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  • How to Write a Strong Hypothesis | Guide & Examples

How to Write a Strong Hypothesis | Guide & Examples

Published on 6 May 2022 by Shona McCombes .

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.

Table of contents

What is a hypothesis, developing a hypothesis (with example), hypothesis examples, 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 variables . An independent variable is something the researcher changes or controls. A dependent variable is something the researcher observes and measures.

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 identify which variables you will study and what you think the relationships are between them. Sometimes, you’ll have to operationalise 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.

Step 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

Step 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.

Step 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 .

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

A hypothesis 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).

A research hypothesis is your proposed answer to your research question. The research hypothesis usually includes an explanation (‘ x affects y because …’).

A statistical hypothesis, on the other hand, is a mathematical statement about a population parameter. Statistical hypotheses always come in pairs: the null and alternative hypotheses. In a well-designed study , the statistical hypotheses correspond logically to the research hypothesis.

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What is and How to Write a Good Hypothesis in Research?

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Table of Contents

One of the most important aspects of conducting research is constructing a strong hypothesis. But what makes a hypothesis in research effective? In this article, we’ll look at the difference between a hypothesis and a research question, as well as the elements of a good hypothesis in research. We’ll also include some examples of effective hypotheses, and what pitfalls to avoid.

What is a Hypothesis in Research?

Simply put, a hypothesis is a research question that also includes the predicted or expected result of the research. Without a hypothesis, there can be no basis for a scientific or research experiment. As such, it is critical that you carefully construct your hypothesis by being deliberate and thorough, even before you set pen to paper. Unless your hypothesis is clearly and carefully constructed, any flaw can have an adverse, and even grave, effect on the quality of your experiment and its subsequent results.

Research Question vs Hypothesis

It’s easy to confuse research questions with hypotheses, and vice versa. While they’re both critical to the Scientific Method, they have very specific differences. Primarily, a research question, just like a hypothesis, is focused and concise. But a hypothesis includes a prediction based on the proposed research, and is designed to forecast the relationship of and between two (or more) variables. Research questions are open-ended, and invite debate and discussion, while hypotheses are closed, e.g. “The relationship between A and B will be C.”

A hypothesis is generally used if your research topic is fairly well established, and you are relatively certain about the relationship between the variables that will be presented in your research. Since a hypothesis is ideally suited for experimental studies, it will, by its very existence, affect the design of your experiment. The research question is typically used for new topics that have not yet been researched extensively. Here, the relationship between different variables is less known. There is no prediction made, but there may be variables explored. The research question can be casual in nature, simply trying to understand if a relationship even exists, descriptive or comparative.

How to Write Hypothesis in Research

Writing an effective hypothesis starts before you even begin to type. Like any task, preparation is key, so you start first by conducting research yourself, and reading all you can about the topic that you plan to research. From there, you’ll gain the knowledge you need to understand where your focus within the topic will lie.

Remember that a hypothesis is a prediction of the relationship that exists between two or more variables. Your job is to write a hypothesis, and design the research, to “prove” whether or not your prediction is correct. A common pitfall is to use judgments that are subjective and inappropriate for the construction of a hypothesis. It’s important to keep the focus and language of your hypothesis objective.

An effective hypothesis in research is clearly and concisely written, and any terms or definitions clarified and defined. Specific language must also be used to avoid any generalities or assumptions.

Use the following points as a checklist to evaluate the effectiveness of your research hypothesis:

  • Predicts the relationship and outcome
  • Simple and concise – avoid wordiness
  • Clear with no ambiguity or assumptions about the readers’ knowledge
  • Observable and testable results
  • Relevant and specific to the research question or problem

Research Hypothesis Example

Perhaps the best way to evaluate whether or not your hypothesis is effective is to compare it to those of your colleagues in the field. There is no need to reinvent the wheel when it comes to writing a powerful research hypothesis. As you’re reading and preparing your hypothesis, you’ll also read other hypotheses. These can help guide you on what works, and what doesn’t, when it comes to writing a strong research hypothesis.

Here are a few generic examples to get you started.

Eating an apple each day, after the age of 60, will result in a reduction of frequency of physician visits.

Budget airlines are more likely to receive more customer complaints. A budget airline is defined as an airline that offers lower fares and fewer amenities than a traditional full-service airline. (Note that the term “budget airline” is included in the hypothesis.

Workplaces that offer flexible working hours report higher levels of employee job satisfaction than workplaces with fixed hours.

Each of the above examples are specific, observable and measurable, and the statement of prediction can be verified or shown to be false by utilizing standard experimental practices. It should be noted, however, that often your hypothesis will change as your research progresses.

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How to Develop a Good Research Hypothesis

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The story of a research study begins by asking a question. Researchers all around the globe are asking curious questions and formulating research hypothesis. However, whether the research study provides an effective conclusion depends on how well one develops a good research hypothesis. Research hypothesis examples could help researchers get an idea as to how to write a good research hypothesis.

This blog will help you understand what is a research hypothesis, its characteristics and, how to formulate a research hypothesis

Table of Contents

What is Hypothesis?

Hypothesis is an assumption or an idea proposed for the sake of argument so that it can be tested. It is a precise, testable statement of what the researchers predict will be outcome of the study.  Hypothesis usually involves proposing a relationship between two variables: the independent variable (what the researchers change) and the dependent variable (what the research measures).

What is a Research Hypothesis?

Research hypothesis is a statement that introduces a research question and proposes an expected result. It is an integral part of the scientific method that forms the basis of scientific experiments. Therefore, you need to be careful and thorough when building your research hypothesis. A minor flaw in the construction of your hypothesis could have an adverse effect on your experiment. In research, there is a convention that the hypothesis is written in two forms, the null hypothesis, and the alternative hypothesis (called the experimental hypothesis when the method of investigation is an experiment).

Characteristics of a Good Research Hypothesis

As the hypothesis is specific, there is a testable prediction about what you expect to happen in a study. You may consider drawing hypothesis from previously published research based on the theory.

A good research hypothesis involves more effort than just a guess. In particular, your hypothesis may begin with a question that could be further explored through background research.

To help you formulate a promising research hypothesis, you should ask yourself the following questions:

  • Is the language clear and focused?
  • What is the relationship between your hypothesis and your research topic?
  • Is your hypothesis testable? If yes, then how?
  • What are the possible explanations that you might want to explore?
  • Does your hypothesis include both an independent and dependent variable?
  • Can you manipulate your variables without hampering the ethical standards?
  • Does your research predict the relationship and outcome?
  • Is your research simple and concise (avoids wordiness)?
  • Is it clear with no ambiguity or assumptions about the readers’ knowledge
  • Is your research observable and testable results?
  • Is it relevant and specific to the research question or problem?

research hypothesis example

The questions listed above can be used as a checklist to make sure your hypothesis is based on a solid foundation. Furthermore, it can help you identify weaknesses in your hypothesis and revise it if necessary.

