statement hypothesis conclusion

Statistics Made Easy

How to Write Hypothesis Test Conclusions (With Examples)

A hypothesis test is used to test whether or not some hypothesis about a population parameter is true.

To perform a hypothesis test in the real world, researchers obtain a random sample from the population and perform a hypothesis test on the sample data, using a null and alternative hypothesis:

Null Hypothesis (H 0 ): The sample data occurs purely from chance.
Alternative Hypothesis (H A ): The sample data is influenced by some non-random cause.

If the p-value of the hypothesis test is less than some significance level (e.g. α = .05), then we reject the null hypothesis .

Otherwise, if the p-value is not less than some significance level then we fail to reject the null hypothesis .

When writing the conclusion of a hypothesis test, we typically include:

Whether we reject or fail to reject the null hypothesis.
The significance level.
A short explanation in the context of the hypothesis test.

For example, we would write:

We reject the null hypothesis at the 5% significance level. There is sufficient evidence to support the claim that…

Or, we would write:

We fail to reject the null hypothesis at the 5% significance level. There is not sufficient evidence to support the claim that…

The following examples show how to write a hypothesis test conclusion in both scenarios.

Example 1: Reject the Null Hypothesis Conclusion

Suppose a biologist believes that a certain fertilizer will cause plants to grow more during a one-month period than they normally do, which is currently 20 inches. To test this, she applies the fertilizer to each of the plants in her laboratory for one month.

She then performs a hypothesis test at a 5% significance level using the following hypotheses:

H 0 : μ = 20 inches (the fertilizer will have no effect on the mean plant growth)
H A : μ > 20 inches (the fertilizer will cause mean plant growth to increase)

Suppose the p-value of the test turns out to be 0.002.

Here is how she would report the results of the hypothesis test:

We reject the null hypothesis at the 5% significance level. There is sufficient evidence to support the claim that this particular fertilizer causes plants to grow more during a one-month period than they normally do.

Example 2: Fail to Reject the Null Hypothesis Conclusion

Suppose the manager of a manufacturing plant wants to test whether or not some new method changes the number of defective widgets produced per month, which is currently 250. To test this, he measures the mean number of defective widgets produced before and after using the new method for one month.

He performs a hypothesis test at a 10% significance level using the following hypotheses:

H 0 : μ after = μ before (the mean number of defective widgets is the same before and after using the new method)
H A : μ after ≠ μ before (the mean number of defective widgets produced is different before and after using the new method)

Suppose the p-value of the test turns out to be 0.27.

Here is how he would report the results of the hypothesis test:

We fail to reject the null hypothesis at the 10% significance level. There is not sufficient evidence to support the claim that the new method leads to a change in the number of defective widgets produced per month.

Additional Resources

The following tutorials provide additional information about hypothesis testing:

Introduction to Hypothesis Testing 4 Examples of Hypothesis Testing in Real Life How to Write a Null Hypothesis

Published by Zach

How to State the Conclusion about a Hypothesis Test

After you have completed the statistical analysis and decided to reject or fail to reject the Null hypothesis, you need to state your conclusion about the claim. To get the correct wording, you need to recall which hypothesis was the claim.

If the claim was the null, then your conclusion is about whether there was sufficient evidence to reject the claim. Remember, we can never prove the null to be true, but failing to reject it is the next best thing. So, it is not correct to say, “Accept the Null.”

If the claim is the alternative hypothesis, your conclusion can be whether there was sufficient evidence to support (prove) the alternative is true.

Use the following table to help you make a good conclusion.

The best way to state the conclusion is to include the significance level of the test and a bit about the claim itself.

For example, if the claim was the alternative that the mean score on a test was greater than 85, and your decision was to Reject then Null , then you could conclude: “ At the 5% significance level, there is sufficient evidence to support the claim that the mean score on the test was greater than 85. ”

The reason you should include the significance level is that the decision, and thus the conclusion, could be different if the significance level was not 5%.

If you are curious why we say “Fail to Reject the Null” instead of “Accept the Null,” this short video might be of interest: Here

2 Responses

It is concluded that the null hypothesis Ho is not rejected proportion p is greater than 0.5, at the 0.05 significance

People living in rural Idaho community live longer than 77 years

The Craft of Writing a Strong Hypothesis

Writing a hypothesis is one of the essential elements of a scientific research paper. It needs to be to the point, clearly communicating what your research is trying to accomplish. A blurry, drawn-out, or complexly-structured hypothesis can confuse your readers. Or worse, the editor and peer reviewers.

A captivating hypothesis is not too intricate. This blog will take you through the process so that, by the end of it, you have a better idea of how to convey your research paper's intent in just one sentence.

What is a Hypothesis?

The first step in your scientific endeavor, a hypothesis, is a strong, concise statement that forms the basis of your research. It is not the same as a thesis statement , which is a brief summary of your research paper .

The sole purpose of a hypothesis is to predict your paper's findings, data, and conclusion. It comes from a place of curiosity and intuition . When you write a hypothesis, you're essentially making an educated guess based on scientific prejudices and evidence, which is further proven or disproven through the scientific method.

The reason for undertaking research is to observe a specific phenomenon. A hypothesis, therefore, lays out what the said phenomenon is. And it does so through two variables, an independent and dependent variable.

The independent variable is the cause behind the observation, while the dependent variable is the effect of the cause. A good example of this is “mixing red and blue forms purple.” In this hypothesis, mixing red and blue is the independent variable as you're combining the two colors at your own will. The formation of purple is the dependent variable as, in this case, it is conditional to the independent variable.

Different Types of Hypotheses‌

Types of hypotheses

Some would stand by the notion that there are only two types of hypotheses: a Null hypothesis and an Alternative hypothesis. While that may have some truth to it, it would be better to fully distinguish the most common forms as these terms come up so often, which might leave you out of context.

Apart from Null and Alternative, there are Complex, Simple, Directional, Non-Directional, Statistical, and Associative and casual hypotheses. They don't necessarily have to be exclusive, as one hypothesis can tick many boxes, but knowing the distinctions between them will make it easier for you to construct your own.

1. Null hypothesis

A null hypothesis proposes no relationship between two variables. Denoted by H 0 , it is a negative statement like “Attending physiotherapy sessions does not affect athletes' on-field performance.” Here, the author claims physiotherapy sessions have no effect on on-field performances. Even if there is, it's only a coincidence.

2. Alternative hypothesis

Considered to be the opposite of a null hypothesis, an alternative hypothesis is donated as H1 or Ha. It explicitly states that the dependent variable affects the independent variable. A good alternative hypothesis example is “Attending physiotherapy sessions improves athletes' on-field performance.” or “Water evaporates at 100 °C. ” The alternative hypothesis further branches into directional and non-directional.

Directional hypothesis: A hypothesis that states the result would be either positive or negative is called directional hypothesis. It accompanies H1 with either the ‘<' or ‘>' sign.
Non-directional hypothesis: A non-directional hypothesis only claims an effect on the dependent variable. It does not clarify whether the result would be positive or negative. The sign for a non-directional hypothesis is ‘≠.'

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. Complex hypothesis

In contrast to a simple hypothesis, a complex hypothesis implies the relationship between multiple independent and dependent variables. For instance, “Individuals who eat more fruits tend to have higher immunity, lesser cholesterol, and high metabolism.” The independent variable is eating more fruits, while the dependent variables are higher immunity, lesser cholesterol, and high metabolism.

5. Associative and casual hypothesis

Associative and casual hypotheses don't exhibit how many variables there will be. They define the relationship between the variables. In an associative hypothesis, changing any one variable, dependent or independent, affects others. In a casual hypothesis, the independent variable directly affects the dependent.

6. Empirical hypothesis

Also referred to as the working hypothesis, an empirical hypothesis claims a theory's validation via experiments and observation. This way, the statement appears justifiable and different from a wild guess.

Say, the hypothesis is “Women who take iron tablets face a lesser risk of anemia than those who take vitamin B12.” This is an example of an empirical hypothesis where the researcher the statement after assessing a group of women who take iron tablets and charting the findings.

7. Statistical hypothesis

The point of a statistical hypothesis is to test an already existing hypothesis by studying a population sample. Hypothesis like “44% of the Indian population belong in the age group of 22-27.” leverage evidence to prove or disprove a particular statement.

Characteristics of a Good Hypothesis

Writing a hypothesis is essential as it can make or break your research for you. That includes your chances of getting published in a journal. So when you're designing one, keep an eye out for these pointers:

A research hypothesis has to be simple yet clear to look justifiable enough.
It has to be testable — your research would be rendered pointless if too far-fetched into reality or limited by technology.
It has to be precise about the results —what you are trying to do and achieve through it should come out in your hypothesis.
A research hypothesis should be self-explanatory, leaving no doubt in the reader's mind.
If you are developing a relational hypothesis, you need to include the variables and establish an appropriate relationship among them.
A hypothesis must keep and reflect the scope for further investigations and experiments.

Separating a Hypothesis from a Prediction

Outside of academia, hypothesis and prediction are often used interchangeably. In research writing, this is not only confusing but also incorrect. And although a hypothesis and prediction are guesses at their core, there are many differences between them.

A hypothesis is an educated guess or even a testable prediction validated through research. It aims to analyze the gathered evidence and facts to define a relationship between variables and put forth a logical explanation behind the nature of events.

Predictions are assumptions or expected outcomes made without any backing evidence. They are more fictionally inclined regardless of where they originate from.

For this reason, a hypothesis holds much more weight than a prediction. It sticks to the scientific method rather than pure guesswork. "Planets revolve around the Sun." is an example of a hypothesis as it is previous knowledge and observed trends. Additionally, we can test it through the scientific method.

Whereas "COVID-19 will be eradicated by 2030." is a prediction. Even though it results from past trends, we can't prove or disprove it. So, the only way this gets validated is to wait and watch if COVID-19 cases end by 2030.

Finally, How to Write a Hypothesis

Quick tips on writing a hypothesis

1. Be clear about your research question

A hypothesis should instantly address the research question or the problem statement. To do so, you need to ask a question. Understand the constraints of your undertaken research topic and then formulate a simple and topic-centric problem. Only after that can you develop a hypothesis and further test for evidence.

2. Carry out a recce

Once you have your research's foundation laid out, it would be best to conduct preliminary research. Go through previous theories, academic papers, data, and experiments before you start curating your research hypothesis. It will give you an idea of your hypothesis's viability or originality.

Making use of references from relevant research papers helps draft a good research hypothesis. SciSpace Discover offers a repository of over 270 million research papers to browse through and gain a deeper understanding of related studies on a particular topic. Additionally, you can use SciSpace Copilot , your AI research assistant, for reading any lengthy research paper and getting a more summarized context of it. A hypothesis can be formed after evaluating many such summarized research papers. Copilot also offers explanations for theories and equations, explains paper in simplified version, allows you to highlight any text in the paper or clip math equations and tables and provides a deeper, clear understanding of what is being said. This can improve the hypothesis by helping you identify potential research gaps.

