hypothesis definition dummies

• More from M-W
• To save this word, you'll need to log in. Log In

Definition of hypothesis

Did you know.

The Difference Between Hypothesis and Theory

A hypothesis is an assumption, an idea that is proposed for the sake of argument so that it can be tested to see if it might be true.

In the scientific method, the hypothesis is constructed before any applicable research has been done, apart from a basic background review. You ask a question, read up on what has been studied before, and then form a hypothesis.

A hypothesis is usually tentative; it's an assumption or suggestion made strictly for the objective of being tested.

A theory , in contrast, is a principle that has been formed as an attempt to explain things that have already been substantiated by data. It is used in the names of a number of principles accepted in the scientific community, such as the Big Bang Theory . Because of the rigors of experimentation and control, it is understood to be more likely to be true than a hypothesis is.

In non-scientific use, however, hypothesis and theory are often used interchangeably to mean simply an idea, speculation, or hunch, with theory being the more common choice.

Since this casual use does away with the distinctions upheld by the scientific community, hypothesis and theory are prone to being wrongly interpreted even when they are encountered in scientific contexts—or at least, contexts that allude to scientific study without making the critical distinction that scientists employ when weighing hypotheses and theories.

The most common occurrence is when theory is interpreted—and sometimes even gleefully seized upon—to mean something having less truth value than other scientific principles. (The word law applies to principles so firmly established that they are almost never questioned, such as the law of gravity.)

This mistake is one of projection: since we use theory in general to mean something lightly speculated, then it's implied that scientists must be talking about the same level of uncertainty when they use theory to refer to their well-tested and reasoned principles.

The distinction has come to the forefront particularly on occasions when the content of science curricula in schools has been challenged—notably, when a school board in Georgia put stickers on textbooks stating that evolution was "a theory, not a fact, regarding the origin of living things." As Kenneth R. Miller, a cell biologist at Brown University, has said , a theory "doesn’t mean a hunch or a guess. A theory is a system of explanations that ties together a whole bunch of facts. It not only explains those facts, but predicts what you ought to find from other observations and experiments.”

While theories are never completely infallible, they form the basis of scientific reasoning because, as Miller said "to the best of our ability, we’ve tested them, and they’ve held up."

• proposition
• supposition

hypothesis , theory , law mean a formula derived by inference from scientific data that explains a principle operating in nature.

hypothesis implies insufficient evidence to provide more than a tentative explanation.

theory implies a greater range of evidence and greater likelihood of truth.

law implies a statement of order and relation in nature that has been found to be invariable under the same conditions.

Examples of hypothesis in a Sentence

These examples are programmatically compiled from various online sources to illustrate current usage of the word 'hypothesis.' Any opinions expressed in the examples do not represent those of Merriam-Webster or its editors. Send us feedback about these examples.

Word History

Greek, from hypotithenai to put under, suppose, from hypo- + tithenai to put — more at do

1641, in the meaning defined at sense 1a

Phrases Containing hypothesis

• counter - hypothesis
• nebular hypothesis
• null hypothesis
• planetesimal hypothesis
• Whorfian hypothesis

Articles Related to hypothesis

This is the Difference Between a...

This is the Difference Between a Hypothesis and a Theory

In scientific reasoning, they're two completely different things

Dictionary Entries Near hypothesis

hypothermia

hypothesize

Cite this Entry

“Hypothesis.” Merriam-Webster.com Dictionary , Merriam-Webster, https://www.merriam-webster.com/dictionary/hypothesis. Accessed 29 Apr. 2024.

Kids Definition

Kids definition of hypothesis, medical definition, medical definition of hypothesis, more from merriam-webster on hypothesis.

Nglish: Translation of hypothesis for Spanish Speakers

Britannica English: Translation of hypothesis for Arabic Speakers

Britannica.com: Encyclopedia article about hypothesis

Subscribe to America's largest dictionary and get thousands more definitions and advanced search—ad free!

Can you solve 4 words at once?

Word of the day.

See Definitions and Examples »

Get Word of the Day daily email!