Source: Educational Hub

How to formulate a research hypothesis.

A testable hypothesis is not a simple statement. It is rather an intricate statement that needs to offer a clear introduction to a scientific experiment, its intentions, and the possible outcomes. However, there are some important things to consider when building a compelling hypothesis.

1. State the problem that you are trying to solve.

Make sure that the hypothesis clearly defines the topic and the focus of the experiment.

2. Try to write the hypothesis as an if-then statement.

Follow this template: If a specific action is taken, then a certain outcome is expected.

3. Define the variables

Independent variables are the ones that are manipulated, controlled, or changed. Independent variables are isolated from other factors of the study.

Dependent variables , as the name suggests are dependent on other factors of the study. They are influenced by the change in independent variable.

4. Scrutinize the hypothesis

Evaluate assumptions, predictions, and evidence rigorously to refine your understanding.

Types of Research Hypothesis

The types of research hypothesis are stated below:

1. Simple Hypothesis

It predicts the relationship between a single dependent variable and a single independent variable.

2. Complex Hypothesis

It predicts the relationship between two or more independent and dependent variables.

3. Directional Hypothesis

It specifies the expected direction to be followed to determine the relationship between variables and is derived from theory. Furthermore, it implies the researcher’s intellectual commitment to a particular outcome.

4. Non-directional Hypothesis

It does not predict the exact direction or nature of the relationship between the two variables. The non-directional hypothesis is used when there is no theory involved or when findings contradict previous research.

5. Associative and Causal Hypothesis

The associative hypothesis defines interdependency between variables. A change in one variable results in the change of the other variable. On the other hand, the causal hypothesis proposes an effect on the dependent due to manipulation of the independent variable.

6. Null Hypothesis

Null hypothesis states a negative statement to support the researcher’s findings that there is no relationship between two variables. There will be no changes in the dependent variable due the manipulation of the independent variable. Furthermore, it states results are due to chance and are not significant in terms of supporting the idea being investigated.

7. Alternative Hypothesis

It states that there is a relationship between the two variables of the study and that the results are significant to the research topic. An experimental hypothesis predicts what changes will take place in the dependent variable when the independent variable is manipulated. Also, it states that the results are not due to chance and that they are significant in terms of supporting the theory being investigated.

Research Hypothesis Examples of Independent and Dependent Variables

Research Hypothesis Example 1 The greater number of coal plants in a region (independent variable) increases water pollution (dependent variable). If you change the independent variable (building more coal factories), it will change the dependent variable (amount of water pollution).
Research Hypothesis Example 2 What is the effect of diet or regular soda (independent variable) on blood sugar levels (dependent variable)? If you change the independent variable (the type of soda you consume), it will change the dependent variable (blood sugar levels)

You should not ignore the importance of the above steps. The validity of your experiment and its results rely on a robust testable hypothesis. Developing a strong testable hypothesis has few advantages, it compels us to think intensely and specifically about the outcomes of a study. Consequently, it enables us to understand the implication of the question and the different variables involved in the study. Furthermore, it helps us to make precise predictions based on prior research. Hence, forming a hypothesis would be of great value to the research. Here are some good examples of testable hypotheses.

More importantly, you need to build a robust testable research hypothesis for your scientific experiments. A testable hypothesis is a hypothesis that can be proved or disproved as a result of experimentation.

Importance of a Testable Hypothesis

To devise and perform an experiment using scientific method, you need to make sure that your hypothesis is testable. To be considered testable, some essential criteria must be met:

  • There must be a possibility to prove that the hypothesis is true.
  • There must be a possibility to prove that the hypothesis is false.
  • The results of the hypothesis must be reproducible.

Without these criteria, the hypothesis and the results will be vague. As a result, the experiment will not prove or disprove anything significant.

What are your experiences with building hypotheses for scientific experiments? What challenges did you face? How did you overcome these challenges? Please share your thoughts with us in the comments section.

Frequently Asked Questions

The steps to write a research hypothesis are: 1. Stating the problem: Ensure that the hypothesis defines the research problem 2. Writing a hypothesis as an 'if-then' statement: Include the action and the expected outcome of your study by following a ‘if-then’ structure. 3. Defining the variables: Define the variables as Dependent or Independent based on their dependency to other factors. 4. Scrutinizing the hypothesis: Identify the type of your hypothesis

Hypothesis testing is a statistical tool which is used to make inferences about a population data to draw conclusions for a particular hypothesis.

Hypothesis in statistics is a formal statement about the nature of a population within a structured framework of a statistical model. It is used to test an existing hypothesis by studying a population.

Research hypothesis is a statement that introduces a research question and proposes an expected result. It forms the basis of scientific experiments.

The different types of hypothesis in research are: • Null hypothesis: Null hypothesis is a negative statement to support the researcher’s findings that there is no relationship between two variables. • Alternate hypothesis: Alternate hypothesis predicts the relationship between the two variables of the study. • Directional hypothesis: Directional hypothesis specifies the expected direction to be followed to determine the relationship between variables. • Non-directional hypothesis: Non-directional hypothesis does not predict the exact direction or nature of the relationship between the two variables. • Simple hypothesis: Simple hypothesis predicts the relationship between a single dependent variable and a single independent variable. • Complex hypothesis: Complex hypothesis predicts the relationship between two or more independent and dependent variables. • Associative and casual hypothesis: Associative and casual hypothesis predicts the relationship between two or more independent and dependent variables. • Empirical hypothesis: Empirical hypothesis can be tested via experiments and observation. • Statistical hypothesis: A statistical hypothesis utilizes statistical models to draw conclusions about broader populations.

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How to Write a Research Hypothesis: Good & Bad Examples

strong hypothesis

What is a research hypothesis?

A research hypothesis is an attempt at explaining a phenomenon or the relationships between phenomena/variables in the real world. Hypotheses are sometimes called “educated guesses”, but they are in fact (or let’s say they should be) based on previous observations, existing theories, scientific evidence, and logic. A research hypothesis is also not a prediction—rather, predictions are ( should be) based on clearly formulated hypotheses. For example, “We tested the hypothesis that KLF2 knockout mice would show deficiencies in heart development” is an assumption or prediction, not a hypothesis. 

The research hypothesis at the basis of this prediction is “the product of the KLF2 gene is involved in the development of the cardiovascular system in mice”—and this hypothesis is probably (hopefully) based on a clear observation, such as that mice with low levels of Kruppel-like factor 2 (which KLF2 codes for) seem to have heart problems. From this hypothesis, you can derive the idea that a mouse in which this particular gene does not function cannot develop a normal cardiovascular system, and then make the prediction that we started with. 