3. Create a 3-dimensional hypothesis

Variables are an essential part of any reasonable hypothesis. So, identify your independent and dependent variable(s) and form a correlation between them. The ideal way to do this is to write the hypothetical assumption in the ‘if-then' form. If you use this form, make sure that you state the predefined relationship between the variables.

In another way, you can choose to present your hypothesis as a comparison between two variables. Here, you must specify the difference you expect to observe in the results.

4. Write the first draft

Now that everything is in place, it's time to write your hypothesis. For starters, create the first draft. In this version, write what you expect to find from your research.

Clearly separate your independent and dependent variables and the link between them. Don't fixate on syntax at this stage. The goal is to ensure your hypothesis addresses the issue.

5. Proof your hypothesis

After preparing the first draft of your hypothesis, you need to inspect it thoroughly. It should tick all the boxes, like being concise, straightforward, relevant, and accurate. Your final hypothesis has to be well-structured as well.

Research projects are an exciting and crucial part of being a scholar. And once you have your research question, you need a great hypothesis to begin conducting research. Thus, knowing how to write a hypothesis is very important.

Now that you have a firmer grasp on what a good hypothesis constitutes, the different kinds there are, and what process to follow, you will find it much easier to write your hypothesis, which ultimately helps your research.

Now it's easier than ever to streamline your research workflow with SciSpace Discover . Its integrated, comprehensive end-to-end platform for research allows scholars to easily discover, write and publish their research and fosters collaboration.

It includes everything you need, including a repository of over 270 million research papers across disciplines, SEO-optimized summaries and public profiles to show your expertise and experience.

If you found these tips on writing a research hypothesis useful, head over to our blog on Statistical Hypothesis Testing to learn about the top researchers, papers, and institutions in this domain.

Frequently Asked Questions (FAQs)

1. what is the definition of hypothesis.

According to the Oxford dictionary, a hypothesis is defined as “An idea or explanation of something that is based on a few known facts, but that has not yet been proved to be true or correct”.

2. What is an example of hypothesis?

The hypothesis is a statement that proposes a relationship between two or more variables. An example: "If we increase the number of new users who join our platform by 25%, then we will see an increase in revenue."

3. What is an example of null hypothesis?

A null hypothesis is a statement that there is no relationship between two variables. The null hypothesis is written as H0. The null hypothesis states that there is no effect. For example, if you're studying whether or not a particular type of exercise increases strength, your null hypothesis will be "there is no difference in strength between people who exercise and people who don't."

4. What are the types of research?

• Fundamental research

• Applied research

• Qualitative research

• Quantitative research

• Mixed research

• Exploratory research

• Longitudinal research

• Cross-sectional research

• Field research

• Laboratory research

• Fixed research

• Flexible research

• Action research

• Policy research

• Classification research

• Comparative research

• Causal research

• Inductive research

• Deductive research

5. How to write a hypothesis?

• Your hypothesis should be able to predict the relationship and outcome.

• Avoid wordiness by keeping it simple and brief.

• Your hypothesis should contain observable and testable outcomes.

• Your hypothesis should be relevant to the research question.

6. What are the 2 types of hypothesis?

• Null hypotheses are used to test the claim that "there is no difference between two groups of data".

• Alternative hypotheses test the claim that "there is a difference between two data groups".

7. Difference between research question and research hypothesis?

A research question is a broad, open-ended question you will try to answer through your research. A hypothesis is a statement based on prior research or theory that you expect to be true due to your study. Example - Research question: What are the factors that influence the adoption of the new technology? Research hypothesis: There is a positive relationship between age, education and income level with the adoption of the new technology.

8. What is plural for hypothesis?

The plural of hypothesis is hypotheses. Here's an example of how it would be used in a statement, "Numerous well-considered hypotheses are presented in this part, and they are supported by tables and figures that are well-illustrated."

9. What is the red queen hypothesis?

The red queen hypothesis in evolutionary biology states that species must constantly evolve to avoid extinction because if they don't, they will be outcompeted by other species that are evolving. Leigh Van Valen first proposed it in 1973; since then, it has been tested and substantiated many times.

10. Who is known as the father of null hypothesis?

The father of the null hypothesis is Sir Ronald Fisher. He published a paper in 1925 that introduced the concept of null hypothesis testing, and he was also the first to use the term itself.

11. When to reject null hypothesis?

You need to find a significant difference between your two populations to reject the null hypothesis. You can determine that by running statistical tests such as an independent sample t-test or a dependent sample t-test. You should reject the null hypothesis if the p-value is less than 0.05.

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Biology library

Course: biology library > unit 1, the scientific method.

Controlled experiments
The scientific method and experimental design

Introduction

Make an observation.
Ask a question.
Form a hypothesis , or testable explanation.
Make a prediction based on the hypothesis.
Test the prediction.
Iterate: use the results to make new hypotheses or predictions.

Scientific method example: Failure to toast

1. make an observation..

Observation: the toaster won't toast.

2. Ask a question.

Question: Why won't my toaster toast?

3. Propose a hypothesis.

Hypothesis: Maybe the outlet is broken.

4. Make predictions.

Prediction: If I plug the toaster into a different outlet, then it will toast the bread.

5. Test the predictions.

Test of prediction: Plug the toaster into a different outlet and try again.
If the toaster does toast, then the hypothesis is supported—likely correct.
If the toaster doesn't toast, then the hypothesis is not supported—likely wrong.

Logical possibility

Practical possibility, building a body of evidence, 6. iterate..

Iteration time!
If the hypothesis was supported, we might do additional tests to confirm it, or revise it to be more specific. For instance, we might investigate why the outlet is broken.
If the hypothesis was not supported, we would come up with a new hypothesis. For instance, the next hypothesis might be that there's a broken wire in the toaster.

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5.2 - writing hypotheses.

The first step in conducting a hypothesis test is to write the hypothesis statements that are going to be tested. For each test you will have a null hypothesis ($H_0$) and an alternative hypothesis ($H_a$).

When writing hypotheses there are three things that we need to know: (1) the parameter that we are testing (2) the direction of the test (non-directional, right-tailed or left-tailed), and (3) the value of the hypothesized parameter.

At this point we can write hypotheses for a single mean ($\mu$), paired means($\mu_d$), a single proportion ($p$), the difference between two independent means ($\mu_1-\mu_2$), the difference between two proportions ($p_1-p_2$), a simple linear regression slope ($\beta$), and a correlation ($\rho$).
The research question will give us the information necessary to determine if the test is two-tailed (e.g., "different from," "not equal to"), right-tailed (e.g., "greater than," "more than"), or left-tailed (e.g., "less than," "fewer than").
The research question will also give us the hypothesized parameter value. This is the number that goes in the hypothesis statements (i.e., $\mu_0$ and $p_0$). For the difference between two groups, regression, and correlation, this value is typically 0.

Hypotheses are always written in terms of population parameters (e.g., $p$ and $\mu$). The tables below display all of the possible hypotheses for the parameters that we have learned thus far. Note that the null hypothesis always includes the equality (i.e., =).

The Scientific Method by Science Made Simple

Understanding and using the scientific method.

The Scientific Method is a process used to design and perform experiments. It's important to minimize experimental errors and bias, and increase confidence in the accuracy of your results.

In the previous sections, we talked about how to pick a good topic and specific question to investigate. Now we will discuss how to carry out your investigation.

Steps of the Scientific Method

Observation/Research
Experimentation

Now that you have settled on the question you want to ask, it's time to use the Scientific Method to design an experiment to answer that question.

If your experiment isn't designed well, you may not get the correct answer. You may not even get any definitive answer at all!

The Scientific Method is a logical and rational order of steps by which scientists come to conclusions about the world around them. The Scientific Method helps to organize thoughts and procedures so that scientists can be confident in the answers they find.

OBSERVATION is first step, so that you know how you want to go about your research.

HYPOTHESIS is the answer you think you'll find.

PREDICTION is your specific belief about the scientific idea: If my hypothesis is true, then I predict we will discover this.

EXPERIMENT is the tool that you invent to answer the question, and

CONCLUSION is the answer that the experiment gives.

Don't worry, it isn't that complicated. Let's take a closer look at each one of these steps. Then you can understand the tools scientists use for their science experiments, and use them for your own.

OBSERVATION

This step could also be called "research." It is the first stage in understanding the problem.

After you decide on topic, and narrow it down to a specific question, you will need to research everything that you can find about it. You can collect information from your own experiences, books, the internet, or even smaller "unofficial" experiments.

Let's continue the example of a science fair idea about tomatoes in the garden. You like to garden, and notice that some tomatoes are bigger than others and wonder why.

Because of this personal experience and an interest in the problem, you decide to learn more about what makes plants grow.

For this stage of the Scientific Method, it's important to use as many sources as you can find. The more information you have on your science fair topic, the better the design of your experiment is going to be, and the better your science fair project is going to be overall.

Also try to get information from your teachers or librarians, or professionals who know something about your science fair project. They can help to guide you to a solid experimental setup.

The next stage of the Scientific Method is known as the "hypothesis." This word basically means "a possible solution to a problem, based on knowledge and research."

The hypothesis is a simple statement that defines what you think the outcome of your experiment will be.

All of the first stage of the Scientific Method -- the observation, or research stage -- is designed to help you express a problem in a single question ("Does the amount of sunlight in a garden affect tomato size?") and propose an answer to the question based on what you know. The experiment that you will design is done to test the hypothesis.

Using the example of the tomato experiment, here is an example of a hypothesis:

TOPIC: "Does the amount of sunlight a tomato plant receives affect the size of the tomatoes?"

HYPOTHESIS: "I believe that the more sunlight a tomato plant receives, the larger the tomatoes will grow.

This hypothesis is based on:

(1) Tomato plants need sunshine to make food through photosynthesis, and logically, more sun means more food, and;

(2) Through informal, exploratory observations of plants in a garden, those with more sunlight appear to grow bigger.

The hypothesis is your general statement of how you think the scientific phenomenon in question works.

Your prediction lets you get specific -- how will you demonstrate that your hypothesis is true? The experiment that you will design is done to test the prediction.

An important thing to remember during this stage of the scientific method is that once you develop a hypothesis and a prediction, you shouldn't change it, even if the results of your experiment show that you were wrong.

An incorrect prediction does NOT mean that you "failed." It just means that the experiment brought some new facts to light that maybe you hadn't thought about before.

Continuing our tomato plant example, a good prediction would be: Increasing the amount of sunlight tomato plants in my experiment receive will cause an increase in their size compared to identical plants that received the same care but less light.

This is the part of the scientific method that tests your hypothesis. An experiment is a tool that you design to find out if your ideas about your topic are right or wrong.