Popular in Grammar & Usage

More commonly misspelled words, commonly misspelled words, how to use em dashes (—), en dashes (–) , and hyphens (-), absent letters that are heard anyway, how to use accents and diacritical marks, popular in wordplay, the words of the week - apr. 26, 9 superb owl words, 'gaslighting,' 'woke,' 'democracy,' and other top lookups, 10 words for lesser-known games and sports, your favorite band is in the dictionary, games & quizzes.

Research Hypothesis In Psychology: Types, & Examples

Saul Mcleod, PhD

Editor-in-Chief for Simply Psychology

BSc (Hons) Psychology, MRes, PhD, University of Manchester

Saul Mcleod, PhD., is a qualified psychology teacher with over 18 years of experience in further and higher education. He has been published in peer-reviewed journals, including the Journal of Clinical Psychology.

Learn about our Editorial Process

Olivia Guy-Evans, MSc

Associate Editor for Simply Psychology

BSc (Hons) Psychology, MSc Psychology of Education

Olivia Guy-Evans is a writer and associate editor for Simply Psychology. She has previously worked in healthcare and educational sectors.

On This Page:

A research hypothesis, in its plural form “hypotheses,” is a specific, testable prediction about the anticipated results of a study, established at its outset. It is a key component of the scientific method .

Hypotheses connect theory to data and guide the research process towards expanding scientific understanding

Some key points about hypotheses:

• A hypothesis expresses an expected pattern or relationship. It connects the variables under investigation.
• It is stated in clear, precise terms before any data collection or analysis occurs. This makes the hypothesis testable.
• A hypothesis must be falsifiable. It should be possible, even if unlikely in practice, to collect data that disconfirms rather than supports the hypothesis.
• Hypotheses guide research. Scientists design studies to explicitly evaluate hypotheses about how nature works.
• For a hypothesis to be valid, it must be testable against empirical evidence. The evidence can then confirm or disprove the testable predictions.
• Hypotheses are informed by background knowledge and observation, but go beyond what is already known to propose an explanation of how or why something occurs.

Predictions typically arise from a thorough knowledge of the research literature, curiosity about real-world problems or implications, and integrating this to advance theory. They build on existing literature while providing new insight.

Types of Research Hypotheses

Alternative hypothesis.

The research hypothesis is often called the alternative or experimental hypothesis in experimental research.

It typically suggests a potential relationship between two key variables: the independent variable, which the researcher manipulates, and the dependent variable, which is measured based on those changes.

The alternative hypothesis states a relationship exists between the two variables being studied (one variable affects the other).

A hypothesis is a testable statement or prediction about the relationship between two or more variables. It is a key component of the scientific method. Some key points about hypotheses:

• Important hypotheses lead to predictions that can be tested empirically. The evidence can then confirm or disprove the testable predictions.

In summary, a hypothesis is a precise, testable statement of what researchers expect to happen in a study and why. Hypotheses connect theory to data and guide the research process towards expanding scientific understanding.

An experimental hypothesis predicts what change(s) will occur in the dependent variable when the independent variable is manipulated.

It states that the results are not due to chance and are significant in supporting the theory being investigated.

The alternative hypothesis can be directional, indicating a specific direction of the effect, or non-directional, suggesting a difference without specifying its nature. It’s what researchers aim to support or demonstrate through their study.

Null Hypothesis

The null hypothesis states no relationship exists between the two variables being studied (one variable does not affect the other). There will be no changes in the dependent variable due to manipulating the independent variable.

It states results are due to chance and are not significant in supporting the idea being investigated.

The null hypothesis, positing no effect or relationship, is a foundational contrast to the research hypothesis in scientific inquiry. It establishes a baseline for statistical testing, promoting objectivity by initiating research from a neutral stance.

Many statistical methods are tailored to test the null hypothesis, determining the likelihood of observed results if no true effect exists.

This dual-hypothesis approach provides clarity, ensuring that research intentions are explicit, and fosters consistency across scientific studies, enhancing the standardization and interpretability of research outcomes.