What is the difference between a hypothesis and a prediction?

You might think that these are very subtle differences, and you will certainly come across many publications that do not contain an actual hypothesis or do not make these distinctions correctly. But considering that the formulation and testing of hypotheses is an integral part of the scientific method, it is good to be aware of the concepts underlying this approach. The two hallmarks of a scientific hypothesis are falsifiability (an evaluation standard that was introduced by the philosopher of science Karl Popper in 1934) and testability —if you cannot use experiments or data to decide whether an idea is true or false, then it is not a hypothesis (or at least a very bad one).

So, in a nutshell, you (1) look at existing evidence/theories, (2) come up with a hypothesis, (3) make a prediction that allows you to (4) design an experiment or data analysis to test it, and (5) come to a conclusion. Of course, not all studies have hypotheses (there is also exploratory or hypothesis-generating research), and you do not necessarily have to state your hypothesis as such in your paper. 

But for the sake of understanding the principles of the scientific method, let’s first take a closer look at the different types of hypotheses that research articles refer to and then give you a step-by-step guide for how to formulate a strong hypothesis for your own paper.

Types of Research Hypotheses

Hypotheses can be simple , which means they describe the relationship between one single independent variable (the one you observe variations in or plan to manipulate) and one single dependent variable (the one you expect to be affected by the variations/manipulation). If there are more variables on either side, you are dealing with a complex hypothesis. You can also distinguish hypotheses according to the kind of relationship between the variables you are interested in (e.g., causal or associative ). But apart from these variations, we are usually interested in what is called the “alternative hypothesis” and, in contrast to that, the “null hypothesis”. If you think these two should be listed the other way round, then you are right, logically speaking—the alternative should surely come second. However, since this is the hypothesis we (as researchers) are usually interested in, let’s start from there.

Alternative Hypothesis

If you predict a relationship between two variables in your study, then the research hypothesis that you formulate to describe that relationship is your alternative hypothesis (usually H1 in statistical terms). The goal of your hypothesis testing is thus to demonstrate that there is sufficient evidence that supports the alternative hypothesis, rather than evidence for the possibility that there is no such relationship. The alternative hypothesis is usually the research hypothesis of a study and is based on the literature, previous observations, and widely known theories. 

Null Hypothesis

The hypothesis that describes the other possible outcome, that is, that your variables are not related, is the null hypothesis ( H0 ). Based on your findings, you choose between the two hypotheses—usually that means that if your prediction was correct, you reject the null hypothesis and accept the alternative. Make sure, however, that you are not getting lost at this step of the thinking process: If your prediction is that there will be no difference or change, then you are trying to find support for the null hypothesis and reject H1. 

Directional Hypothesis

While the null hypothesis is obviously “static”, the alternative hypothesis can specify a direction for the observed relationship between variables—for example, that mice with higher expression levels of a certain protein are more active than those with lower levels. This is then called a one-tailed hypothesis. 

Another example for a directional one-tailed alternative hypothesis would be that 

H1: Attending private classes before important exams has a positive effect on performance. 

Your null hypothesis would then be that

H0: Attending private classes before important exams has no/a negative effect on performance.

Nondirectional Hypothesis

A nondirectional hypothesis does not specify the direction of the potentially observed effect, only that there is a relationship between the studied variables—this is called a two-tailed hypothesis. For instance, if you are studying a new drug that has shown some effects on pathways involved in a certain condition (e.g., anxiety) in vitro in the lab, but you can’t say for sure whether it will have the same effects in an animal model or maybe induce other/side effects that you can’t predict and potentially increase anxiety levels instead, you could state the two hypotheses like this:

H1: The only lab-tested drug (somehow) affects anxiety levels in an anxiety mouse model.

You then test this nondirectional alternative hypothesis against the null hypothesis:

H0: The only lab-tested drug has no effect on anxiety levels in an anxiety mouse model.

hypothesis in a research paper

How to Write a Hypothesis for a Research Paper

Now that we understand the important distinctions between different kinds of research hypotheses, let’s look at a simple process of how to write a hypothesis.

Writing a Hypothesis Step:1

Ask a question, based on earlier research. Research always starts with a question, but one that takes into account what is already known about a topic or phenomenon. For example, if you are interested in whether people who have pets are happier than those who don’t, do a literature search and find out what has already been demonstrated. You will probably realize that yes, there is quite a bit of research that shows a relationship between happiness and owning a pet—and even studies that show that owning a dog is more beneficial than owning a cat ! Let’s say you are so intrigued by this finding that you wonder: 

What is it that makes dog owners even happier than cat owners? 

Let’s move on to Step 2 and find an answer to that question.

Writing a Hypothesis Step 2:

Formulate a strong hypothesis by answering your own question. Again, you don’t want to make things up, take unicorns into account, or repeat/ignore what has already been done. Looking at the dog-vs-cat papers your literature search returned, you see that most studies are based on self-report questionnaires on personality traits, mental health, and life satisfaction. What you don’t find is any data on actual (mental or physical) health measures, and no experiments. You therefore decide to make a bold claim come up with the carefully thought-through hypothesis that it’s maybe the lifestyle of the dog owners, which includes walking their dog several times per day, engaging in fun and healthy activities such as agility competitions, and taking them on trips, that gives them that extra boost in happiness. You could therefore answer your question in the following way:

Dog owners are happier than cat owners because of the dog-related activities they engage in.

Now you have to verify that your hypothesis fulfills the two requirements we introduced at the beginning of this resource article: falsifiability and testability . If it can’t be wrong and can’t be tested, it’s not a hypothesis. We are lucky, however, because yes, we can test whether owning a dog but not engaging in any of those activities leads to lower levels of happiness or well-being than owning a dog and playing and running around with them or taking them on trips.  

Writing a Hypothesis Step 3:

Make your predictions and define your variables. We have verified that we can test our hypothesis, but now we have to define all the relevant variables, design our experiment or data analysis, and make precise predictions. You could, for example, decide to study dog owners (not surprising at this point), let them fill in questionnaires about their lifestyle as well as their life satisfaction (as other studies did), and then compare two groups of active and inactive dog owners. Alternatively, if you want to go beyond the data that earlier studies produced and analyzed and directly manipulate the activity level of your dog owners to study the effect of that manipulation, you could invite them to your lab, select groups of participants with similar lifestyles, make them change their lifestyle (e.g., couch potato dog owners start agility classes, very active ones have to refrain from any fun activities for a certain period of time) and assess their happiness levels before and after the intervention. In both cases, your independent variable would be “ level of engagement in fun activities with dog” and your dependent variable would be happiness or well-being . 

Examples of a Good and Bad Hypothesis

Let’s look at a few examples of good and bad hypotheses to get you started.