It is absolutely necessary to design a science fair experiment that will accurately test your hypothesis. The experiment is the most important part of the scientific method. It's the logical process that lets scientists learn about the world.

On the next page, we'll discuss the ways that you can go about designing a science fair experiment idea.

The final step in the scientific method is the conclusion. This is a summary of the experiment's results, and how those results match up to your hypothesis.

You have two options for your conclusions: based on your results, either:

(1) YOU CAN REJECT the hypothesis, or

(2) YOU CAN NOT REJECT the hypothesis.

This is an important point!

You can not PROVE the hypothesis with a single experiment, because there is a chance that you made an error somewhere along the way.

What you can say is that your results SUPPORT the original hypothesis.

If your original hypothesis didn't match up with the final results of your experiment, don't change the hypothesis.

Instead, try to explain what might have been wrong with your original hypothesis. What information were you missing when you made your prediction? What are the possible reasons the hypothesis and experimental results didn't match up?

Remember, a science fair experiment isn't a failure simply because does not agree with your hypothesis. No one will take points off if your prediction wasn't accurate. Many important scientific discoveries were made as a result of experiments gone wrong!

A science fair experiment is only a failure if its design is flawed. A flawed experiment is one that (1) doesn't keep its variables under control, and (2) doesn't sufficiently answer the question that you asked of it.

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In a short paper—even a research paper—you don’t need to provide an exhaustive summary as part of your conclusion. But you do need to make some kind of transition between your final body paragraph and your concluding paragraph. This may come in the form of a few sentences of summary. Or it may come in the form of a sentence that brings your readers back to your thesis or main idea and reminds your readers where you began and how far you have traveled.

So, for example, in a paper about the relationship between ADHD and rejection sensitivity, Vanessa Roser begins by introducing readers to the fact that researchers have studied the relationship between the two conditions and then provides her explanation of that relationship. Here’s her thesis: “While socialization may indeed be an important factor in RS, I argue that individuals with ADHD may also possess a neurological predisposition to RS that is exacerbated by the differing executive and emotional regulation characteristic of ADHD.”

In her final paragraph, Roser reminds us of where she started by echoing her thesis: “This literature demonstrates that, as with many other conditions, ADHD and RS share a delicately intertwined pattern of neurological similarities that is rooted in the innate biology of an individual’s mind, a connection that cannot be explained in full by the behavioral mediation hypothesis.”

Highlight the “so what”

At the beginning of your paper, you explain to your readers what’s at stake—why they should care about the argument you’re making. In your conclusion, you can bring readers back to those stakes by reminding them why your argument is important in the first place. You can also draft a few sentences that put those stakes into a new or broader context.

In the conclusion to her paper about ADHD and RS, Roser echoes the stakes she established in her introduction—that research into connections between ADHD and RS has led to contradictory results, raising questions about the “behavioral mediation hypothesis.”

She writes, “as with many other conditions, ADHD and RS share a delicately intertwined pattern of neurological similarities that is rooted in the innate biology of an individual’s mind, a connection that cannot be explained in full by the behavioral mediation hypothesis.”

Leave your readers with the “now what”

After the “what” and the “so what,” you should leave your reader with some final thoughts. If you have written a strong introduction, your readers will know why you have been arguing what you have been arguing—and why they should care. And if you’ve made a good case for your thesis, then your readers should be in a position to see things in a new way, understand new questions, or be ready for something that they weren’t ready for before they read your paper.

In her conclusion, Roser offers two “now what” statements. First, she explains that it is important to recognize that the flawed behavioral mediation hypothesis “seems to place a degree of fault on the individual. It implies that individuals with ADHD must have elicited such frequent or intense rejection by virtue of their inadequate social skills, erasing the possibility that they may simply possess a natural sensitivity to emotion.” She then highlights the broader implications for treatment of people with ADHD, noting that recognizing the actual connection between rejection sensitivity and ADHD “has profound implications for understanding how individuals with ADHD might best be treated in educational settings, by counselors, family, peers, or even society as a whole.”

To find your own “now what” for your essay’s conclusion, try asking yourself these questions:

What can my readers now understand, see in a new light, or grapple with that they would not have understood in the same way before reading my paper? Are we a step closer to understanding a larger phenomenon or to understanding why what was at stake is so important?
What questions can I now raise that would not have made sense at the beginning of my paper? Questions for further research? Other ways that this topic could be approached?
Are there other applications for my research? Could my questions be asked about different data in a different context? Could I use my methods to answer a different question?
What action should be taken in light of this argument? What action do I predict will be taken or could lead to a solution?
What larger context might my argument be a part of?

What to avoid in your conclusion

a complete restatement of all that you have said in your paper.
a substantial counterargument that you do not have space to refute; you should introduce counterarguments before your conclusion.
an apology for what you have not said. If you need to explain the scope of your paper, you should do this sooner—but don’t apologize for what you have not discussed in your paper.
fake transitions like “in conclusion” that are followed by sentences that aren’t actually conclusions. (“In conclusion, I have now demonstrated that my thesis is correct.”)
picture_as_pdf Conclusions

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Statement of the Conclusion

When writing your results, you’re going to write the decision regarding the null, but you also want to state the results in layman’s terms. Tie the statistical results back to the original claim and interpret what those statistics mean, without all the quantitative jargon.

1) Claim : Females run faster than males. Results of the test : t o > t c Decision : Reject Null Hypothesis. Conclusion : There is sufficient evidence to suggest that females run faster than males.

2) Claim : There is a difference in the highest level of education obtained based on socioeconomic status. Results of the test : p > α Decision : Fail to Reject Null Hypothesis. Conclusion : There is not enough evidence to suggest that highest level of education differs based on socioeconomic status.

3) Claim : The number of calories consumed and the number of hours spent exercising each week are significant predictors of weight. Results of the test : p < α Decision : Reject Null Hypothesis. Conclusion : The results of the hypothesis test suggest that a person’s weight can be predicted given caloric intake and the number of hours spent exercising each week.

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Last Updated: Apr 19, 2024 3:09 PM
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How to Write Hypothesis Test Conclusions (With Examples)

A hypothesis test is used to test whether or not some hypothesis about a population parameter is true.

To perform a hypothesis test in the real world, researchers obtain a random sample from the population and perform a hypothesis test on the sample data, using a null and alternative hypothesis:

Null Hypothesis (H 0 ): The sample data occurs purely from chance.
Alternative Hypothesis (H A ): The sample data is influenced by some non-random cause.

If the p-value of the hypothesis test is less than some significance level (e.g. α = .05), then we reject the null hypothesis .

Otherwise, if the p-value is not less than some significance level then we fail to reject the null hypothesis .

When writing the conclusion of a hypothesis test, we typically include:

Whether we reject or fail to reject the null hypothesis.
The significance level.
A short explanation in the context of the hypothesis test.

For example, we would write:

We reject the null hypothesis at the 5% significance level. There is sufficient evidence to support the claim that…

Or, we would write:

We fail to reject the null hypothesis at the 5% significance level. There is not sufficient evidence to support the claim that…

The following examples show how to write a hypothesis test conclusion in both scenarios.

Example 1: Reject the Null Hypothesis Conclusion

She then performs a hypothesis test at a 5% significance level using the following hypotheses:

H 0 : μ = 20 inches (the fertilizer will have no effect on the mean plant growth)
H A : μ > 20 inches (the fertilizer will cause mean plant growth to increase)

Suppose the p-value of the test turns out to be 0.002.

Here is how she would report the results of the hypothesis test:

We reject the null hypothesis at the 5% significance level. There is sufficient evidence to support the claim that this particular fertilizer causes plants to grow more during a one-month period than they normally do.

Example 2: Fail to Reject the Null Hypothesis Conclusion

He performs a hypothesis test at a 10% significance level using the following hypotheses:

H 0 : μ after = μ before (the mean number of defective widgets is the same before and after using the new method)
H A : μ after ≠ μ before (the mean number of defective widgets produced is different before and after using the new method)

Suppose the p-value of the test turns out to be 0.27.

Here is how he would report the results of the hypothesis test:

We fail to reject the null hypothesis at the 10% significance level. There is not sufficient evidence to support the claim that the new method leads to a change in the number of defective widgets produced per month.

Additional Resources

The following tutorials provide additional information about hypothesis testing:

Introduction to Hypothesis Testing 4 Examples of Hypothesis Testing in Real Life How to Write a Null Hypothesis

10 Examples of Using Probability in Real Life

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Education and Communications
Science Writing

How to Write a Good Lab Conclusion in Science

Last Updated: March 21, 2024 Fact Checked

This article was co-authored by Bess Ruff, MA . Bess Ruff is a Geography PhD student at Florida State University. She received her MA in Environmental Science and Management from the University of California, Santa Barbara in 2016. She has conducted survey work for marine spatial planning projects in the Caribbean and provided research support as a graduate fellow for the Sustainable Fisheries Group. There are 11 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 1,760,296 times.

A lab report describes an entire experiment from start to finish, outlining the procedures, reporting results, and analyzing data. The report is used to demonstrate what has been learned, and it will provide a way for other people to see your process for the experiment and understand how you arrived at your conclusions. The conclusion is an integral part of the report; this is the section that reiterates the experiment’s main findings and gives the reader an overview of the lab trial. Writing a solid conclusion to your lab report will demonstrate that you’ve effectively learned the objectives of your assignment.

Outlining Your Conclusion

Restate : Restate the lab experiment by describing the assignment.
Explain : Explain the purpose of the lab experiment. What were you trying to figure out or discover? Talk briefly about the procedure you followed to complete the lab.
Results : Explain your results. Confirm whether or not your hypothesis was supported by the results.
Uncertainties : Account for uncertainties and errors. Explain, for example, if there were other circumstances beyond your control that might have impacted the experiment’s results.
New : Discuss new questions or discoveries that emerged from the experiment.

Your assignment may also have specific questions that need to be answered. Make sure you answer these fully and coherently in your conclusion.

Discussing the Experiment and Hypothesis

Step 1 Introduce the experiment in your conclusion.

If you tried the experiment more than once, describe the reasons for doing so. Discuss changes that you made in your procedures.
Brainstorm ways to explain your results in more depth. Go back through your lab notes, paying particular attention to the results you observed. [5] X Trustworthy Source University of North Carolina Writing Center UNC's on-campus and online instructional service that provides assistance to students, faculty, and others during the writing process Go to source

Step 3 Describe what you discovered briefly.

Start this section with wording such as, “The results showed that…”
You don’t need to give the raw data here. Just summarize the main points, calculate averages, or give a range of data to give an overall picture to the reader.
Make sure to explain whether or not any statistical analyses were significant, and to what degree, such as 1%, 5%, or 10%.

Step 4 Comment on whether or not your hypothesis is supported.

Use simple language such as, “The results supported the hypothesis,” or “The results did not support the hypothesis.”