Nondirectional Hypothesis

A non-directional hypothesis, also known as a two-tailed hypothesis, predicts that there is a difference or relationship between two variables but does not specify the direction of this relationship.

It merely indicates that a change or effect will occur without predicting which group will have higher or lower values.

For example, “There is a difference in performance between Group A and Group B” is a non-directional hypothesis.

Directional Hypothesis

A directional (one-tailed) hypothesis predicts the nature of the effect of the independent variable on the dependent variable. It predicts in which direction the change will take place. (i.e., greater, smaller, less, more)

It specifies whether one variable is greater, lesser, or different from another, rather than just indicating that there’s a difference without specifying its nature.

For example, “Exercise increases weight loss” is a directional hypothesis.

Falsifiability

The Falsification Principle, proposed by Karl Popper , is a way of demarcating science from non-science. It suggests that for a theory or hypothesis to be considered scientific, it must be testable and irrefutable.

Falsifiability emphasizes that scientific claims shouldn’t just be confirmable but should also have the potential to be proven wrong.

It means that there should exist some potential evidence or experiment that could prove the proposition false.

However many confirming instances exist for a theory, it only takes one counter observation to falsify it. For example, the hypothesis that “all swans are white,” can be falsified by observing a black swan.

For Popper, science should attempt to disprove a theory rather than attempt to continually provide evidence to support a research hypothesis.

Can a Hypothesis be Proven?

Hypotheses make probabilistic predictions. They state the expected outcome if a particular relationship exists. However, a study result supporting a hypothesis does not definitively prove it is true.

All studies have limitations. There may be unknown confounding factors or issues that limit the certainty of conclusions. Additional studies may yield different results.

In science, hypotheses can realistically only be supported with some degree of confidence, not proven. The process of science is to incrementally accumulate evidence for and against hypothesized relationships in an ongoing pursuit of better models and explanations that best fit the empirical data. But hypotheses remain open to revision and rejection if that is where the evidence leads.

• Disproving a hypothesis is definitive. Solid disconfirmatory evidence will falsify a hypothesis and require altering or discarding it based on the evidence.
• However, confirming evidence is always open to revision. Other explanations may account for the same results, and additional or contradictory evidence may emerge over time.

We can never 100% prove the alternative hypothesis. Instead, we see if we can disprove, or reject the null hypothesis.

If we reject the null hypothesis, this doesn’t mean that our alternative hypothesis is correct but does support the alternative/experimental hypothesis.

Upon analysis of the results, an alternative hypothesis can be rejected or supported, but it can never be proven to be correct. We must avoid any reference to results proving a theory as this implies 100% certainty, and there is always a chance that evidence may exist which could refute a theory.

How to Write a Hypothesis

• Identify variables . The researcher manipulates the independent variable and the dependent variable is the measured outcome.
• Operationalized the variables being investigated . Operationalization of a hypothesis refers to the process of making the variables physically measurable or testable, e.g. if you are about to study aggression, you might count the number of punches given by participants.
• Decide on a direction for your prediction . If there is evidence in the literature to support a specific effect of the independent variable on the dependent variable, write a directional (one-tailed) hypothesis. If there are limited or ambiguous findings in the literature regarding the effect of the independent variable on the dependent variable, write a non-directional (two-tailed) hypothesis.
• Make it Testable : Ensure your hypothesis can be tested through experimentation or observation. It should be possible to prove it false (principle of falsifiability).
• Clear & concise language . A strong hypothesis is concise (typically one to two sentences long), and formulated using clear and straightforward language, ensuring it’s easily understood and testable.

Consider a hypothesis many teachers might subscribe to: students work better on Monday morning than on Friday afternoon (IV=Day, DV= Standard of work).

Now, if we decide to study this by giving the same group of students a lesson on a Monday morning and a Friday afternoon and then measuring their immediate recall of the material covered in each session, we would end up with the following:

• The alternative hypothesis states that students will recall significantly more information on a Monday morning than on a Friday afternoon.
• The null hypothesis states that there will be no significant difference in the amount recalled on a Monday morning compared to a Friday afternoon. Any difference will be due to chance or confounding factors.