Good Hypothesis Examples

Bad hypothesis examples, tips for writing a research hypothesis.

If you understood the distinction between a hypothesis and a prediction we made at the beginning of this article, then you will have no problem formulating your hypotheses and predictions correctly. To refresh your memory: We have to (1) look at existing evidence, (2) come up with a hypothesis, (3) make a prediction, and (4) design an experiment. For example, you could summarize your dog/happiness study like this:

(1) While research suggests that dog owners are happier than cat owners, there are no reports on what factors drive this difference. (2) We hypothesized that it is the fun activities that many dog owners (but very few cat owners) engage in with their pets that increases their happiness levels. (3) We thus predicted that preventing very active dog owners from engaging in such activities for some time and making very inactive dog owners take up such activities would lead to an increase and decrease in their overall self-ratings of happiness, respectively. (4) To test this, we invited dog owners into our lab, assessed their mental and emotional well-being through questionnaires, and then assigned them to an “active” and an “inactive” group, depending on… 

Note that you use “we hypothesize” only for your hypothesis, not for your experimental prediction, and “would” or “if – then” only for your prediction, not your hypothesis. A hypothesis that states that something “would” affect something else sounds as if you don’t have enough confidence to make a clear statement—in which case you can’t expect your readers to believe in your research either. Write in the present tense, don’t use modal verbs that express varying degrees of certainty (such as may, might, or could ), and remember that you are not drawing a conclusion while trying not to exaggerate but making a clear statement that you then, in a way, try to disprove . And if that happens, that is not something to fear but an important part of the scientific process.

Similarly, don’t use “we hypothesize” when you explain the implications of your research or make predictions in the conclusion section of your manuscript, since these are clearly not hypotheses in the true sense of the word. As we said earlier, you will find that many authors of academic articles do not seem to care too much about these rather subtle distinctions, but thinking very clearly about your own research will not only help you write better but also ensure that even that infamous Reviewer 2 will find fewer reasons to nitpick about your manuscript. 

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Now that you know how to write a strong research hypothesis for your research paper, you might be interested in our free AI proofreader , Wordvice AI, which finds and fixes errors in grammar, punctuation, and word choice in academic texts. Or if you are interested in human proofreading , check out our English editing services , including research paper editing and manuscript editing .

On the Wordvice academic resources website , you can also find many more articles and other resources that can help you with writing the other parts of your research paper , with making a research paper outline before you put everything together, or with writing an effective cover letter once you are ready to submit.

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How To Write a Strong Research Hypothesis

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Are you looking to take your research project to the next level? Have you heard of the power of a hypothesis but need to figure out how to formulate one that will unlock potential discoveries? We can help!

So get ready; it's time to dive into unlocking the power of research! This blog post will explore what makes a well-crafted and powerful hypothesis - from identifying a research question to developing supporting evidence.

By learning how to craft a compelling hypothesis, you'll have more tremendous success in every step of your research project.

What are hypotheses, and why are they important?

A hypothesis is an educated guess or a proposition based on limited evidence as a starting point for further investigation. It provides a framework for research and allows researchers to refine their ideas, collect data, and draw conclusions. Hypotheses are essential to the process because they will enable us to organize our thoughts and test theories properly.

Hypotheses are used in many fields , from medicine to psychology to economics. In each area, developing hypotheses based on observations enable researchers to make predictions about their data and guide them toward finding meaningful results.

For example, in medicine, hypotheses can be used to predict which treatments will be most effective for particular conditions or which drugs may have adverse effects when taken together. This allows doctors to make better decisions when caring for patients.

In psychology, hypotheses are often used in experiments to determine whether certain variables influence behavior or mental processes. By testing different combinations of variables, psychologists can identify patterns and understand why people behave the way they do.

In economics, hypotheses provide economists with a framework for analyzing the relationship between economic variables such as wages and consumer spending habits. By understanding these relationships, economists can better understand how economic forces affect the economy.

Overall, hypotheses play an essential role in helping scientists develop new ideas and draw meaningful conclusions from the collected data. Without taking the step to create hypotheses, it would be difficult for researchers to make sense of the vast amounts of information available today and use it effectively in their investigations.

How to determine an effective research question to form your hypothesis

When conducting research, having a compelling research question is critical . Properly formulating this question will allow the researcher to develop their hypothesis. A research question provides a clear and focused goal for your research study and also gives direction on how to get there. A compelling research question should be specific, answerable in the context of your field of study, significant, novel (not already answered by previous studies), and timely – that is, relevant to current events or trends.

Before determining the best research question, you must first understand your topic. Think about the area of knowledge that interests you most and narrow it down to a single theme or concept within this topic. Focus on what interests you most within this theme, and make sure there is room for further exploration and analysis. Once you have chosen a specific topic and narrowed down your focus, you can begin formulating questions related to your project.

To ensure relevance and impact to your field of study, choose questions that address essential issues in the literature or suggest solutions to existing problems. Avoid overly broad topics with unclear objectives; instead, opt for focused questions to enable targeted data collection and analysis with concrete results.

Additionally, consider time frames when formulating questions. If the issue has been discussed extensively in the past but has not been revisited recently, then it's likely not worthy of a new investigation.

Once you have developed some potential questions related to your topic, review them carefully and decide which question best captures the essence of what you want to learn through researching this topic.

Ask yourself:

  • Is this question answerable?
  • Does it fit within my field of study?
  • Is it significant enough?
  • Would its findings be novel?

If so, then congratulations! You have identified a compelling research question.

Tips for crafting a well-crafted hypothesis

Once you have formulated the official research question, you may develop the formal hypothesis. When composing a hypothesis, it's essential to think carefully about the question you are trying to answer.

A solid hypothesis should be testable, meaning that it can be verified or disproved through research. It should also be specific and focused on one issue at a time. Here are some tips for crafting a well-crafted hypothesis:

  • Consider the goal of your research: Think about what it is that you want to learn or determine from your experiment and make sure that your hypothesis reflects this goal.
  • Create an educated guess as to why something is happening: Your hypothesis should explain why something is occurring based on what evidence you already have and direct further investigation into the matter. For example, if you hypothesize that increased carbon dioxide levels in the atmosphere will lead to global warming, your research should focus on examining this relationship further.
  • Define any variables or parameters involved in the experiment: This includes things like temperature or chemical composition that could potentially affect the outcome of any experiments done in pursuit of testing your hypothesis.
  • Use clear and precise language: Make sure your hypothesis is written with clear and precise language so that anyone reading it can understand exactly what you are attempting to investigate or explain. Avoid complex words and keep sentences short whenever possible.

Following these simple tips will help ensure that your hypothesis is well-crafted and ready for testing!