Step 5 Link your results to your hypothesis.

Demonstrating What You Have Learned

Step 1 Describe what you learned in the lab.

If it’s not clear in your conclusion what you learned from the lab, start off by writing, “In this lab, I learned…” This will give the reader a heads up that you will be describing exactly what you learned.
Add details about what you learned and how you learned it. Adding dimension to your learning outcomes will convince your reader that you did, in fact, learn from the lab. Give specifics about how you learned that molecules will act in a particular environment, for example.
Describe how what you learned in the lab could be applied to a future experiment.

Step 2 Answer specific questions given in the assignment.

On a new line, write the question in italics. On the next line, write the answer to the question in regular text.

Step 3 Explain whether you achieved the experiment’s objectives.

If your experiment did not achieve the objectives, explain or speculate why not.

Wrapping Up Your Conclusion

Step 1 Describe possible errors that may have occurred.

If your experiment raised questions that your collected data can’t answer, discuss this here.

Describe what is new or innovative about your research.
This can often set you apart from your classmates, many of whom will just write up the barest of discussion and conclusion.

Finalizing Your Lab Report

Community Q&A

If you include figures or tables in your conclusion, be sure to include a brief caption or label so that the reader knows what the figures refer to. Also, discuss the figures briefly in the text of your report. Thanks Helpful 0 Not Helpful 0
Once again, avoid using personal pronouns (I, myself, we, our group) in a lab report. The first-person point-of-view is often seen as subjective, whereas science is based on objectivity. Thanks Helpful 0 Not Helpful 0
Ensure the language used is straightforward with specific details. Try not to drift off topic. Thanks Helpful 0 Not Helpful 0

Take care with writing your lab report when working in a team setting. While the lab experiment may be a collaborative effort, your lab report is your own work. If you copy sections from someone else’s report, this will be considered plagiarism. Thanks Helpful 3 Not Helpful 0

↑ https://phoenixcollege.libguides.com/LabReportWriting/introduction
↑ https://www.hcs-k12.org/userfiles/354/Classes/18203/conclusionwriting.pdf
↑ https://www.education.vic.gov.au/school/teachers/teachingresources/discipline/english/literacy/Pages/puttingittogether.aspx
↑ https://writingcenter.unc.edu/tips-and-tools/brainstorming/
↑ https://advice.writing.utoronto.ca/types-of-writing/lab-report/
↑ http://www.socialresearchmethods.net/kb/hypothes.php
↑ https://libguides.usc.edu/writingguide/conclusion
↑ https://libguides.usc.edu/writingguide/introduction/researchproblem
↑ http://writingcenter.unc.edu/handouts/scientific-reports/
↑ https://phoenixcollege.libguides.com/LabReportWriting/labreportstyle
↑ https://writingcenter.unc.edu/tips-and-tools/editing-and-proofreading/

About This Article

To write a good lab conclusion in science, start with restating the lab experiment by describing the assignment. Next, explain what you were trying to discover or figure out by doing the experiment. Then, list your results and explain how they confirmed or did not confirm your hypothesis. Additionally, include any uncertainties, such as circumstances beyond your control that may have impacted the results. Finally, discuss any new questions or discoveries that emerged from the experiment. For more advice, including how to wrap up your lab report with a final statement, keep reading. Did this summary help you? Yes No

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Writing Tips

How to Write a Conclusion for a Research Paper

3-minute read

29th August 2023

If you’re writing a research paper, the conclusion is your opportunity to summarize your findings and leave a lasting impression on your readers. In this post, we’ll take you through how to write an effective conclusion for a research paper and how you can:

· Reword your thesis statement

· Highlight the significance of your research

· Discuss limitations

· Connect to the introduction

· End with a thought-provoking statement

Rewording Your Thesis Statement

Begin your conclusion by restating your thesis statement in a way that is slightly different from the wording used in the introduction. Avoid presenting new information or evidence in your conclusion. Just summarize the main points and arguments of your essay and keep this part as concise as possible. Remember that you’ve already covered the in-depth analyses and investigations in the main body paragraphs of your essay, so it’s not necessary to restate these details in the conclusion.

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Highlighting the Significance of Your Research

The conclusion is a good place to emphasize the implications of your research . Avoid ambiguous or vague language such as “I think” or “maybe,” which could weaken your position. Clearly explain why your research is significant and how it contributes to the broader field of study.

Here’s an example from a (fictional) study on the impact of social media on mental health:

Discussing Limitations

Although it’s important to emphasize the significance of your study, you can also use the conclusion to briefly address any limitations you discovered while conducting your research, such as time constraints or a shortage of resources. Doing this demonstrates a balanced and honest approach to your research.

Connecting to the Introduction

In your conclusion, you can circle back to your introduction , perhaps by referring to a quote or anecdote you discussed earlier. If you end your paper on a similar note to how you began it, you will create a sense of cohesion for the reader and remind them of the meaning and significance of your research.

Ending With a Thought-Provoking Statement

Consider ending your paper with a thought-provoking and memorable statement that relates to the impact of your research questions or hypothesis. This statement can be a call to action, a philosophical question, or a prediction for the future (positive or negative). Here’s an example that uses the same topic as above (social media and mental health):

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Ensure that your essay ends on a high note by having our experts proofread your research paper. Our team has experience with a wide variety of academic fields and subjects and can help make your paper stand out from the crowd – get started today and see the difference it can make in your work.

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1.1: Statements and Conditional Statements

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Ted Sundstrom
Grand Valley State University via ScholarWorks @Grand Valley State University

Much of our work in mathematics deals with statements. In mathematics, a statement is a declarative sentence that is either true or false but not both. A statement is sometimes called a proposition . The key is that there must be no ambiguity. To be a statement, a sentence must be true or false, and it cannot be both. So a sentence such as "The sky is beautiful" is not a statement since whether the sentence is true or not is a matter of opinion. A question such as "Is it raining?" is not a statement because it is a question and is not declaring or asserting that something is true.

Some sentences that are mathematical in nature often are not statements because we may not know precisely what a variable represents. For example, the equation 2$x$+5 = 10 is not a statement since we do not know what $x$ represents. If we substitute a specific value for $x$ (such as $x$ = 3), then the resulting equation, 2$\cdot$3 +5 = 10 is a statement (which is a false statement). Following are some more examples:

There exists a real number $x$ such that 2$x$+5 = 10. This is a statement because either such a real number exists or such a real number does not exist. In this case, this is a true statement since such a real number does exist, namely $x$ = 2.5.
For each real number $x$, $2x +5 = 2 \left( x + \dfrac{5}{2}\right)$. This is a statement since either the sentence $2x +5 = 2 \left( x + \dfrac{5}{2}\right)$ is true when any real number is substituted for $x$ (in which case, the statement is true) or there is at least one real number that can be substituted for $x$ and produce a false statement (in which case, the statement is false). In this case, the given statement is true.
Solve the equation $x^2 - 7x +10 =0$. This is not a statement since it is a directive. It does not assert that something is true.
$(a+b)^2 = a^2+b^2$ is not a statement since it is not known what $a$ and $b$ represent. However, the sentence, “There exist real numbers $a$ and $b$ such that $(a+b)^2 = a^2+b^2$" is a statement. In fact, this is a true statement since there are such integers. For example, if $a=1$ and $b=0$, then $(a+b)^2 = a^2+b^2$.
Compare the statement in the previous item to the statement, “For all real numbers $a$ and $b$, $(a+b)^2 = a^2+b^2$." This is a false statement since there are values for $a$ and $b$ for which $(a+b)^2 \ne a^2+b^2$. For example, if $a=2$ and $b=3$, then $(a+b)^2 = 5^2 = 25$ and $a^2 + b^2 = 2^2 +3^2 = 13$.

Progress Check 1.1: Statements

Which of the following sentences are statements? Do not worry about determining whether a statement is true or false; just determine whether each sentence is a statement or not.

2$\cdot$7 + 8 = 22.
$(x-1) = \sqrt(x + 11)$.
$2x + 5y = 7$.
There are integers $x$ and $y$ such that $2x + 5y = 7$.
There are integers $x$ and $y$ such that $23x + 27y = 52$.
Given a line $L$ and a point $P$ not on that line, there is a unique line through $P$ that does not intersect $L$.
$(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$.
$(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$ for all real numbers $a$ and $b$.
The derivative of $f(x) = \sin x$ is $f' (x) = \cos x$.
Does the equation $3x^2 - 5x - 7 = 0$ have two real number solutions?
If $ABC$ is a right triangle with right angle at vertex $B$, and if $D$ is the midpoint of the hypotenuse, then the line segment connecting vertex $B$ to $D$ is half the length of the hypotenuse.
There do not exist three integers $x$, $y$, and $z$ such that $x^3 + y^2 = z^3$.

Add texts here. Do not delete this text first.

How Do We Decide If a Statement Is True or False?

In mathematics, we often establish that a statement is true by writing a mathematical proof. To establish that a statement is false, we often find a so-called counterexample. (These ideas will be explored later in this chapter.) So mathematicians must be able to discover and construct proofs. In addition, once the discovery has been made, the mathematician must be able to communicate this discovery to others who speak the language of mathematics. We will be dealing with these ideas throughout the text.

For now, we want to focus on what happens before we start a proof. One thing that mathematicians often do is to make a conjecture beforehand as to whether the statement is true or false. This is often done through exploration. The role of exploration in mathematics is often difficult because the goal is not to find a specific answer but simply to investigate. Following are some techniques of exploration that might be helpful.

Techniques of Exploration

Guesswork and conjectures . Formulate and write down questions and conjectures. When we make a guess in mathematics, we usually call it a conjecture.

For example, if someone makes the conjecture that $\sin(2x) = 2 \sin(x)$, for all real numbers $x$, we can test this conjecture by substituting specific values for $x$. One way to do this is to choose values of $x$ for which $\sin(x)$is known. Using $x = \frac{\pi}{4}$, we see that

$\sin(2(\frac{\pi}{4})) = \sin(\frac{\pi}{2}) = 1,$ and

$2\sin(\frac{\pi}{4}) = 2(\frac{\sqrt2}{2}) = \sqrt2$.

Since $1 \ne \sqrt2$, these calculations show that this conjecture is false. However, if we do not find a counterexample for a conjecture, we usually cannot claim the conjecture is true. The best we can say is that our examples indicate the conjecture is true. As an example, consider the conjecture that

If $x$ and $y$ are odd integers, then $x + y$ is an even integer.

We can do lots of calculation, such as $3 + 7 = 10$ and $5 + 11 = 16$, and find that every time we add two odd integers, the sum is an even integer. However, it is not possible to test every pair of odd integers, and so we can only say that the conjecture appears to be true. (We will prove that this statement is true in the next section.)

