenon specific to only one person or at one time, but instead can explain a phenomenon for a broader group of people.

To complete this experiment, you pick the month of October to observe the quad. Every day (except on the days where it’s raining)from 3 to 4 PM, when most classes have released for the day, you observe how many people are on the quad. You measure how many people come  and how many leave. You also write down the temperature on the hour. 

After writing down all of your observations and putting them on a graph, you find that the most students study on the quad when it is 70 degrees outside, and that the number of students drops a lot once the temperature reaches 60 degrees or below. In this case, your research report would state that you accept or “failed to reject” your first hypothesis with your findings.

Experiment #2: The Cupcake Store (Forming a Simple Experiment)

Let’s say that you work at a bakery. You specialize in cupcakes, and you make only two colors of frosting: yellow and purple. You want to know what kind of customers are more likely to buy what kind of cupcake, so you set up an experiment. Your independent variable is the customer’s gender, and the dependent variable is the color of the frosting. What is an example of a hypothesis that might answer the question of this study?

Here’s what your hypotheses might look like: 

If-then: “If customers’ gender is female, then they will buy more yellow cupcakes than purple cupcakes.”

Null: “If customers’ gender is female, then they will be just as likely to buy purple cupcakes as yellow cupcakes.”

This is a pretty simple experiment! It passes the test of plausibility (there could easily be a difference), defined concepts (there’s nothing complicated about cupcakes!), observability (both color and gender can be easily observed), and general explanation ( this would potentially help you make better business decisions ).

Experiment #3: Backyard Bird Feeders (Integrating Multiple Variables and Rejecting the If-Then Hypothesis)

While watching your backyard bird feeder, you realized that different birds come on the days when you change the types of seeds. You decide that you want to see more cardinals in your backyard, so you decide to see what type of food they like the best and set up an experiment. 

However, one morning, you notice that, while some cardinals are present, blue jays are eating out of your backyard feeder filled with millet. You decide that, of all of the other birds, you would like to see the blue jays the least. This means you'll have more than one variable in your hypothesis. Your new hypotheses might look like this: 

If-then: “If sunflower seeds are placed in the bird feeders, then more cardinals will come than blue jays. If millet is placed in the bird feeders, then more blue jays will come than cardinals.”

Null: “If either sunflower seeds or millet are placed in the bird, equal numbers of cardinals and blue jays will come.”

Through simple observation, you actually find that cardinals come as often as blue jays when sunflower seeds or millet is in the bird feeder. In this case, you would reject your “if-then” hypothesis and “fail to reject” your null hypothesis . You cannot accept your first hypothesis, because it’s clearly not true. Instead you found that there was actually no relation between your different variables. Consequently, you would need to run more experiments with different variables to see if the new variables impact the results.

Experiment #4: In-Class Survey (Including an Alternative Hypothesis)

You’re about to give a speech in one of your classes about the importance of paying attention. You want to take this opportunity to test a hypothesis you’ve had for a while: 

If-then: If students sit in the first two rows of the classroom, then they will listen better than students who do not.

Null: If students sit in the first two rows of the classroom, then they will not listen better or worse than students who do not.

You give your speech and then ask your teacher if you can hand out a short survey to the class. On the survey, you’ve included questions about some of the topics you talked about. When you get back the results, you’re surprised to see that not only do the students in the first two rows not pay better attention, but they also scored worse than students in other parts of the classroom! Here, both your if-then and your null hypotheses are not representative of your findings. What do you do?

This is when you reject both your if-then and null hypotheses and instead create an alternative hypothesis . This type of hypothesis is used in the rare circumstance that neither of your hypotheses is able to capture your findings . Now you can use what you’ve learned to draft new hypotheses and test again! 

Key Takeaways: Hypothesis Writing

The more comfortable you become with writing hypotheses, the better they will become. The structure of hypotheses is flexible and may need to be changed depending on what topic you are studying. The most important thing to remember is the purpose of your hypothesis and the difference between the if-then and the null . From there, in forming your hypothesis, you should constantly be asking questions, making observations, doing secondary research, and considering your variables. After you have written your hypothesis, be sure to edit it so that it is plausible, clearly defined, observable, and helpful in explaining a general phenomenon.

Writing a hypothesis is something that everyone, from elementary school children competing in a science fair to professional scientists in a lab, needs to know how to do. Hypotheses are vital in experiments and in properly executing the scientific method . When done correctly, hypotheses will set up your studies for success and help you to understand the world a little better, one experiment at a time.

What’s Next?

If you’re studying for the science portion of the ACT, there’s definitely a lot you need to know. We’ve got the tools to help, though! Start by checking out our ultimate study guide for the ACT Science subject test. Once you read through that, be sure to download our recommended ACT Science practice tests , since they’re one of the most foolproof ways to improve your score. (And don’t forget to check out our expert guide book , too.)