Examples of evidence that can support your hypothesis

When it comes to developing a hypothesis, supporting evidence is essential for making sure it holds up. This evidence helps strengthen the argument that is being driven by providing facts and logical reasoning that support the hypothesis.

Examples of evidence that can be used to back up a hypothesis include using data from experiments, case studies, and other research projects. Data from experiments can provide insight into how certain variables interact to form a particular outcome.

Case studies may offer greater depth in understanding a specific phenomenon's cause and effect; research projects may yield results that confirm or refute existing theories on a subject.

In addition to these traditional forms of evidence, personal experiences or observations can also help to support a hypothesis. For example, if someone's daily commute has been consistently faster since they changed routes, they could use their personal experience to argue that making this change resulted in shorter commutes.

Similarly, suppose someone has witnessed how two variables consistently coincide (i.e., when one goes up, another goes down). In that case, this could be used to support the notion that there is some correlation between these two aspects.

Overall, evidence to support your hypothesis is crucial for ensuring its validity and credibility. While conducting experiments or researching may seem like time-consuming processes, having solid supporting evidence will make it much easier to defend your ideas convincingly when challenged.

Therefore, it is crucial to take the time necessary to gather credible sources of information to provide the most substantial possible backing for your hypotheses.

Understanding the potential of hypotheses and how they can help your research project progress

The power of research lies in the ability to develop and test hypotheses. A hypothesis is a statement or an idea that can be tested to determine its validity.

Essentially, it is a form of educated guesswork that helps researchers form conclusions about their data. By developing a hypothesis for a research project, you are effectively setting up the framework for further exploration.

When developing a hypothesis, you must consider both the expected outcomes and possible alternative explanations. This will help you focus on testing the possible results without getting sidetracked by irrelevant information. Once you have established a concrete hypothesis, it can then be used as a basis for further research and experimentation.

The process of testing hypotheses is an integral part of the scientific method and can help researchers build confidence in their findings and conclusions. Through careful observation and experimentation, researchers can compare their results against what they initially hypothesized, allowing them to draw more accurate conclusions about their data. As such, hypotheses play an essential role in helping researchers connect the dots between different pieces of evidence and form meaningful conclusions.

Overall, understanding how hypotheses can be used in research projects can be immensely beneficial in helping progress towards reaching meaningful insights from their data. By setting up expectations ahead of time and then testing them against real-world conditions, researchers can gain valuable insights that could potentially change the way we understand our world – now that's something worth exploring!

Final thoughts

A hypothesis is a proposed explanation for an observable phenomenon. It's important to note that hypotheses are not the same thing as theories–a theory is a much broader and well-established frame of reference that explains multiple phenomena.

Generally, scientists form a research question and then narrow it down to a testable hypothesis. After making observations and conducting experiments to gather data, researchers can use evidence to support or reject the hypothesis.

By following these steps to formulate a solid hypothesis, you will be on your way to developing a successful research project. Happy researching!

Header image by Bnenin .

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How to Write a Great Hypothesis

Hypothesis Definition, Format, Examples, and Tips

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

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Amy Morin, LCSW, is a psychotherapist and international bestselling author. Her books, including "13 Things Mentally Strong People Don't Do," have been translated into more than 40 languages. Her TEDx talk,  "The Secret of Becoming Mentally Strong," is one of the most viewed talks of all time.

strong hypothesis

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."

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Hypothesis Examples

Hypothesis Examples

A hypothesis is a prediction of the outcome of a test. It forms the basis for designing an experiment in the scientific method . A good hypothesis is testable, meaning it makes a prediction you can check with observation or experimentation. Here are different hypothesis examples.

Null Hypothesis Examples

The null hypothesis (H 0 ) is also known as the zero-difference or no-difference hypothesis. It predicts that changing one variable ( independent variable ) will have no effect on the variable being measured ( dependent variable ). Here are null hypothesis examples:

  • Plant growth is unaffected by temperature.
  • If you increase temperature, then solubility of salt will increase.
  • Incidence of skin cancer is unrelated to ultraviolet light exposure.
  • All brands of light bulb last equally long.
  • Cats have no preference for the color of cat food.
  • All daisies have the same number of petals.

Sometimes the null hypothesis shows there is a suspected correlation between two variables. For example, if you think plant growth is affected by temperature, you state the null hypothesis: “Plant growth is not affected by temperature.” Why do you do this, rather than say “If you change temperature, plant growth will be affected”? The answer is because it’s easier applying a statistical test that shows, with a high level of confidence, a null hypothesis is correct or incorrect.

Research Hypothesis Examples

A research hypothesis (H 1 ) is a type of hypothesis used to design an experiment. This type of hypothesis is often written as an if-then statement because it’s easy identifying the independent and dependent variables and seeing how one affects the other. If-then statements explore cause and effect. In other cases, the hypothesis shows a correlation between two variables. Here are some research hypothesis examples:

  • If you leave the lights on, then it takes longer for people to fall asleep.
  • If you refrigerate apples, they last longer before going bad.
  • If you keep the curtains closed, then you need less electricity to heat or cool the house (the electric bill is lower).
  • If you leave a bucket of water uncovered, then it evaporates more quickly.
  • Goldfish lose their color if they are not exposed to light.
  • Workers who take vacations are more productive than those who never take time off.

Is It Okay to Disprove a Hypothesis?

Yes! You may even choose to write your hypothesis in such a way that it can be disproved because it’s easier to prove a statement is wrong than to prove it is right. In other cases, if your prediction is incorrect, that doesn’t mean the science is bad. Revising a hypothesis is common. It demonstrates you learned something you did not know before you conducted the experiment.

Test yourself with a Scientific Method Quiz .

  • Mellenbergh, G.J. (2008). Chapter 8: Research designs: Testing of research hypotheses. In H.J. Adèr & G.J. Mellenbergh (eds.), Advising on Research Methods: A Consultant’s Companion . Huizen, The Netherlands: Johannes van Kessel Publishing.
  • Popper, Karl R. (1959). The Logic of Scientific Discovery . Hutchinson & Co. ISBN 3-1614-8410-X.
  • Schick, Theodore; Vaughn, Lewis (2002). How to think about weird things: critical thinking for a New Age . Boston: McGraw-Hill Higher Education. ISBN 0-7674-2048-9.
  • Tobi, Hilde; Kampen, Jarl K. (2018). “Research design: the methodology for interdisciplinary research framework”. Quality & Quantity . 52 (3): 1209–1225. doi: 10.1007/s11135-017-0513-8

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What Makes a Strong Hypothesis for Scientific Research?

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What Are Examples of a Hypothesis?

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A hypothesis is an explanation for a set of observations. Here are examples of a scientific hypothesis.