Use of prior knowledge. This also is very important. We cannot start from square one every time we explore a statement. We must make use of our acquired mathematical knowledge. For the conjecture that $\sin (2x) = 2 \sin(x)$, for all real numbers $x$, we might recall that there are trigonometric identities called “double angle identities.” We may even remember the correct identity for $\sin (2x)$, but if we do not, we can always look it up. We should recall (or find) that for all real numbers $x$, \[\sin(2x) = 2 \sin(x)\cos(x).\]
We could use this identity to argue that the conjecture “for all real numbers $x$, $\sin (2x) = 2 \sin(x)$” is false, but if we do, it is still a good idea to give a specific counterexample as we did before.
Cooperation and brainstorming . Working together is often more fruitful than working alone. When we work with someone else, we can compare notes and articulate our ideas. Thinking out loud is often a useful brainstorming method that helps generate new ideas.

Progress Check 1.2: Explorations

Use the techniques of exploration to investigate each of the following statements. Can you make a conjecture as to whether the statement is true or false? Can you determine whether it is true or false?

$(a + b)^2 = a^2 + b^2$, for all real numbers a and b.
There are integers $x$ and $y$ such that $2x + 5y = 41$.
If $x$ is an even integer, then $x^2$ is an even integer.
If $x$ and $y$ are odd integers, then $x \cdot y$ is an odd integer.

Conditional Statements

One of the most frequently used types of statements in mathematics is the so-called conditional statement. Given statements $P$ and $Q$, a statement of the form “If $P$ then $Q$” is called a conditional statement . It seems reasonable that the truth value (true or false) of the conditional statement “If $P$ then $Q$” depends on the truth values of $P$ and $Q$. The statement “If $P$ then $Q$” means that $Q$ must be true whenever $P$ is true. The statement $P$ is called the hypothesis of the conditional statement, and the statement $Q$ is called the conclusion of the conditional statement. Since conditional statements are probably the most important type of statement in mathematics, we give a more formal definition.

A conditional statement is a statement that can be written in the form “If $P$ then $Q$,” where $P$ and $Q$ are sentences. For this conditional statement, $P$ is called the hypothesis and $Q$ is called the conclusion .

Intuitively, “If $P$ then $Q$” means that $Q$ must be true whenever $P$ is true. Because conditional statements are used so often, a symbolic shorthand notation is used to represent the conditional statement “If $P$ then $Q$.” We will use the notation $P \to Q$ to represent “If $P$ then $Q$.” When $P$ and $Q$ are statements, it seems reasonable that the truth value (true or false) of the conditional statement $P \to Q$ depends on the truth values of $P$ and $Q$. There are four cases to consider:

$P$ is true and $Q$ is true.
$P$ is false and $Q$ is true.
$P$ is true and $Q$ is false.
$P$ is false and $Q$ is false.

The conditional statement $P \to Q$ means that $Q$ is true whenever $P$ is true. It says nothing about the truth value of $Q$ when $P$ is false. Using this as a guide, we define the conditional statement $P \to Q$ to be false only when $P$ is true and $Q$ is false, that is, only when the hypothesis is true and the conclusion is false. In all other cases, $P \to Q$ is true. This is summarized in Table 1.1 , which is called a truth table for the conditional statement $P \to Q$. (In Table 1.1 , T stands for “true” and F stands for “false.”)

Table 1.1: Truth Table for $P \to Q$

The important thing to remember is that the conditional statement $P \to Q$ has its own truth value. It is either true or false (and not both). Its truth value depends on the truth values for $P$ and $Q$, but some find it a bit puzzling that the conditional statement is considered to be true when the hypothesis P is false. We will provide a justification for this through the use of an example.

Example 1.3:

Suppose that I say

“If it is not raining, then Daisy is riding her bike.”

We can represent this conditional statement as $P \to Q$ where $P$ is the statement, “It is not raining” and $Q$ is the statement, “Daisy is riding her bike.”

Although it is not a perfect analogy, think of the statement $P \to Q$ as being false to mean that I lied and think of the statement $P \to Q$ as being true to mean that I did not lie. We will now check the truth value of $P \to Q$ based on the truth values of $P$ and $Q$.

Suppose that both $P$ and $Q$ are true. That is, it is not raining and Daisy is riding her bike. In this case, it seems reasonable to say that I told the truth and that$P \to Q$ is true.
Suppose that $P$ is true and $Q$ is false or that it is not raining and Daisy is not riding her bike. It would appear that by making the statement, “If it is not raining, then Daisy is riding her bike,” that I have not told the truth. So in this case, the statement $P \to Q$ is false.
Now suppose that $P$ is false and $Q$ is true or that it is raining and Daisy is riding her bike. Did I make a false statement by stating that if it is not raining, then Daisy is riding her bike? The key is that I did not make any statement about what would happen if it was raining, and so I did not tell a lie. So we consider the conditional statement, “If it is not raining, then Daisy is riding her bike,” to be true in the case where it is raining and Daisy is riding her bike.
Finally, suppose that both $P$ and $Q$ are false. That is, it is raining and Daisy is not riding her bike. As in the previous situation, since my statement was $P \to Q$, I made no claim about what would happen if it was raining, and so I did not tell a lie. So the statement $P \to Q$ cannot be false in this case and so we consider it to be true.

Progress Check 1.4: xplorations with Conditional Statements

1 . Consider the following sentence:

If $x$ is a positive real number, then $x^2 + 8x$ is a positive real number.

Although the hypothesis and conclusion of this conditional sentence are not statements, the conditional sentence itself can be considered to be a statement as long as we know what possible numbers may be used for the variable $x$. From the context of this sentence, it seems that we can substitute any positive real number for $x$. We can also substitute 0 for $x$ or a negative real number for x provided that we are willing to work with a false hypothesis in the conditional statement. (In Chapter 2 , we will learn how to be more careful and precise with these types of conditional statements.)

(a) Notice that if $x = -3$, then $x^2 + 8x = -15$, which is negative. Does this mean that the given conditional statement is false?

(b) Notice that if $x = 4$, then $x^2 + 8x = 48$, which is positive. Does this mean that the given conditional statement is true?

(c) Do you think this conditional statement is true or false? Record the results for at least five different examples where the hypothesis of this conditional statement is true.

2 . “If $n$ is a positive integer, then $n^2 - n +41$ is a prime number.” (Remember that a prime number is a positive integer greater than 1 whose only positive factors are 1 and itself.) To explore whether or not this statement is true, try using (and recording your results) for $n = 1$, $n = 2$, $n = 3$, $n = 4$, $n = 5$, and $n = 10$. Then record the results for at least four other values of $n$. Does this conditional statement appear to be true?

Further Remarks about Conditional Statements

Suppose that Ed has exactly $52 in his wallet. The following four statements will use the four possible truth combinations for the hypothesis and conclusion of a conditional statement.

If Ed has exactly $52 in his wallet, then he has $20 in his wallet. This is a true statement. Notice that both the hypothesis and the conclusion are true.
If Ed has exactly $52 in his wallet, then he has $100 in his wallet. This statement is false. Notice that the hypothesis is true and the conclusion is false.
If Ed has $100 in his wallet, then he has at least $50 in his wallet. This statement is true regardless of how much money he has in his wallet. In this case, the hypothesis is false and the conclusion is true.

This is admittedly a contrived example but it does illustrate that the conventions for the truth value of a conditional statement make sense. The message is that in order to be complete in mathematics, we need to have conventions about when a conditional statement is true and when it is false.

If $n$ is a positive integer, then $(n^2 - n + 41)$ is a prime number.

Perhaps for all of the values you tried for $n$, $(n^2 - n + 41)$ turned out to be a prime number. However, if we try $n = 41$, we ge $n^2 - n + 41 = 41^2 - 41 + 41$ $n^2 - n + 41 = 41^2$ So in the case where $n = 41$, the hypothesis is true (41 is a positive integer) and the conclusion is false $41^2$ is not prime. Therefore, 41 is a counterexample for this conjecture and the conditional statement “If $n$ is a positive integer, then $(n^2 - n + 41)$ is a prime number” is false. There are other counterexamples (such as $n = 42$, $n = 45$, and $n = 50$), but only one counterexample is needed to prove that the statement is false.

Although one example can be used to prove that a conditional statement is false, in most cases, we cannot use examples to prove that a conditional statement is true. For example, in Progress Check 1.4 , we substituted values for $x$ for the conditional statement “If $x$ is a positive real number, then $x^2 + 8x$ is a positive real number.” For every positive real number used for $x$, we saw that $x^2 + 8x$ was positive. However, this does not prove the conditional statement to be true because it is impossible to substitute every positive real number for $x$. So, although we may believe this statement is true, to be able to conclude it is true, we need to write a mathematical proof. Methods of proof will be discussed in Section 1.2 and Chapter 3 .

Progress Check 1.5: Working with a Conditional Statement

The following statement is a true statement, which is proven in many calculus texts.

If the function $f$ is differentiable at $a$, then the function $f$ is continuous at $a$.

Using only this true statement, is it possible to make a conclusion about the function in each of the following cases?

It is known that the function $f$, where $f(x) = \sin x$, is differentiable at 0.
It is known that the function $f$, where $f(x) = \sqrt[3]x$, is not differentiable at 0.
It is known that the function $f$, where $f(x) = |x|$, is continuous at 0.
It is known that the function $f$, where $f(x) = \dfrac{|x|}{x}$ is not continuous at 0.

Closure Properties of Number Systems

The primary number system used in algebra and calculus is the real number system . We usually use the symbol R to stand for the set of all real numbers. The real numbers consist of the rational numbers and the irrational numbers. The rational numbers are those real numbers that can be written as a quotient of two integers (with a nonzero denominator), and the irrational numbers are those real numbers that cannot be written as a quotient of two integers. That is, a rational number can be written in the form of a fraction, and an irrational number cannot be written in the form of a fraction. Some common irrational numbers are $\sqrt2$, $\pi$ and $e$. We usually use the symbol $\mathbb{Q}$ to represent the set of all rational numbers. (The letter $\mathbb{Q}$ is used because rational numbers are quotients of integers.) There is no standard symbol for the set of all irrational numbers.

Perhaps the most basic number system used in mathematics is the set of natural numbers . The natural numbers consist of the positive whole numbers such as 1, 2, 3, 107, and 203. We will use the symbol $\mathbb{N}$ to stand for the set of natural numbers. Another basic number system that we will be working with is the set of integers . The integers consist of zero, the positive whole numbers, and the negatives of the positive whole numbers. If $n$ is an integer, we can write $n = \dfrac{n}{1}$. So each integer is a rational number and hence also a real number.

We will use the letter $\mathbb{Z}$ to stand for the set of integers. (The letter $\mathbb{Z}$ is from the German word, $Zahlen$, for numbers.) Three of the basic properties of the integers are that the set $\mathbb{Z}$ is closed under addition , the set $\mathbb{Z}$ is closed under multiplication , and the set of integers is closed under subtraction. This means that

If $x$ and $y$ are integers, then $x + y$ is an integer;
If $x$ and $y$ are integers, then $x \cdot y$ is an integer; and
If $x$ and $y$ are integers, then $x - y$ is an integer.