If you love science and want to major in a scientific field, you should start preparing in high school . Here are the science classes you should take to set yourself up for success.

If you’re trying to think of science experiments you can do for class (or for a science fair!), here’s a list of 37 awesome science experiments you can do at home

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Ashley Sufflé Robinson has a Ph.D. in 19th Century English Literature. As a content writer for PrepScholar, Ashley is passionate about giving college-bound students the in-depth information they need to get into the school of their dreams.

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Writing a Proof by Induction

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• Shanthanu Rai
• Hua Zhi Vee
• Rajdeep Dhingra

While writing a proof by induction , there are certain fundamental terms and mathematical jargon which must be used, as well as a certain format which has to be followed. These norms can never be ignored. Some of the basic contents of a proof by induction are as follows:

• a given proposition \(P_n\) (what is to be proved);
• a given domain for the proposition \((\)for example, for all positive integers \(n);\)
• a base case \((\)where we usually try to prove the proposition \(P_n\) holds true for \(n=1);\)
• an induction hypothesis \((\)which assumes that \(P_k\) is true for any \(k\) in the domain of \(P_n\); this is then used for the case \(P_{k+1});\)
• the conclusion.

Based on these, we have a rough format for a proof by Induction:

Statement: Let \(P_n\) be the proposition induction hypothesis for \(n\) in the domain. Base Case: Consider the base case: \(\hspace{0.5cm}\) LHS = LHS \(\hspace{0.5cm}\) RHS = RHS. Since LHS = RHS, the base case is true. Induction Step: Assume \(P_k\) is true for some \(k\) in the domain. Consider \(P_{k+1}\): \(\hspace{0.5cm}\) LHS \(P_{k+1}\) = RHS \(P_{k+1}\) (using induction hypothesis). Hence the fact that \(P_k\) is true implies that \(P_{k+1}\) is true. Conclusion: By mathematical induction, since both the base case and \(P_{k}\) being true implies \(P_{k+1}\) is true, \(P_{n}\) is true for all \(n\) in the domain.

Let us elucidate this through some examples.

Show that for all positive integers \(n\), \[ \sum_{i=1} ^n i^2 = \frac {n(n+1)(2n+1)} {6}. \] This is a well-known statement about the sum of the first \(n\) squares, and we will show this via induction. Statement: Let \(P_{n} \) be the proposition that \( \displaystyle \sum_{i=1} ^n i^2 = \frac {n(n+1)(2n+1)} {6}\) for all positive integers. (We have stated what the induction hypothesis is, and specified the domain in which the statement is true.) Base Case: Consider \(n=1\). \(\hspace{0.5cm}\) LHS = \( \displaystyle \sum_{i=1} ^ 1 1^2 = 1 \). \(\hspace{0.5cm}\) RHS = \( \frac{ 1 \times (1+1) \times (2 + 1) } {6} = 1 \). Since LHS=RHS, the base case is true. (We have identified the base case, and shown that it is true.) Induction Hypothesis: Assume \(P_k\) is true for some positive integer \(k\). Consider \(P_{k+1} \): We have \[\begin{align} \mbox{LHS of } P_{k+1} & = \sum_{i=1}^{k+1} i^2 \\ & = \left( \displaystyle \sum_{i=1}^k i^2 \right) + (k+1) ^2 \\ & = \frac{ k(k+1)(2k+1) } { 6} + (k+1) ^2 \\ & = (k+1) \left( \frac{ 2k^2 + k } { 6} + \frac{ 6 (k+1) } { 6} \right) \\ & = (k+1) \frac{ 2k^2 + 7k + 6 } { 6} \\ & = \frac { (k+1) (k+2) ( 2k + 3) }{6} \\ & = \mbox{RHS of } P_{k+1}. \end{align} \] Hence, the fact that \(P_{k}\) is true implies that \(P_{k+1}\) is true. \((\)We successfully used the induction hypothesis \((\)in the 3\(^\text{rd}\) equation\()\) along with algebraic manipulation to show that the next statement is true.\()\) Conclusion: By mathematical induction, since the fact that \(P_1\) is true and so is \(P_{k}\) implies \(P_{k+1}\) is true, \(P_{n}\) is true for all positive integers \(n\). \(_\square\)