Although you could state a scientific hypothesis in various ways, most hypotheses are either "If, then" statements or forms of the null hypothesis . The null hypothesis is sometimes called the "no difference" hypothesis. The null hypothesis is good for experimentation because it's simple to disprove. If you disprove a null hypothesis, that is evidence for a relationship between the variables you are examining.

Examples of Null Hypotheses

  • Hyperactivity is unrelated to eating sugar.
  • All daisies have the same number of petals.
  • The number of pets in a household is unrelated to the number of people living in it.
  • A person's preference for a shirt is unrelated to its color.

Examples of If, Then Hypotheses

  • If you get at least 6 hours of sleep, you will do better on tests than if you get less sleep.
  • If you drop a ball, it will fall toward the ground.
  • If you drink coffee before going to bed, then it will take longer to fall asleep.
  • If you cover a wound with a bandage, then it will heal with less scarring.

Improving a Hypothesis to Make It Testable

You may wish to revise your first hypothesis in order to make it easier to design an experiment to test. For example, let's say you have a bad breakout the morning after eating a lot of greasy food. You may wonder if there is a correlation between eating greasy food and getting pimples. You propose the hypothesis:

Eating greasy food causes pimples.

Next, you need to design an experiment to test this hypothesis. Let's say you decide to eat greasy food every day for a week and record the effect on your face. Then, as a control, you'll avoid greasy food for the next week and see what happens. Now, this is not a good experiment because it does not take into account other factors such as hormone levels, stress, sun exposure, exercise, or any number of other variables that might conceivably affect your skin.

The problem is that you cannot assign cause to your effect . If you eat french fries for a week and suffer a breakout, can you definitely say it was the grease in the food that caused it? Maybe it was the salt. Maybe it was the potato. Maybe it was unrelated to diet. You can't prove your hypothesis. It's much easier to disprove a hypothesis.

So, let's restate the hypothesis to make it easier to evaluate the data:

Getting pimples is unaffected by eating greasy food.

So, if you eat fatty food every day for a week and suffer breakouts and then don't break out the week that you avoid greasy food, you can be pretty sure something is up. Can you disprove the hypothesis? Probably not, since it is so hard to assign cause and effect. However, you can make a strong case that there is some relationship between diet and acne.

If your skin stays clear for the entire test, you may decide to accept your hypothesis . Again, you didn't prove or disprove anything, which is fine

  • Null Hypothesis Examples
  • Difference Between Independent and Dependent Variables
  • Examples of Independent and Dependent Variables
  • Null Hypothesis Definition and Examples
  • What Is a Hypothesis? (Science)
  • What Are the Elements of a Good Hypothesis?
  • Understanding Simple vs Controlled Experiments
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Strong Induction

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Strong induction is a variant of induction, in which we assume that the statement holds for all values preceding \(k\). This provides us with more information to use when trying to prove the statement.

Proof of Strong Induction

Additional problems.

Now that we know how standard induction works, it's time to look at a variant of it, strong induction. In many ways, strong induction is similar to normal induction. There is, however, a difference in the inductive hypothesis. Normally, when using induction, we assume that \(P(k)\) is true to prove \(P(k+1)\). In strong induction, we assume that all of \(P(1), P(2), . . . , P(k)\) are true to prove \(P(k + 1)\).

Why would we need to do that? Let's go back to our domino analogy. Say that you have infinitely many dominoes arranged in a line. But this time, the weight of the \(k^\text{th}\) domino isn't enough to knock down the \((k+1)^\text{th}\) domino. Knocking down the \((k+1)^\text{th}\) domino requires the weight of all the dominoes before it. Even now, if you are able to knock down the first domino, you can prove that all the dominoes will eventually fall.

The reason why this is called "strong induction" is that we use more statements in the inductive hypothesis. Let's write what we've learned till now a bit more formally.

Proof by strong induction Step 1. Demonstrate the base case: This is where you verify that \(P(k_0)\) is true. In most cases, \(k_0=1.\) Step 2. Prove the inductive step: This is where you assume that all of \(P(k_0)\), \(P(k_0+1), P(k_0+2), \ldots, P(k)\) are true (our inductive hypothesis). Then you show that \(P(k+1)\) is true.

The proof of why this works is similar to that of standard induction.

The Fibonacci sequence is defined by \( F_{n+2} = F_{n+1} + F_n \) for integers \( n \geq 0 \), with starting values \( F_1 = F_ 2 = 1 \). Show that \[ F_n = \frac{ 1}{ \sqrt{5} } \left [ \left ( \frac{1 + \sqrt{5} } { 2} \right)^n - \left ( \frac{1 - \sqrt{5} } { 2} \right)^n \right]. \] Base case: For \( n = 1 \), we have \( LHS: F_1 =1 \) and \( RHS: \frac{ 1}{ \sqrt{5} } \left [ \left ( \frac{1 + \sqrt{5} } { 2} \right)^1 - \left ( \frac{1 - \sqrt{5} } { 2} \right)^1 \right] = \frac{ 1 } { \sqrt{5} } \left[ \frac{ 2 \sqrt{5} } { 2} \right] = 1 \). For \( n = 2 \), we have \( LHS: F_2 = 1 \) and \( RHS: \frac{ 1}{ \sqrt{5} } \left [ \left ( \frac{1 + \sqrt{5} } { 2} \right)^2 - \left ( \frac{1 - \sqrt{5} } { 2} \right)^2 \right] = \frac{ 1 } { \sqrt{5} } \left[ \frac{ 4 \sqrt{5} } { 4} \right] = 1 \). Induction step: Suppose that the statement is true for \( n = k-1 \) and \( k \). Then, we have \[\begin{align} F_{n+1} & = F_n + F_{n-1} \\\\ & = \frac{ 1}{ \sqrt{5} } \left [ \left ( \frac{1 + \sqrt{5} } { 2} \right)^n - \left ( \frac{1 - \sqrt{5} } { 2} \right)^n \right] + \frac{ 1}{ \sqrt{5} } \left [ \left ( \frac{1 + \sqrt{5} } { 2} \right)^{n-1} - \left ( \frac{1 - \sqrt{5} } { 2} \right)^{n-1} \right] \\\\ & = \frac{ 1}{ \sqrt{5} } \left [ \left ( \frac{1 + \sqrt{5} } { 2} \right)^n + \left ( \frac{1 + \sqrt{5} } { 2} \right)^{n-1} \right] - \frac{ 1}{ \sqrt{5} } \left [ \left ( \frac{1 - \sqrt{5} } { 2} \right)^n + \left ( \frac{1 - \sqrt{5} } { 2} \right)^{n-1} \right] \\\\ & = \frac{ 1}{ \sqrt{5} } \left [ \left ( \frac{1 + \sqrt{5} } { 2} \right)^{n+1} - \left ( \frac{1 - \sqrt{5} } { 2} \right)^{n+1} \right]. \end{align} \] Hence, the proposition is true. \(_\square\)

Note that, in this case, we did not need to use all of prior statements, but just the previous 2.