Notice that these so-called closure properties are defined in terms of conditional statements. This means that if we can find one instance where the hypothesis is true and the conclusion is false, then the conditional statement is false.

Example 1.6: Closure

In order for the set of natural numbers to be closed under subtraction, the following conditional statement would have to be true: If $x$ and $y$ are natural numbers, then $x - y$ is a natural number. However, since 5 and 8 are natural numbers, $5 - 8 = -3$, which is not a natural number, this conditional statement is false. Therefore, the set of natural numbers is not closed under subtraction.
We can use the rules for multiplying fractions and the closure rules for the integers to show that the rational numbers are closed under multiplication. If $\dfrac{a}{b}$ and $\dfrac{c}{d}$ are rational numbers (so $a$, $b$, $c$, and $d$ are integers and $b$ and $d$ are not zero), then $\dfrac{a}{b} \cdot \dfrac{c}{d} = \dfrac{ac}{bd}.$ Since the integers are closed under multiplication, we know that $ac$ and $bd$ are integers and since $b \ne 0$ and $d \ne 0$, $bd \ne 0$. Hence, $\dfrac{ac}{bd}$ is a rational number and this shows that the rational numbers are closed under multiplication.

Progress Check 1.7: Closure Properties

Answer each of the following questions.

Is the set of rational numbers closed under addition? Explain.
Is the set of integers closed under division? Explain.
Is the set of rational numbers closed under subtraction? Explain.
Which of the following sentences are statements? (a) $3^2 + 4^2 = 5^2.$ (b) $a^2 + b^2 = c^2.$ (c) There exists integers $a$, $b$, and $c$ such that $a^2 + b^2 = c^2.$ (d) If $x^2 = 4$, then $x = 2.$ (e) For each real number $x$, if $x^2 = 4$, then $x = 2.$ (f) For each real number $t$, $\sin^2t + \cos^2t = 1.$ (g) $\sin x < \sin (\frac{\pi}{4}).$ (h) If $n$ is a prime number, then $n^2$ has three positive factors. (i) 1 + $\tan^2 \theta = \text{sec}^2 \theta.$ (j) Every rectangle is a parallelogram. (k) Every even natural number greater than or equal to 4 is the sum of two prime numbers.
Identify the hypothesis and the conclusion for each of the following conditional statements. (a) If $n$ is a prime number, then $n^2$ has three positive factors. (b) If $a$ is an irrational number and $b$ is an irrational number, then $a \cdot b$ is an irrational number. (c) If $p$ is a prime number, then $p = 2$ or $p$ is an odd number. (d) If $p$ is a prime number and $p \ne 2$ or $p$ is an odd number. (e) $p \ne 2$ or $p$ is a even number, then $p$ is not prime.
Determine whether each of the following conditional statements is true or false. (a) If 10 < 7, then 3 = 4. (b) If 7 < 10, then 3 = 4. (c) If 10 < 7, then 3 + 5 = 8. (d) If 7 < 10, then 3 + 5 = 8.
Determine the conditions under which each of the following conditional sentences will be a true statement. (a) If a + 2 = 5, then 8 < 5. (b) If 5 < 8, then a + 2 = 5.
Let $P$ be the statement “Student X passed every assignment in Calculus I,” and let $Q$ be the statement “Student X received a grade of C or better in Calculus I.” (a) What does it mean for $P$ to be true? What does it mean for $Q$ to be true? (b) Suppose that Student X passed every assignment in Calculus I and received a grade of B-, and that the instructor made the statement $P \to Q$. Would you say that the instructor lied or told the truth? (c) Suppose that Student X passed every assignment in Calculus I and received a grade of C-, and that the instructor made the statement $P \to Q$. Would you say that the instructor lied or told the truth? (d) Now suppose that Student X did not pass two assignments in Calculus I and received a grade of D, and that the instructor made the statement $P \to Q$. Would you say that the instructor lied or told the truth? (e) How are Parts ( 5b ), ( 5c ), and ( 5d ) related to the truth table for $P \to Q$?

Theorem If f is a quadratic function of the form $f(x) = ax^2 + bx + c$ and a < 0, then the function f has a maximum value when $x = \dfrac{-b}{2a}$. Using only this theorem, what can be concluded about the functions given by the following formulas? (a) $g (x) = -8x^2 + 5x - 2$ (b) $h (x) = -\dfrac{1}{3}x^2 + 3x$ (c) $k (x) = 8x^2 - 5x - 7$ (d) $j (x) = -\dfrac{71}{99}x^2 +210$ (e) $f (x) = -4x^2 - 3x + 7$ (f) $F (x) = -x^4 + x^3 + 9$

Theorem If $f$ is a quadratic function of the form $f(x) = ax^2 + bx + c$ and ac < 0, then the function $f$ has two x-intercepts.

Using only this theorem, what can be concluded about the functions given by the following formulas? (a) $g (x) = -8x^2 + 5x - 2$ (b) $h (x) = -\dfrac{1}{3}x^2 + 3x$ (c) $k (x) = 8x^2 - 5x - 7$ (d) $j (x) = -\dfrac{71}{99}x^2 +210$ (e) $f (x) = -4x^2 - 3x + 7$ (f) $F (x) = -x^4 + x^3 + 9$

Theorem A. If $f$ is a cubic function of the form $f (x) = x^3 - x + b$ and b > 1, then the function $f$ has exactly one $x$-intercept. Following is another theorem about $x$-intercepts of functions: Theorem B . If $f$ and $g$ are functions with $g (x) = k \cdot f (x)$, where $k$ is a nonzero real number, then $f$ and $g$ have exactly the same $x$-intercepts.

Using only these two theorems and some simple algebraic manipulations, what can be concluded about the functions given by the following formulas? (a) $f (x) = x^3 -x + 7$ (b) $g (x) = x^3 + x +7$ (c) $h (x) = -x^3 + x - 5$ (d) $k (x) = 2x^3 + 2x + 3$ (e) $r (x) = x^4 - x + 11$ (f) $F (x) = 2x^3 - 2x + 7$

(a) Is the set of natural numbers closed under division? (b) Is the set of rational numbers closed under division? (c) Is the set of nonzero rational numbers closed under division? (d) Is the set of positive rational numbers closed under division? (e) Is the set of positive real numbers closed under subtraction? (f) Is the set of negative rational numbers closed under division? (g) Is the set of negative integers closed under addition? Explorations and Activities
Exploring Propositions . In Progress Check 1.2 , we used exploration to show that certain statements were false and to make conjectures that certain statements were true. We can also use exploration to formulate a conjecture that we believe to be true. For example, if we calculate successive powers of $2, (2^1, 2^2, 2^3, 2^4, 2^5, ...)$ and examine the units digits of these numbers, we could make the following conjectures (among others): $\bullet$ If $n$ is a natural number, then the units digit of $2^n$ must be 2, 4, 6, or 8. $\bullet$ The units digits of the successive powers of 2 repeat according to the pattern “2, 4, 8, 6.” (a) Is it possible to formulate a conjecture about the units digits of successive powers of $4 (4^1, 4^2, 4^3, 4^4, 4^5,...)$? If so, formulate at least one conjecture. (b) Is it possible to formulate a conjecture about the units digit of numbers of the form $7^n - 2^n$, where $n$ is a natural number? If so, formulate a conjecture in the form of a conditional statement in the form “If $n$ is a natural number, then ... .” (c) Let $f (x) = e^(2x)$. Determine the first eight derivatives of this function. What do you observe? Formulate a conjecture that appears to be true. The conjecture should be written as a conditional statement in the form, “If n is a natural number, then ... .”

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2.11: If Then Statements

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Hypothesis followed by a conclusion in a conditional statement.

Conditional Statements

A conditional statement (also called an if-then statement ) is a statement with a hypothesis followed by a conclusion . The hypothesis is the first, or “if,” part of a conditional statement. The conclusion is the second, or “then,” part of a conditional statement. The conclusion is the result of a hypothesis.

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If-then statements might not always be written in the “if-then” form. Here are some examples of conditional statements:

Statement 1: If you work overtime, then you’ll be paid time-and-a-half.
Statement 2: I’ll wash the car if the weather is nice.
Statement 3: If 2 divides evenly into $x$, then $x$ is an even number.
Statement 4: I’ll be a millionaire when I win the lottery.
Statement 5: All equiangular triangles are equilateral.

Statements 1 and 3 are written in the “if-then” form. The hypothesis of Statement 1 is “you work overtime.” The conclusion is “you’ll be paid time-and-a-half.” Statement 2 has the hypothesis after the conclusion. If the word “if” is in the middle of the statement, then the hypothesis is after it. The statement can be rewritten: If the weather is nice, then I will wash the car. Statement 4 uses the word “when” instead of “if” and is like Statement 2. It can be written: If I win the lottery, then I will be a millionaire. Statement 5 “if” and “then” are not there. It can be rewritten: If a triangle is equiangular, then it is equilateral.

What if you were given a statement like "All squares are rectangles"? How could you determine the hypothesis and conclusion of this statement?

Example $\PageIndex{1}$

Determine the hypothesis and conclusion: I'll bring an umbrella if it rains.

Hypothesis: "It rains." Conclusion: "I'll bring an umbrella."

Example $\PageIndex{2}$

Determine the hypothesis and conclusion: All right angles are $90^{\circ}$.

Hypothesis: "An angle is right." Conclusion: "It is $90^{\circ}$."

Example $\PageIndex{3}$

Use the statement: I will graduate when I pass Calculus.

Rewrite in if-then form and determine the hypothesis and conclusion.

This statement can be rewritten as If I pass Calculus, then I will graduate. The hypothesis is “I pass Calculus,” and the conclusion is “I will graduate.”

Example $\PageIndex{4}$

Use the statement: All prime numbers are odd.

Rewrite in if-then form, determine the hypothesis and conclusion, and determine whether this is a true statement.

This statement can be rewritten as If a number is prime, then it is odd. The hypothesis is "a number is prime" and the conclusion is "it is odd". This is not a true statement (remember that not all conditional statements will be true!) since 2 is a prime number but it is not odd.

Example $\PageIndex{5}$

Determine the hypothesis and conclusion: Sarah will go to the store if Riley does the laundry.

The statement can be rewritten as "If Riley does the laundry then Sarah will go to the store." The hypothesis is "Riley does the laundry" and the conclusion is "Sarah will go to the store."

Determine the hypothesis and the conclusion for each statement.