Show that \(n^3 - n \) is divisible by \(3\ \forall n\in \mathbb{N}\). We have to prove that \(n^3-n=3\alpha\) in the given domain, where \(\alpha\in \mathbb{Z}\). Statement: Let \(P_{n} \) be the proposition that \( n^3-n = 3\alpha\ \forall n\in \mathbb{N}\). (We have stated what the induction hypothesis is, and specified the domain in which the statement is true.) Base Case: Consider \(n=1\). \(\hspace{0.5cm}\) LHS = \( 1^3-1 = 0 \). \(\hspace{0.5cm}\) RHS = \(3\alpha = 0 \) for \(\alpha=0\). Since LHS=RHS, the base case is true. (We have identified the base case, and shown that it is true.) Induction Hypothesis: Assume \(P_k\) is true for some \(k\in \mathbb{N}\). Consider \(P_{k+1} \): We have \[\begin{align} \mbox{LHS of } P_{k+1} & = (k+1)^3 - (k+1) \\ & = k^3 +1 + 3k^2 +3k - k -1 \\ & = (k^3 - k) + 3k^2 + 3k\\ & = 3a + 3(k^2 +k) \qquad (\text{by the induction hypothesis } \forall a\in \mathbb{Z})\\ & = 3(a+k^2+k) \\ & = 3\alpha \\ & = \mbox{RHS of } P_{k+1}. \end{align} \] Hence the fact that \(P_{k}\) is true implies that \(P_{k+1}\) is true. \((\)We successfully used the induction hypothesis \((\)in the 3\(^\text{rd}\) equation\()\) along with algebraic manipulation to show that the next statement is true.\()\) Conclusion: By mathematical induction, since the fact that \(P_1\) is true and so is \(P_{k}\) implies \(P_{k+1}\) is true, \(P_{n}\) is true for all positive integers \(n\). \(_\square\)

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How we edit science part 1: the scientific method

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We take science seriously at The Conversation and we work hard to report it accurately. This series of five posts is adapted from an internal presentation on how to understand and edit science by our Australian Science & Technology Editor, Tim Dean. We thought you might also find it useful.

Introduction

If I told you that science was a truth-seeking endeavour that uses a single robust method to prove scientific facts about the world, steadily and inexorably driving towards objective truth, would you believe me?

Many would. But you shouldn’t.

The public perception of science is often at odds with how science actually works. Science is often seen to be a separate domain of knowledge, framed to be superior to other forms of knowledge by virtue of its objectivity, which is sometimes referred to as it having a “ view from nowhere ”.

But science is actually far messier than this - and far more interesting. It is not without its limitations and flaws, but it’s still the most effective tool we have to understand the workings of the natural world around us.

In order to report or edit science effectively - or to consume it as a reader - it’s important to understand what science is, how the scientific method (or methods) work, and also some of the common pitfalls in practising science and interpreting its results.

This guide will give a short overview of what science is and how it works, with a more detailed treatment of both these topics in the final post in the series.

What is science?

Science is special, not because it claims to provide us with access to the truth, but because it admits it can’t provide truth .

Other means of producing knowledge, such as pure reason, intuition or revelation, might be appealing because they give the impression of certainty , but when this knowledge is applied to make predictions about the world around us, reality often finds them wanting.

Rather, science consists of a bunch of methods that enable us to accumulate evidence to test our ideas about how the world is, and why it works the way it does. Science works precisely because it enables us to make predictions that are borne out by experience.

Science is not a body of knowledge. Facts are facts, it’s just that some are known with a higher degree of certainty than others. What we often call “scientific facts” are just facts that are backed by the rigours of the scientific method, but they are not intrinsically different from other facts about the world.

What makes science so powerful is that it’s intensely self-critical. In order for a hypothesis to pass muster and enter a textbook, it must survive a battery of tests designed specifically to show that it could be wrong. If it passes, it has cleared a high bar.

The scientific method(s)

Despite what some philosophers have stated , there is a method for conducting science. In fact, there are many. And not all revolve around performing experiments.

One method involves simple observation, description and classification, such as in taxonomy. (Some physicists look down on this – and every other – kind of science, but they’re only greasing a slippery slope .)

However, when most of us think of The Scientific Method, we’re thinking of a particular kind of experimental method for testing hypotheses.

This begins with observing phenomena in the world around us, and then moves on to positing hypotheses for why those phenomena happen the way they do. A hypothesis is just an explanation, usually in the form of a causal mechanism: X causes Y. An example would be: gravitation causes the ball to fall back to the ground.

A scientific theory is just a collection of well-tested hypotheses that hang together to explain a great deal of stuff.

Crucially, a scientific hypothesis needs to be testable and falsifiable .

An untestable hypothesis would be something like “the ball falls to the ground because mischievous invisible unicorns want it to”. If these unicorns are not detectable by any scientific instrument, then the hypothesis that they’re responsible for gravity is not scientific.