A chocolate bar consists of unit squares arranged in an \( n \times m \) rectangular grid. You may split the bar into individual unit squares, by breaking along the lines. What is the number of breaks required? We will show that the number of breaks needed is \( nm - 1 \). Base Case: For a \( 1 \times 1 \) square, we are already done, so no steps are needed. \( 1 \times 1 - 1 = 0 \), so the base case is true. Induction Step: Let \( P(n,m) \) denote the number of breaks needed to split up an \( n \times m \) square. WLOG, we may assume that the first break is along a row, and we get an \( n_1 \times m \) and an \( n_2 \times m \) bar, where \( n_1 + n_2 = n \). By the induction hypothesis, the number of further breaks that we need is \( n_1 \times m - 1 \) and \( n_2 \times m - 1 \). Hence, the total number of breaks that we need is \[ 1 + ( n_1 \times m -1 ) + ( n_2 \times m - 1 ) = (n_1 + n_2) \times m - 1 = n \times m - 1.\ _\square \]

Note: This problem can also be approached using Invariance principle .

A country has \(n\) cities. Any two cities are connected by a one-way road. Show that there is a route that passes through every city. If you have already read the wiki on standard induction , this problem may seem familiar. Yes, we did prove this in that article (if you haven't read that wiki, now would be a good time to do that). We'll see how stronger induction produces a shorter and cleaner solution. As we've already seen, our base case for this is true. Now we make the "strong hypothesis." We assume that our statement is true for any set of \(k\) or fewer cities. Now, for a set of \((k+1)\) cities, take out the \((k+1)^\text{th}\) city \(C_{k+1}\) and split the rest of them into two sets \(A\) and \(B\). \(A\) will contain all the cities that lead to \(C_{k+1}\) and \(B\) will contain all the cities \(C_{k+1}\) leads to. Since \(A\) has \(k\) or fewer cities in it, by the inductive hypothesis, there is a route that passes through every city in \(A\). The same argument holds for \(B\). Now start with the route that passes through every city in \(A\). Then go to \(C_{k+1}\). You can do that because all the cities in \(A\) lead to \(C_{k+1}\). After that, go to the route that passes through every city in \(B\). Again, you can do that because \(C_{k+1}\) leads to every city in \(B\). And just like that, our proof is complete! \(_\square\)

This proof is almost identical to the proof of standard induction . Can you spot the differences?

Let \(S\) be a set of positive integers with the following properties: The integer 1 belongs to the set. Whenever the integers \(1, 2, 3, \ldots, k\) are in \(S\), the next integer \(k+1\) must also be in \(S\). Then \(S\) is the set of all positive integers.
We will prove this theorem by contradiction. Let \(T\) be the set of all positive integers not in \(S\). By assumption, \(T\) is non-empty. Hence it must contain a smallest element, which we will denote by \(\alpha\). By (1), \(0 < \alpha-1 < \alpha\). Since \( \alpha\) is the smallest integer in \(T\), this implies that \( 1, 2, \ldots, \alpha - 1 \not \in T \implies 1, 2, \ldots, \alpha -1 \in S \). By (2), \(S\) must also contain \( (\alpha-1)+1=\alpha\). This contradicts the assumption that \(\alpha\subset\) \(T\). Hence set \(T\) is empty, and set \(S\) contains all positive integers. \(_\square \)

Show that every integer \(N \neq 0\) can be written in the form \(N=2^k l\), where \(k\) is a non-negative integer and \(l\) is an odd integer.

[APMO '99] Let \(\{a_i\}\) be a sequence of real numbers that satisfy \( a_{i+j} \leq a_i + a_j\) \(\forall i, j\). Prove that \[\frac{ a_1}{1} + \frac {a_2}{2} + \cdots+ \frac {a_n}{n} \geq a_n. \]

Prove that every positive integer \(n\) has a binary expression. Namely, that there exists integers \( c_i \in \{ 0, 1 \} \) such that \[ n = c_r 2^r + c_{r-1} 2^{r-1} + \cdots + c_2 2^2 + c_1 2^1 + c_0 2^ 0. \]

Consider the sequence defined as \( d_1 = 1, d_2 = 2, d_3 = 3, \) and \( d_{n+3} = d_{n+2} + d_{n+1} + d_n \) for all positive integers \(n\). Show that \( d_n < 2 ^ n \).

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Quantitative Biology > Neurons and Cognition

Title: spatial cognition: a wave hypothesis.

Abstract: Animals build Bayesian 3D models of their surroundings, to control their movements. There is strong selection pressure to make these models as precise as possible, given their sense data. A previous paper has described how a precise 3D model of space can be built by object tracking. This only works if 3D locations are stored with high spatial precision. Neural models of 3D spatial memory have large random errors; too large to support the tracking model. An alternative is described, in which neurons couple to a wave excitation in the brain, representing 3-D space. This can give high spatial precision, fast response, and other benefits. Three lines of evidence support the wave hypothesis: (1) it has better precision and speed than neural spatial memory, and is good enough to support object tracking; (2) the central body of the insect brain, whose form is highly conserved across all insect species, is well suited to hold a wave; and (3) the thalamus, whose round shape is conserved across all mammal species, is well suited to hold a wave. These lines of evidence strongly support the wave hypothesis.

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GenTranslate: Large Language Models are Generative Multilingual Speech and Machine Translators

Recent advances in large language models (LLMs) have stepped forward the development of multilingual speech and machine translation by its reduced representation errors and incorporated external knowledge. However, both translation tasks typically utilize beam search decoding and top-1 hypothesis selection for inference. These techniques struggle to fully exploit the rich information in the diverse N-best hypotheses, making them less optimal for translation tasks that require a single, high-quality output sequence. In this paper, we propose a new generative paradigm for translation tasks, namely GenTranslate, which builds upon LLMs to generate better results from the diverse translation versions in N-best list. Leveraging the rich linguistic knowledge and strong reasoning abilities of LLMs, our new paradigm can integrate the rich information in N-best candidates to generate a higher-quality translation result. Furthermore, to support LLM finetuning, we build and release a HypoTranslate dataset that contains over 592K hypotheses-translation pairs in 11 languages. Experiments on various speech and machine translation benchmarks (e.g., FLEURS, CoVoST-2, WMT) demonstrate that our GenTranslate significantly outperforms the state-of-the-art model.

Huck Yang

  • Large Language Models are Efficient Learners of Noise-Robust Speech Recognition
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COMMENTS

  1. How to Write a Strong Hypothesis

    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. If a first-year student starts attending more lectures, then their exam scores will improve.