If 5 divides evenly into $x$, then $x$ ends in 0 or 5.
If a triangle has three congruent sides, it is an equilateral triangle.
Three points are coplanar if they all lie in the same plane.
If $x=3$, then $x^2=9$.
If you take yoga, then you are relaxed.
All baseball players wear hats.
I'll learn how to drive when I am 16 years old.
If you do your homework, then you can watch TV.
Alternate interior angles are congruent if lines are parallel.
All kids like ice cream.

Additional Resources

Video: If-Then Statements Principles - Basic

Activities: If-Then Statements Discussion Questions

Study Aids: Conditional Statements Study Guide

Practice: If Then Statements

Real World: If Then Statements

Understanding Logical Statements

Learning Objectives

Identify the hypothesis and conclusion in a logical statement.
Determine whether mathematical statements involving linear, quadratic, absolute value expressions, equations, or inequalities are always, sometimes, or never true.
Use counterexamples to show that a statement is false, and recognize that a single counterexample is sufficient.

Introduction

Logic is an essential part of the study of mathematics. Much of mathematics is concerned with the characteristics of numbers and other mathematical objects (such as geometric figures or variables), and being able to make decisions about what must be true based on known characteristics and other facts is vital.

The Parts of a Logical Statement

A logical statement A statement that allows drawing a conclusion or result based on a hypothesis or premise. is a statement that, when true, allows us to take a known set of facts and infer (or assume) a new fact from them. Logical statements have two parts: The hypothesis The part of a logical statement that provides the premise on which the conclusion is based. In a statement “If `x` then `y` ,” the hypothesis is `x` . , which is the premise or set of facts that we start with, and the conclusion The part of a logical statement that provides the result or consequences of the hypothesis. In a statement “If `x` then `y` ,” the conclusion is `y` . , which is the new fact that we can infer when the hypothesis is true. (Note: If you've used hypothesis in science class, you've probably noticed that this is a fairly different definition. Be careful not to get confused!)

Consider this statement:

If you go outside without any rain gear or cover when it’s pouring rain, you will get wet.

Here, the hypothesis is “you go outside without any rain gear or cover when it’s pouring rain.” The hypothesis must be completely true before we can use the statement to infer anything new from it. What does this statement say about someone who doesn’t go outside? About someone who uses an umbrella? About what happens to someone when it’s not pouring rain? Nothing. This statement doesn't apply to anyone in those cases.

The conclusion of this statement is “you will get wet.” Suppose it’s raining, and someone walks outside, and doesn't have any rain gear or other kind of cover—what will happen? All the parts of the hypothesis have been met, so—if the statement is true—we can infer that the person is going to get wet. It certainly seems reasonable that they would!

Note that in this example, if the hypothesis isn’t true, the person still could get wet. On a sunny day with no rain, someone might go outside to wash his car and get sprayed by the hose. Someone else might go swimming, and then they would really get wet! The statement says nothing when the hypothesis is false. It's only helpful when the hypothesis is true.

Not all logical statements are written as “If (something is true) then (something else is true).” To identify the hypothesis and conclusion, you may need to try to rewrite a statement in an “if-then” format.

Logical statements can also be about mathematics, of course! Anything that lets us infer a new fact about something mathematical from given information is a logical statement. For example, “The diagonals of a rectangle have the same length” is a logical statement. The hypothesis is the part that can help us if we know it’s true. When could this statement be useful?

With algebraic statements, the hypothesis is often an assumption about what values are allowed for a variable. For example, you might have seen a statement like “ `a + b = b + a` , where `a` and `b` are real numbers.” Let's treat this equation as a logical statement:

Testing for Truth

Critical thinking is important, not just in mathematics but in everyday life. Have you ever heard someone make a statement and then thought, “Wait. Is that true?” Sometimes people have reasons for thinking something is true even though it isn’t. Determining if a statement is true is a great skill to have!

When determining if a statement is true, most people start by looking for examples A situation that suggests a logical statement may be true. , which are situations for which the statement does turn out to be true (both the hypothesis and the conclusion are true). A more powerful situation to find, if one exists, is a counterexample A situation that provides evidence that a logical statement is false. , a situation for which the statement turns out to be false (the hypothesis is true, but the conclusion is false). Why are counterexamples so powerful?

Consider a person who sees the moon many times at night and then thinks: I’ve never seen the moon during the day. The person might then make the statement, “The moon only comes out at night.” As an if-then statement, this is the same as “If the moon is out, then it’s night.” We can all probably think of many times when we saw the moon out, and it was nighttime. These are examples, situations when the statement was true.

But in fact, the statement is not always true, and we only need to see the moon during the day once —only one counterexample—to know that the statement is not true. Many, many examples cannot prove the statement is true, but we only need one counterexample to prove it’s not!

For a given statement, then, we have three possibilities:

So how can we be sure if something is true (or never true for that matter), if we can’t rely on lots of examples? With algebraic statements, sometimes we can turn to a graph for help:

Another way to decide if something is always, sometimes, or never true is reasoning from other things we know are true. We can start with something we know is true and try to create the original statement. Let’s try this with the same example above:

When we try to put together logical arguments like that, the biggest problem can be knowing where to start. There's a good chance our first attempt(s) will run into a dead end, and we'll need to start over. Practice does make it easier. Sometimes it helps to work backward: start by assuming the conclusion is true, try to think of a related statement known to be true (or false), and then connect them. Let’s take one last look at the example above:

Another way to test the truth of a statement is to look for counterexamples. Graphs can help us there, too:

Although we know `|-x| = -x` is not always true if `x` is any real number, we also know that it is sometimes true. In fact, we can specify when it’s true—using the graph from the example, we can see it’s true when `x<=0` . We can use that fact to create a new statement:

If `x<=0` , then `|-x| = -x` .

Because of the narrower hypothesis, this statement is always true.

Special Cases

When we look for examples, and particularly for counterexamples, there are some special cases that are easy to overlook. Keeping these cases in mind is often very helpful. Look through some special cases and consider if any of them provides a counterexample for this statement: “When two numbers are multiplied, the product is larger than each of the factors.”

Here are some more examples. Consider the special cases above as you read through these.

Logical statements have two parts, a hypothesis that presents facts that the statement needs to be true, and a conclusion that presents a new fact we can infer when the hypothesis is true.

For a statement to be always true, there must be no counterexamples for which the hypothesis is true and the conclusion is false. If there are examples for which the statement is true, but there are also counterexamples, then the statement is sometimes true. These sometimes true statements can be made into always true statements by changing the hypothesis. A statement is never true if there are no examples for which both the hypothesis and the conclusion are true. When looking for counterexamples and examples, there are some special cases (such as negative numbers and fractions) that should be considered.

$SplashLearn$

Conditional Statement – Definition, Truth Table, Examples, FAQs

What is a conditional statement, how to write a conditional statement, what is a biconditional statement, solved examples on conditional statements, practice problems on conditional statements, frequently asked questions about conditional statements.

A conditional statement is a statement that is written in the “If p, then q” format. Here, the statement p is called the hypothesis and q is called the conclusion. It is a fundamental concept in logic and mathematics.

Conditional statement symbol : p → q

A conditional statement consists of two parts.

The “if” clause, which presents a condition or hypothesis.
The “then” clause, which indicates the consequence or result that follows if the condition is true.

Example : If you brush your teeth, then you won’t get cavities.

Hypothesis (Condition): If you brush your teeth

Conclusion (Consequence): then you won’t get cavities

$Conditional statement$

Conditional Statement: Definition

A conditional statement is characterized by the presence of “if” as an antecedent and “then” as a consequent. A conditional statement, also known as an “if-then” statement consists of two parts:

The “if” clause (hypothesis): This part presents a condition, situation, or assertion. It is the initial condition that is being considered.
The “then” clause (conclusion): This part indicates the consequence, result, or action that will occur if the condition presented in the “if” clause is true or satisfied.

Related Worksheets

$Complete the Statements Using Addition Sentence Worksheet$

Representation of Conditional Statement

The conditional statement of the form ‘If p, then q” is represented as p → q.

It is pronounced as “p implies q.”

Different ways to express a conditional statement are:

p implies q
p is sufficient for q
q is necessary for p

Parts of a Conditional Statement

There are two parts of conditional statements, hypothesis and conclusion. The hypothesis or condition will begin with the “if” part, and the conclusion or action will begin with the “then” part. A conditional statement is also called “implication.”

Conditional Statements Examples:

Example 1: If it is Sunday, then you can go to play.

Hypothesis: If it is Sunday

Conclusion: then you can go to play.

Example 2: If you eat all vegetables, then you can have the dessert.

Condition: If you eat all vegetables

Conclusion: then you can have the dessert

To form a conditional statement, follow these concise steps:

Step 1 : Identify the condition (antecedent or “if” part) and the consequence (consequent or “then” part) of the statement.

Step 2 : Use the “if… then…” structure to connect the condition and consequence.

Step 3 : Ensure the statement expresses a logical relationship where the condition leads to the consequence.

Example 1 : “If you study (condition), then you will pass the exam (consequence).”

This conditional statement asserts that studying leads to passing the exam. If you study (condition is true), then you will pass the exam (consequence is also true).

Example 2 : If you arrange the numbers from smallest to largest, then you will have an ascending order.

Hypothesis: If you arrange the numbers from smallest to largest

Conclusion: then you will have an ascending order

Truth Table for Conditional Statement

The truth table for a conditional statement is a table used in logic to explore the relationship between the truth values of two statements. It lists all possible combinations of truth values for “p” and “q” and determines whether the conditional statement is true or false for each combination.

The truth value of p → q is false only when p is true and q is False.

If the condition is false, the consequence doesn’t affect the truth of the conditional; it’s always true.

In all the other cases, it is true.

The truth table is helpful in the analysis of possible combinations of truth values for hypothesis or condition and conclusion or action. It is useful to understand the presence of truth or false statements.

Converse, Inverse, and Contrapositive

The converse, inverse, and contrapositive are three related conditional statements that are derived from an original conditional statement “p → q.”

Consider a conditional statement: If I run, then I feel great.

Converse:

The converse of “p → q” is “q → p.” It reverses the order of the original statement. While the original statement says “if p, then q,” the converse says “if q, then p.”

Converse: If I feel great, then I run.

Inverse:

The inverse of “p → q” is “~p → ~q,” where “” denotes negation (opposite). It negates both the antecedent (p) and the consequent (q). So, if the original statement says “if p, then q,” the inverse says “if not p, then not q.”

Inverse : If I don’t run, then I don’t feel great.

Contrapositive:

The contrapositive of “p → q” is “~q → ~p.” It reverses the order and also negates both the statements. So, if the original statement says “if p, then q,” the contrapositive says “if not q, then not p.”

Contrapositive: If I don’t feel great, then I don’t run.

A biconditional statement is a type of compound statement in logic that expresses a bidirectional or two-way relationship between two statements. It asserts that “p” is true if and only if “q” is true, and vice versa. In symbolic notation, a biconditional statement is represented as “p ⟺ q.”