An unfalsifiable hypothesis is one where no amount of testing can prove it wrong. An example might be the psychic who claims the experiment to test their powers of ESP failed because the scientific instruments were interfering with their abilities.

(Caveat: there are some hypotheses that are untestable because we choose not to test them. That doesn’t make them unscientific in principle, it’s just that they’ve been denied by an ethics committee or other regulation.)

Experimentation

There are often many hypotheses that could explain any particular phenomenon. Does the rock fall to the ground because an invisible force pulls on the rock? Or is it because the mass of the Earth warps spacetime , and the rock follows the lowest-energy path, thus colliding with the ground? Or is it that all substances have a natural tendency to fall towards the centre of the Universe , which happens to be at the centre of the Earth?

The trick is figuring out which hypothesis is the right one. That’s where experimentation comes in.

A scientist will take their hypothesis and use that to make a prediction, and they will construct an experiment to see if that prediction holds. But any observation that confirms one hypothesis will likely confirm several others as well. If I lift and drop a rock, it supports all three of the hypotheses on gravity above.

Furthermore, you can keep accumulating evidence to confirm a hypothesis, and it will never prove it to be absolutely true. This is because you can’t rule out the possibility of another similar hypothesis being correct, or of making some new observation that shows your hypothesis to be false. But if one day you drop a rock and it shoots off into space, that ought to cast doubt on all of the above hypotheses.

So while you can never prove a hypothesis true simply by making more confirmatory observations, you only one need one solid contrary observation to prove a hypothesis false. This notion is at the core of the hypothetico-deductive model of science.

This is why a great deal of science is focused on testing hypotheses, pushing them to their limits and attempting to break them through experimentation. If the hypothesis survives repeated testing, our confidence in it grows.

So even crazy-sounding theories like general relativity and quantum mechanics can become well accepted, because both enable very precise predictions, and these have been exhaustively tested and come through unscathed.

The next post will cover hypothesis testing in greater detail.

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What does proving the Riemann Hypothesis accomplish?

I've recently been reading about the Millennium Prize problems, specifically the Riemann Hypothesis. I'm not near qualified to even fully grasp the problem, but seeing the hypothesis and the other problems I wonder: what practical use will a solution have?

Many researchers have spent a lot of time on it, trying to prove it, but why is it important to solve the problem?

I've tried relating the situation to problems in my field. For instance, solving the $P \ vs. NP$ problem has important implications if $P = NP$ is shown, and important implications if $P \neq NP$ is shown. For instance, there would be implications regarding the robustness or security of cryptographic protocols and algorithms. However, it's hard to say WHY the Riemann Hypothesis is important.

Given that the Poincaré Conjecture has been resolved, perhaps a hint about what to expect if and when the Riemann Hypothesis is resolved could be obtained by seeing what a proof of the Poincaré Conjecture has led to.

• number-theory
• riemann-hypothesis

• 7 $\begingroup$ Define "practical use". And it is important because it is there and it is fun to know stuff, though this would probably not fit within the "practical use" tag... $\endgroup$ –  DonAntonio Commented May 28, 2013 at 9:51
• $\begingroup$ It is very likely to be a duplicate isn't it? $\endgroup$ –  Dominic Michaelis Commented May 28, 2013 at 9:52
• $\begingroup$ @DominicMichaelis; I don't follow, a duplicate of what? $\endgroup$ –  Mythio Commented May 28, 2013 at 10:00
• 4 $\begingroup$ Well it is a very famous hypothesis so you won't be the first one to ask it, partial answers are for example here $\endgroup$ –  Dominic Michaelis Commented May 28, 2013 at 10:14
• 3 $\begingroup$ immortality ... in the sense of Pythagoras ($a^2 + b^2 = c^2$) and Einstein ($E=mc^2$) $\endgroup$ –  zerosofthezeta Commented Sep 14, 2013 at 7:43

5 Answers 5

Proving the Riemann Hypothesis will get you tenure, pretty much anywhere you want it.

• 16 $\begingroup$ Worth the downvotes :) $\endgroup$ –  Nathaniel Bubis Commented Dec 29, 2013 at 2:51
• 16 $\begingroup$ Not to mention the 1M on your bank account... $\endgroup$ –  Matemáticos Chibchas Commented Dec 29, 2013 at 3:44
• 6 $\begingroup$ I've already proven it, I just need to check my calculations... ;) $\endgroup$ –  King Squirrel Commented May 4, 2014 at 17:26
• 37 $\begingroup$ @KingSquirrel If only that damn margin were to be a bit wider... ;-) $\endgroup$ –  triple_sec Commented Aug 26, 2014 at 7:46
• 1 $\begingroup$ @KingSquirrel I have proved it too, just waiting for the prize to be increased to $10M.;) $\endgroup$ –  ishandutta2007 Commented Aug 12, 2022 at 8:55

The Millennium problems are not necessarily problems whose solution will lead to curing cancer. These are problems in mathematics and were chosen for their importance in mathematics rather for their potential in applications.