  2. How to Write a Strong Hypothesis

    Step 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. If a first-year student starts attending more lectures, then their exam scores will improve.

  3. How to Write a Hypothesis w/ Strong Examples

    Learn how to write a strong hypothesis with our comprehensive guide. Step-by-step techniques with examples to formulate clear, testable hypotheses that lay the foundation for successful research. Ideal for students, academics, and aspiring researchers.

  4. A Strong Hypothesis

    Good Hypothesis : Poor Hypothesis: When there is less oxygen in the water, rainbow trout suffer more lice. Kristin says: "This hypothesis is good because it is testable, simple, written as a statement, and establishes the participants (trout), variables (oxygen in water, and numbers of lice), and predicts effect (as oxygen levels go down, the numbers of lice go up)."

  5. Formulating Strong Hypotheses

    There are some important things to consider when building a compelling, testable hypothesis. Clearly state the prediction you are proposing. Make sure that the hypothesis clearly defines the topic and the focus of the study. Mask wearing and its effect on virus case load. Aim to write the hypothesis as an if-then statement.

  6. How to Write a Strong Hypothesis in 6 Simple Steps

    Learning how to write a hypothesis comes down to knowledge and strategy. So where do you start? Learn how to make your hypothesis strong step-by-step here.

  7. The Craft of Writing a Strong Hypothesis

    3. Simple hypothesis. A simple hypothesis is a statement made to reflect the relation between exactly two variables. One independent and one dependent. Consider the example, "Smoking is a prominent cause of lung cancer." The dependent variable, lung cancer, is dependent on the independent variable, smoking. 4.

  8. What is and How to Write a Good Hypothesis in Research?

    An effective hypothesis in research is clearly and concisely written, and any terms or definitions clarified and defined. Specific language must also be used to avoid any generalities or assumptions. Use the following points as a checklist to evaluate the effectiveness of your research hypothesis: Predicts the relationship and outcome.

  9. Writing a Strong Hypothesis Statement

    Make sure your hypothesis clearly posits a relationship between variables. Make sure your hypothesis is testable considering your available time and resources. Before writing a thesis, it is important to create a strong hypothesis statement. This statement is a prediction of what you think will happen in your research study.

  10. How To Write An A-Grade Research Hypothesis (+ Examples ...

    Learn what exactly a research (or scientific) hypothesis is and how to write high-quality hypothesis statements for any dissertation, thesis, or research pro...

  11. What is a Research Hypothesis and How to Write a Hypothesis

    The steps to write a research hypothesis are: 1. Stating the problem: Ensure that the hypothesis defines the research problem. 2. Writing a hypothesis as an 'if-then' statement: Include the action and the expected outcome of your study by following a 'if-then' structure. 3.

  12. How to write a strong hypothesis

    They must be mentioned in the hypothesis. Use simple, clear language. Avoid overly technical jargon and descriptive terminology. The hypothesis should not be confusing and must state what your research will investigate and how. Focus your hypothesis only on what should be tested. Don't forget to mention the specific variables used.

  13. How to Write a Research Hypothesis: Good & Bad Examples

    Writing a Hypothesis Step 2: Formulate a strong hypothesis by answering your own question. Again, you don't want to make things up, take unicorns into account, or repeat/ignore what has already been done. Looking at the dog-vs-cat papers your literature search returned, you see that most studies are based on self-report questionnaires on ...

  14. Research Hypothesis In Psychology: Types, & Examples

    A strong hypothesis is concise (typically one to two sentences long), and formulated using clear and straightforward language, ensuring it's easily understood and testable. Examples. Consider a hypothesis many teachers might subscribe to: students work better on Monday morning than on Friday afternoon (IV=Day, DV= Standard of work).

  15. How To Write a Strong Research Hypothesis

    Here are some tips for crafting a well-crafted hypothesis: Consider the goal of your research: Think about what it is that you want to learn or determine from your experiment and make sure that your hypothesis reflects this goal. Create an educated guess as to why something is happening: Your hypothesis should explain why something is occurring ...

  16. Hypothesis: Definition, Examples, and Types

    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 ...

  17. 6 Steps to Formulate a STRONG Hypothesis

    Start typing, then use the up and down arrows to select an option from the list.?

  18. What Are the Elements of a Good Hypothesis?

    A hypothesis is an educated guess or prediction of what will happen. In science, a hypothesis proposes a relationship between factors called variables. A good hypothesis relates an independent variable and a dependent variable. The effect on the dependent variable depends on or is determined by what happens when you change the independent variable.

  19. Hypothesis Examples

    A hypothesis proposes a relationship between the independent and dependent variable. A hypothesis is a prediction of the outcome of a test. It forms the basis for designing an experiment in the scientific method.A good hypothesis is testable, meaning it makes a prediction you can check with observation or experimentation.

  20. What Makes a Strong Hypothesis for Scientific Research?

    A strong hypothesis is concise, clear, and defines an expected relationship between the dependent and independent variables. This relationship should be stated as explicitly as possible and must be testable. Having a concise hypothesis makes it obvious to the reader when you transition from background information to a research question without ...

  21. What Are Examples of a Hypothesis?

    Here are examples of a scientific hypothesis. Although you could state a scientific hypothesis in various ways, most hypotheses are either "If, then" statements or forms of the null hypothesis. The null hypothesis is sometimes called the "no difference" hypothesis. The null hypothesis is good for experimentation because it's simple to disprove.

  22. Crafting a Hypothesis: Key Steps for Researchers

    Crafting a hypothesis is a critical step in the research process, serving as a foundation for your inquiry. It's not merely an educated guess but a testable prediction that guides your research ...

  23. Strong Induction

    The reason why this is called "strong induction" is that we use more statements in the inductive hypothesis. Let's write what we've learned till now a bit more formally. Proof by strong induction. Step 1. Demonstrate the base case: This is where you verify that \(P(k_0)\) is true. In most cases, \(k_0=1.\) Step 2. Prove the inductive step:

  24. [2405.10112] Spatial Cognition: a Wave Hypothesis

    Animals build Bayesian 3D models of their surroundings, to control their movements. There is strong selection pressure to make these models as precise as possible, given their sense data. A previous paper has described how a precise 3D model of space can be built by object tracking. This only works if 3D locations are stored with high spatial precision. Neural models of 3D spatial memory have ...

  25. GenTranslate: Large Language Models are Generative Multilingual Speech

    Recent advances in large language models (LLMs) have stepped forward the development of multilingual speech and machine translation by its reduced representation errors and incorporated external knowledge. However, both translation tasks typically utilize beam search decoding and top-1 hypothesis selection for inference. These techniques struggle to fully exploit the rich information in the ...