In simpler terms, a biconditional statement means that the truth of “p” and “q” are interdependent.

If “p” is true, then “q” must also be true, and if “q” is true, then “p” must be true. Conversely, if “p” is false, then “q” must be false, and if “q” is false, then “p” must be false.

Biconditional statements are often used to express equality, equivalence, or conditions where two statements are mutually dependent for their truth values.

Examples :

I will stop my bike if and only if the traffic light is red.
I will stay if and only if you play my favorite song.

Facts about Conditional Statements

The negation of a conditional statement “p → q” is expressed as “p and not q.” It is denoted as “𝑝 ∧ ∼𝑞.”
The conditional statement is not logically equivalent to its converse and inverse.
The conditional statement is logically equivalent to its contrapositive.
Thus, we can write p → q ∼q → ∼p

In this article, we learned about the fundamentals of conditional statements in mathematical logic, including their structure, parts, truth tables, conditional logic examples, and various related concepts. Understanding conditional statements is key to logical reasoning and problem-solving. Now, let’s solve a few examples and practice MCQs for better comprehension.

Example 1: Identify the hypothesis and conclusion.

If you sing, then I will dance.

Solution :

Given statement: If you sing, then I will dance.

Here, the antecedent or the hypothesis is “if you sing.”

The conclusion is “then I will dance.”

Example 2: State the converse of the statement: “If the switch is off, then the machine won’t work.”

Here, p: The switch is off

q: The machine won’t work.

The conditional statement can be denoted as p → q.

Converse of p → q is written by reversing the order of p and q in the original statement.

Converse of p → q is q → p.

Converse of p → q: q → p: If the machine won’t work, then the switch is off.

Example 3: What is the truth value of the given conditional statement?

If 2+2=5 , then pigs can fly.

Solution:

q: Pigs can fly.

The statement p is false. Now regardless of the truth value of statement q, the overall statement will be true.

F → F = T

Hence, the truth value of the statement is true.

Conditional Statement - Definition, Truth Table, Examples, FAQs

Attend this quiz & Test your knowledge.

What is the antecedent in the given conditional statement? If it’s sunny, then I’ll go to the beach.

A conditional statement can be expressed as, what is the converse of “a → b”, when the antecedent is true and the consequent is false, the conditional statement is.

What is the meaning of conditional statements?

Conditional statements, also known as “if-then” statements, express a cause-and-effect or logical relationship between two propositions.

When does the truth value of a conditional statement is F?

A conditional statement is considered false when the antecedent is true and the consequent is false.

What is the contrapositive of a conditional statement?

The contrapositive reverses the order of the statements and also negates both the statements. It is equivalent in truth value to the original statement.

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How to Write Hypothesis Test Conclusions (With Examples)
A hypothesis test is used to test whether or not some hypothesis about a population parameter is true.. To perform a hypothesis test in the real world, researchers obtain a random sample from the population and perform a hypothesis test on the sample data, using a null and alternative hypothesis:. Null Hypothesis (H 0): The sample data occurs purely from chance.
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.
How to State the Conclusion about a Hypothesis Test
Use the following table to help you make a good conclusion. The best way to state the conclusion is to include the significance level of the test and a bit about the claim itself. " At the 5% significance level, there is sufficient evidence to support the claim that the mean score on the test was greater than 85. The reason you should include ...
How to Write a Hypothesis w/ Strong Examples
In conclusion, understanding and effectively formulating a solid hypothesis is what scientific research and inquiry is all about—regardless of the type of work you're doing. It may be a simple, complex, directional, non-directional, null, associative, or causal hypothesis—no matter: each type has its own specific purpose and guides the ...
Hypothesis Testing
Table of contents. Step 1: State your null and alternate hypothesis. Step 2: Collect data. Step 3: Perform a statistical test. Step 4: Decide whether to reject or fail to reject your null hypothesis. Step 5: Present your findings. Other interesting articles. Frequently asked questions about hypothesis testing.
Research Hypothesis: Definition, Types, Examples and Quick Tips
What is a Hypothesis? The first step in your scientific endeavor, a hypothesis, is a strong, concise statement that forms the basis of your research. It is not the same as a thesis statement, which is a brief summary of your research paper. The sole purpose of a hypothesis is to predict your paper's findings, data, and conclusion.
The scientific method (article)
The scientific method. At the core of biology and other sciences lies a problem-solving approach called the scientific method. The scientific method has five basic steps, plus one feedback step: Make an observation. Ask a question. Form a hypothesis, or testable explanation. Make a prediction based on the hypothesis.
5.2
5.2 - Writing Hypotheses. The first step in conducting a hypothesis test is to write the hypothesis statements that are going to be tested. For each test you will have a null hypothesis ( H 0) and an alternative hypothesis ( H a ). Null Hypothesis. The statement that there is not a difference in the population (s), denoted as H 0.
The Scientific Method
CONCLUSION. The final step in the scientific method is the conclusion. This is a summary of the experiment's results, and how those results match up to your hypothesis. You have two options for your conclusions: based on your results, either: (1) YOU CAN REJECT the hypothesis, or (2) YOU CAN NOT REJECT the hypothesis. This is an important point!
Writing a Research Paper Conclusion
Step 1: Restate the problem. The first task of your conclusion is to remind the reader of your research problem. You will have discussed this problem in depth throughout the body, but now the point is to zoom back out from the details to the bigger picture. While you are restating a problem you've already introduced, you should avoid phrasing ...
Conclusions
Highlight the "so what". At the beginning of your paper, you explain to your readers what's at stake—why they should care about the argument you're making. In your conclusion, you can bring readers back to those stakes by reminding them why your argument is important in the first place. You can also draft a few sentences that put ...
Statement of Conclusion
Examples: 1) Claim: Females run faster than males. Results of the test: to > tc. Decision: Reject Null Hypothesis. Conclusion: There is sufficient evidence to suggest that females run faster than males. 2) Claim: There is a difference in the highest level of education obtained based on socioeconomic status. Results of the test: p > α.
How to Write Hypothesis Test Conclusions (With Examples)
A hypothesis test is used to test whether or not some hypothesis about a population parameter is true.. To perform a hypothesis test in the real world, researchers obtain a random sample from the population and perform a hypothesis test on the sample data, using a null and alternative hypothesis:. Null Hypothesis (H 0): The sample data occurs purely from chance.
5 Ways to Write a Good Lab Conclusion in Science
1. Introduce the experiment in your conclusion. Start out the conclusion by providing a brief overview of the experiment. Describe the experiment in 1-2 sentences and discuss the objective of the experiment. Also, make sure to include your manipulated (independent), controlled and responding (dependent) variables. [3] 2.
How to Write a Conclusion for a Research Paper
Begin your conclusion by restating your thesis statement in a way that is slightly different from the wording used in the introduction. Avoid presenting new information or evidence in your conclusion. Just summarize the main points and arguments of your essay and keep this part as concise as possible. Remember that you've already covered the ...
1.1: Statements and Conditional Statements
A conditional statement is a statement that can be written in the form "If P then Q ," where P and Q are sentences. For this conditional statement, P is called the hypothesis and Q is called the conclusion. Intuitively, "If P then Q " means that Q must be true whenever P is true.
How to Conclude an Essay
Step 1: Return to your thesis. To begin your conclusion, signal that the essay is coming to an end by returning to your overall argument. Don't just repeat your thesis statement —instead, try to rephrase your argument in a way that shows how it has been developed since the introduction. Example: Returning to the thesis.
Understanding Logical Statements
A logical statement A statement that allows drawing a conclusion or result based on a hypothesis or premise. is a statement that, when true, allows us to take a known set of facts and infer (or assume) a new fact from them. Logical statements have two parts: The hypothesis The part of a logical statement that provides the premise on which the conclusion is based.
2.11: If Then Statements
The conclusion is the result of a hypothesis. Figure 2.11.1 2.11. 1. If-then statements might not always be written in the "if-then" form. Here are some examples of conditional statements: Statement 1: If you work overtime, then you'll be paid time-and-a-half. Statement 2: I'll wash the car if the weather is nice.
Understanding Logical Statements
A logical statement A statement that allows drawing a conclusion or result based on a hypothesis or premise. is a statement that, when true, allows us to take a known set of facts and infer (or assume) a new fact from them. Logical statements have two parts: The hypothesis The part of a logical statement that provides the premise on which the conclusion is based.
How to Write a Thesis or Dissertation Conclusion
Step 1: Answer your research question. Step 2: Summarize and reflect on your research. Step 3: Make future recommendations. Step 4: Emphasize your contributions to your field. Step 5: Wrap up your thesis or dissertation. Full conclusion example. Conclusion checklist. Other interesting articles.
Conditional Statement: Definition, Truth Table, Examples
A conditional statement is a statement that is written in the "If p, then q" format. Here, the statement p is called the hypothesis and q is called the conclusion. It is a fundamental concept in logic and mathematics. Conditional statement symbol: p → q. A conditional statement consists of two parts.

How to Write Hypothesis Test Conclusions (With Examples)

Example 1: Reject the Null Hypothesis Conclusion

Example 2: Fail to Reject the Null Hypothesis Conclusion

Additional Resources

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How to State the Conclusion about a Hypothesis Test

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The Craft of Writing a Strong Hypothesis

Table of Contents

What is a Hypothesis?

Different Types of Hypotheses‌

1. Null hypothesis

2. Alternative hypothesis

3. Simple hypothesis

4. Complex hypothesis

5. Associative and casual hypothesis

6. Empirical hypothesis

7. Statistical hypothesis

Characteristics of a Good Hypothesis

Separating a Hypothesis from a Prediction

Finally, How to Write a Hypothesis

1. Be clear about your research question

2. Carry out a recce

3. Create a 3-dimensional hypothesis

4. Write the first draft

5. Proof your hypothesis

Frequently Asked Questions (FAQs)

2. What is an example of hypothesis?

3. What is an example of null hypothesis?

4. What are the types of research?

5. How to write a hypothesis?

6. What are the 2 types of hypothesis?

7. Difference between research question and research hypothesis?

8. What is plural for hypothesis?

9. What is the red queen hypothesis?

10. Who is known as the father of null hypothesis?

11. When to reject null hypothesis?

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Introduction

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2. Ask a question.

3. Propose a hypothesis.

4. Make predictions.

5. Test the predictions.

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How to Write Hypothesis Test Conclusions (With Examples)

Example 1: Reject the Null Hypothesis Conclusion

Example 2: Fail to Reject the Null Hypothesis Conclusion

Additional Resources

10 Examples of Using Probability in Real Life

How to Write a Good Lab Conclusion in Science

Outlining Your Conclusion

Discussing the Experiment and Hypothesis

Demonstrating What You Have Learned

Wrapping Up Your Conclusion

Finalizing Your Lab Report

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