There are plenty of important open problems in mathematics, and the Clay Institute had to narrow it down to seven. Whatever the reasons may be, it is clear such a short list is incomplete and does not claim to be a comprehensive list of the most important problems to solve. However, each of the problems solved is extremely central, important, interesting, and hard. Some of these problems have direct consequences, for instance the Riemann hypothesis. There are many (many many) theorems in number theory that go like "if the Riemann hypothesis is true, then blah blah", so knowing it is true will immediately validate the consequences in these theorems as true.

In contrast, a solution to some of the other Millennium problems is (highly likely) not going to lead to anything dramatic. For instance, the $P$ vs. $NP$ problem. I personally doubt it is probable that $P=NP$ . The reason it's an important question is not because we don't (philosophically) already know the answer, but rather that we don't have a bloody clue how to prove it. It means that there are fundamental issues in computability (which is a hell of an important subject these days) that we just don't understand. Solving $P \ne NP$ will be important not for the result but for the techniques that will be used. (Of course, in the unlikely event that $P=NP$ , enormous consequences will follow. But that is about as likely as it is that the Hitchhiker's Guide to the Galaxy is based on true events.)

The Poincaré conjecture is an extremely basic problem about three-dimensional space. I think three-dimensional space is very important, so if we can't answer a very fundamental question about it, then we don't understand it well. I'm not an expert on Perelman's solution, nor the field to which it belongs, so I can't tell what consequences his techniques have for better understanding three-dimensional space, but I'm sure there are.

• 16 $\begingroup$ about Perelman's solution and the Poincaré conjecture : in theoretical physics, more precisely in general relativity, there are equations relating the distribution of mass in the universe to the curvature of space-time. Perelman's solution used curvature (via the Ricci flow) to get topological information, i.e. the shape of the compact simply connected manifold (homeomorphic to a sphere). these kind of techniques are related to determnining the shape of the universe (space-time) which is a 4-manifold, depending on the curvature (which is determined by mass repartition, which we can observe) $\endgroup$ –  Albert Commented May 28, 2013 at 13:22
• 2 $\begingroup$ @Glougloubarbaki - it would be great if you could expound your comment as an answer. $\endgroup$ –  Nathaniel Bubis Commented Dec 29, 2013 at 2:45
• 2 $\begingroup$ Is it really obvious that "enormous consequences will follow" from a proof of P = NP? $\endgroup$ –  bof Commented Nov 14, 2014 at 4:53
• $\begingroup$ @bof highly likely, yes. At the very least lots and lots of expert computer scientists will look at the proof in disbelief as it shatters something they really felt very confident about. And, unless the proof will be completely unusable computationally, some really hard to solve problems will be found polynomial solutions. $\endgroup$ –  Ittay Weiss Commented Nov 14, 2014 at 4:57
• 1 $\begingroup$ P=NP. I'd love an opportunity to demonstrate my NTM. Nonetheless, the argument that no-one believes P=NP is just patently false. P=NP via BPP. $\endgroup$ –  Jonathan Charlton Commented Jan 18, 2016 at 6:44

Explaining the true mathematics behind the Riemann Hypothesis requires more text that I'm allotted (took most of my undergraduate degree in mathematics to even touch the surface; required all of graduate school to fully appreciate the beauty).

In very simple terms, the Riemann Hypothesis is mostly about the distribution of prime numbers. The idea is that mathematicians have some very good approximations (emphasis on approximate) for the density of the primes (so you give me an integer, and I can use these approximate functions to tell you roughly how many primes are between 0 [really 2] and that integer). The reason we use these approximations is that no [known] function exists that efficiently and perfectly computes the number of primes less than a given integer (we're talking numbers with literally millions of zeros). Since we can't determine the exact values (again, I'm simplifying a lot of this) the problem mathematicians want to know is exactly HOW good are these approximations.

This is where the Riemann Hypothesis comes in to play. For well over a century, mathematicians have known that a special form of the polylogarithm function (again, more fun math if you're bored) is a really great approximation for the prime counting function (and it's way easier to compute). The Riemann Hypothesis, if true, would guarantee a far greater bound on the difference between this approximation and the real value. In other words, the importance of the Riemann Hypothesis is that it tells us a lot about how chaotic the primes numbers really are. That's an incredibly high-level explanation and the Riemann Hypothesis deals with literally hundreds of other concepts, but the main point is understanding the distribution of the primes.

The Riemann hypothesis is a conjecture about the Riemann zeta function $$\zeta(s)=\sum_{n=1}^{\infty}\dfrac{1}{n^s}$$ This is a function $\mathbb{C} \rightarrow \mathbb{C}$. With the definition I have provided the zeta function is only defined for $\Re(s)\gt1$. With some complex analysis you can proof that there is a continuous (actually holomorphic if you know what it means) extension of the function so that it is defined in whole $\mathbb{C}$. The Riemann zeta function has some trivial zero points like $-2,-4,-6.$ The hypothesis says that the other zero points lie on the critical line $\Re(s)=\dfrac12$. This hypothesis had many application in analysis and number theory. The first proof of the prime number theorem used this conjecture.

In order to give an anwer to your question a would like to refer to this website, where you can find tons of applications of the Riemann hypothesis.

• 38 $\begingroup$ This is not an answer to the question, but a (very) brief introduction to what the RH is. OP asked about what applications proving RH would have. $\endgroup$ –  Lord_Farin Commented May 28, 2013 at 9:58
• 2 $\begingroup$ the link was interesting (+1) $\endgroup$ –  zerosofthezeta Commented Sep 14, 2013 at 7:46
• $\begingroup$ And we want the zeros of $\zeta(s)$ because $\frac{\zeta'(s)}{s\zeta(s)}$ is the Laplace transform of $\Psi(u) = \sum_{p^k < e^u} \log p$ a function showing us the distribution of primes numbers. If the RH is true, then $\frac{\Psi(u) -e^u}{e^{u/2}}= o(e^{-u/2})+\log 2\pi + \sum_\gamma \frac{e^{i \gamma u}}{1/2+i\gamma}$ where $\gamma$ are the imaginary parts of the non-trivial zeros $\endgroup$ –  reuns Commented Dec 23, 2016 at 7:33

The techniques used in the proofs of some of the most difficult theorems are used to prove so many other theorems. A proof of 1 of these theorems will give us access to an incredible amount of new techniques that will definitely make mathematics shorter,simpler and easier to understand.

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Is it possible to prove a null hypothesis?

As the question states - Is it possible to prove the null hypothesis? From my (limited) understanding of hypothesis, the answer is no but I can't come up with a rigorous explanation for it. Does the question have a definitive answer?

• hypothesis-testing
• equivalence

• 5 $\begingroup$ It depends on what you mean by "prove." As stated, this is a philosophical question, not a statistical one, and has no definitive answer (although, at least since David Hume's time, most people would answer "no"). $\endgroup$ –  whuber ♦ Commented Jan 13, 2011 at 16:54
• 1 $\begingroup$ This is somewhat of an ill-posed question. We need to know the conditions under which this "proof" is to occur. $\endgroup$ –  probabilityislogic Commented Jan 16, 2011 at 15:39
• 2 $\begingroup$ Perhaps a better posed question is "Under what conditions/assumptions is it possible to prove the null hypothesis?" $\endgroup$ –  probabilityislogic Commented Jan 16, 2011 at 15:40
• $\begingroup$ Related: Why do statisticians say a non-significant result means “you can't reject the null” as opposed to accepting the null hypothesis? $\endgroup$ –  amoeba Commented Feb 14, 2016 at 22:40

8 Answers 8

If you are talking about the real world & not formal logic, the answer is of course. "Proof" of anything by empirical means depends on the strength of the inference one can make, which in turn is determined by validity of the testing process as evaluated in light of everything one knows about how the world works (i.e., theory). Whenever one accepts that certain empirical results justify rejecting the "null" hypothesis, one is necessarily making judgments of this sort (validity of design; world works in certain way), so having to make the analogous assumptions necessary to justify inferring "proof of the null" is not problematic at all.

So what are the analogous assumptions? Here is an example of "proving the null" that is commonplace in health science & in social science. (1) Define "null" or "no effect" in some way that is practically meaningful. Let's say that I believe that I should conduct myself as if there is no meaningful difference between 2 treatments, t1 & t2, for a disease unless one gives a 3% better chance of recovery than the other. (2) Figure out a valid design for testing whether there is any effect-- in this case, whether there is a difference in recovery likelihood between t1 & t2. (3) Do a power analysis to determine whether what sample size is necessary to generate a sufficiently high likelihood-- one that I am confident relying on given what's at stake -- that I would see the meaningful effect, 3% in my example, assuming it exists. Usually people say power is sufficient if the likelihood of observing a specified effect at a specified alpha is at least 0.80, but the right level of confidence is really a matter of how averse you are to error -- same as it is when you select p-value threshold for "rejecting the null."(4) Perform the empirical test & observe the effect. If it is below the specified "meaningful difference" value -- 3% in my example -- you've "proven" that there is "no effect."

For a good treatment of this matter, see Streiner, D.L. Unicorns Do Exist: A Tutorial on “Proving” the Null Hypothesis . Canadian Journal of Psychiatry 48, 756-761 (2003).

• 1 $\begingroup$ +1. This is a nice example of the importance of being clear about one's standard of "proof." In many applications the one you invoke here--the "act as if" standard, if I may call it that--is so weak that nobody would accept it as "proof." I do not deny its utility, though, and advocate this kind of approach to support rational decision making. (But maybe Bayesian methods are better... :-) $\endgroup$ –  whuber ♦ Commented Jan 14, 2011 at 0:47
• 1 $\begingroup$ (+1) Nice answer. I added a link to an online version of Streiner's article; I hope you don't mind (feel free to remove). $\endgroup$ –  chl Commented Jan 14, 2011 at 9:18
• 1 $\begingroup$ couple more things: (1) Treating failure to reject the null as evidence in support of null is a shockingly common error & usual occasion for Streiner's point. This mistake essentially turns the strong aversion to type 1 error in "p < 0.05" norm into license to make type 2. S says, "wait--you need power..." (2) Whuber cites Hume's famous argument. H's pt is actually just as subversive of empirical proofs rejecting the null as of proofs of the null. H says induction can't support causal inference. Ok; but there's no alternative for empirical study! Go Pearl (& Bayes), not Hume, on causality! $\endgroup$ –  dmk38 Commented Jan 14, 2011 at 23:49
• 1 $\begingroup$ this question on equivalence testing also has some good suggestions stats.stackexchange.com/questions/3038/… $\endgroup$ –  Jeromy Anglim Commented Feb 11, 2011 at 4:46
• $\begingroup$ Is this equivalent to assuming "not the null" as the new null hypothesis and then rejecting this new null hypothesis? $\endgroup$ –  user37312 Commented Feb 12, 2014 at 12:17

Answer from the mathematical side : it is possible if and only if "hypotheses are mutually singular".

If by "prove" you mean have a rule that can "accept" (should I say that:) ) $H_0$ with a probability to make a mistake that is zero, then you are searching what could be called "ideal test" and this exists:

If you are testing wether a random variable $X$ is drawn from $P_0$ or from $P_1$ (i.e testing $H_0: X\leadsto P_0$ versus $H_1: X\leadsto P_1$) then there exists an ideal test if and only if $P_1\bot P_0$ ($P_1$ and $P_0$ are "mutually singular").

If you don't know what "mutually singular" means I can give you an example: $\mathcal{U}[0,1]$ and $\mathcal{U}[3,4]$ (uniforms on $[0,1]$ and $[3,4]$) are mutually singular. This means if you want to test

$H_0: X\leadsto \mathcal{U}[0,1]$ versus $H_1: X\leadsto \mathcal{U}[3,4]$

then there exist an ideal test (guess what it is :) ) : a test that is never wrong !

If $P_1$ and $P_0$ are not mutually singular, then this does not exist (this results from the "only if part")!

In non mathematical terms this means that you can prove the null if and only if the proof is already in your assumptions (i.e. if and only if you have chosen the hypothesis $H_0$ and $H_1$ that are so different that a single observation from $H_0$ cannot be identifyed as one from $H_1$ and vise versa).

• 4 $\begingroup$ +1 Nice answer. A simple rendering of the math is that the null and its alternatives are assumed to yield disjoint sets of outcomes; e.g., either there is a zebra in this room or there isn't. Of course "prove" here implicitly includes "conditional on the model," which itself is never established with the same rigor as, say, a mathematical theorem; it implicitly includes "conditional on the accuracy of the observations;" and it implicitly includes that the hypotheses can be unambiguously interpreted. (For criticism of the latter, see George Lakoff's Women, Fire, and Dangerous Things. ) $\endgroup$ –  whuber ♦ Commented Jan 13, 2011 at 21:29

Yes there is a definitive answer. That answer is: No, there isn't a way to prove a null hypothesis. The best you can do, as far as I know, is throw confidence intervals around your estimate and demonstrate that the effect is so small that it might as well be essentially non-existent.

• 5 $\begingroup$ More generally the problem in statistics is not that you are unable to prove the null hypothesis, it is that you are not able to make any point estimates with certainty. That is, just as you can't say "there is no effect of the variable" you are unable to say that "the effect size of the variable is 1.95". Statistics always have confidence intervals. $\endgroup$ –  russellpierce Commented Jan 20, 2011 at 3:41
• 1 $\begingroup$ Agreed that the answer is a big NO, and for a very strong reason: by construction of statistical hypothesis. The fact that the accepted answer is claiming otherwise is absolutely tragic. What hypothesis testing provides as an answer is: assuming my hypothesis is true , are the data that I sampled consistent with it? And by no means the other way around. It does not take much reasoning to understand that you cannot deduce from that whether the hypothesis is true or not. $\endgroup$ –  Christophe Commented Mar 27, 2019 at 7:56

For me, the decision theoretical framework presents the easiest way to understand the "null hypothesis". It basically says that there must be at least two alternatives: the Null hypothesis, and at least one alternative. Then the "decision problem" is to accept one of the alternatives, and reject the others (although we need to be precise about what we mean by "accepting" and "rejecting" the hypothesis). I see the question of "can we prove the null hypothesis?" as analogous to "can we always make the correct decision?". From a decision theory perspective the answer is clearly yes if

1)there is no uncertainty in the decision making process, for then it is a mathematical exercise to work out what the correct decision is.

2)we accept all the other premises/assumptions of the problem. The most critical one (I think) is that the hypothesis we are deciding between are exhaustive, and one (and only one) of them must be true, and the others must be false.

From a more philosophical standpoint, it is not possible to "prove" anything, in the sense that the "proof" depends entirely on the assumptions / axioms which lead to that "proof". I see proof as a kind of logical equivalence rather than a "fact" or "truth" in the sense that if the proof is wrong, the assumptions which led to it are also wrong.

Applying this to the "proving the null hypothesis" I can "prove" it to be true by simply assuming that it is true, or by assuming that it is true if certain conditions are meet (such as the value of a statistic).

Technically, no, a null hypothesis cannot be proven. For any fixed, finite sample size, there will always be some small but nonzero effect size for which your statistical test has virtually no power. More practically, though, you can prove that you're within some small epsilon of the null hypothesis, such that deviations less than this epsilon are not practically significant.

There is a case where a proof is possible. Suppose you have a school and your null hypothesis is that the numbers of boys and of girls is equal. As the sample size increases, the uncertainty in the ratio of boys to girls tends to reduce, eventually reaching certainty (which is what I assume you mean by proof) when the whole pupil population is sampled.

But if you do not have a finite population, or if you are sampling with replacement and cannot spot resampled individuals, then you cannot reduce the uncertainty to zero with a finite sample.

Yes, it is possible to prove the null--in exactly the same sense that it is possible to prove any alternative to the null. In a Bayesian analysis, it is perfectly possible for the odds in favor of the null versus any of the proposed alternatives to it to become arbitrarily large. Moreover, it is false to assert, as some of the above answers assert, that one can only prove the null if the alternatives to it are disjoint (do not overlap with the null). In a Bayesian analysis every hypothesis has a prior probability distribution. This distribution spreads a unit mass of prior probability out over the proposed alternatives. The null hypothesis puts all of the prior probability on a single alternative. In principle, alternatives to the null may put all of the prior probability on some non-null alternative (on another "point"), but this is rare. In general, alternatives hedge, that is, they spread the same mass of prior probability out over other alternatives--either to the exclusion of the null alternative, or, more commonly, including the null alternative. The question then becomes which hypothesis puts the most prior probability where the experimental data actually fall. If the data fall tightly around where the null says they should fall, then it will be the odds-on favority (among the proposed hypotheses) EVEN THOUGH IT IS INCLUDED IN (NESTED IN, NOT MUTUALLY EXCLUSIVE WITH) THE ALTERNATIVES TO IT. The believe that it is not possible for a nested alternative to be more likely than the set in which it is nested reflects the failure to distinguish between probability and likelihood. While it is impossible for a component of a set to be less probable than the entire set, it is perfectly possible for the posterior likelihood of a component of a set of hypotheses to be greater than the posterior likelihood of the set as a whole. The posterior likelihood of an hypothesis is the product of the likelihood function and the prior probability distribution that the hypothesis posits. If an hypothesis puts all of the prior probability in the right place (e.g., on the null), then it will have a higher posterior likelihood than an hypothesis that puts some of the prior probability in the wrong place (not on the null).

I would like to discuss here a point a lot of users are somewhat confused. What is the real meaning of the Null Hypothesis statement H0: p=0? Are we trying to determine if the parameter p is zero? Of course not, there is no way to achieve such a goal.

What we intend to establish is that, given the data set, the evaluated parameter value is (or not) indiscernible from zero. Remember that NHST is "unfair" towards the alternative hypotheses: the null is ascribed a 95% Confidence Level, and only 5% to the alternative. In consequence a “non-significant" result does not mean that H0 holds but simply that we did not found sufficient evidence that the alternative is likely.

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