More Examples

• Memory : Participants exposed to classical music during study sessions will recall more items from a list than those who studied in silence.
• Social Psychology : Individuals who frequently engage in social media use will report higher levels of perceived social isolation compared to those who use it infrequently.
• Developmental Psychology : Children who engage in regular imaginative play have better problem-solving skills than those who don’t.
• Clinical Psychology : Cognitive-behavioral therapy will be more effective in reducing symptoms of anxiety over a 6-month period compared to traditional talk therapy.
• Cognitive Psychology : Individuals who multitask between various electronic devices will have shorter attention spans on focused tasks than those who single-task.
• Health Psychology : Patients who practice mindfulness meditation will experience lower levels of chronic pain compared to those who don’t meditate.
• Organizational Psychology : Employees in open-plan offices will report higher levels of stress than those in private offices.
• Behavioral Psychology : Rats rewarded with food after pressing a lever will press it more frequently than rats who receive no reward.

What is a scientific hypothesis?

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

Hypothesis basics

What makes a hypothesis testable.

• Types of hypotheses
• Hypothesis versus theory

Additional resources

Bibliography.

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

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

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

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

Here are some examples of hypothesis statements:

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

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

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

Types of scientific hypotheses

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

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

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

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

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

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

Scientific theory vs. scientific hypothesis

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

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

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

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

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

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

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

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

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

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

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

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

Sign up for the Live Science daily newsletter now

Get the world’s most fascinating discoveries delivered straight to your inbox.

Eerie, orange skies loom over Athens as dust storm engulfs southern Greece

Hidden 'biosphere' of extreme microbes discovered 13 feet below Atacama Desert is deepest found there to date

Eclipse from space: Paths of 2024 and 2017 eclipses collide over US in new satellite image

Most Popular

• 2 James Webb telescope confirms there is something seriously wrong with our understanding of the universe
• 3 Giant, 82-foot lizard fish discovered on UK beach could be largest marine reptile ever found
• 4 Global 'time signals' subtly shifted as the total solar eclipse reshaped Earth's upper atmosphere, new data shows
• 5 'I nearly fell out of my chair': 1,800-year-old mini portrait of Alexander the Great found in a field in Denmark
• 2 New UTI vaccine wards off infection for years, early studies suggest
• 3 Tweak to Schrödinger's cat equation could unite Einstein's relativity and quantum mechanics, study hints
• 4 Plato's burial place finally revealed after AI deciphers ancient scroll carbonized in Mount Vesuvius eruption
• 5 George Washington's stash of centuries-old cherries found hidden under Mount Vernon floor

If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

To log in and use all the features of Khan Academy, please enable JavaScript in your browser.

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.

Want to join the conversation?

• Upvote Button navigates to signup page
• Downvote Button navigates to signup page
• Flag Button navigates to signup page

• Cambridge Dictionary +Plus

Meaning of hypothesis in English

Your browser doesn't support HTML5 audio

• abstraction
• afterthought
• anthropocentrism
• anti-Darwinian
• exceptionalism
• foundation stone
• great minds think alike idiom
• non-dogmatic
• non-empirical
• non-material
• non-practical
• social Darwinism
• supersensible
• the domino theory

hypothesis | American Dictionary

Hypothesis | business english, examples of hypothesis, translations of hypothesis.

Get a quick, free translation!

Word of the Day

anonymously

without the name of someone who has done a particular thing being known or made public

Dead ringers and peas in pods (Talking about similarities, Part 2)

Learn more with +Plus

• Recent and Recommended {{#preferredDictionaries}} {{name}} {{/preferredDictionaries}}
• Definitions Clear explanations of natural written and spoken English English Learner’s Dictionary Essential British English Essential American English
• Grammar and thesaurus Usage explanations of natural written and spoken English Grammar Thesaurus
• Pronunciation British and American pronunciations with audio English Pronunciation
• English–Chinese (Simplified) Chinese (Simplified)–English
• English–Chinese (Traditional) Chinese (Traditional)–English
• English–Dutch Dutch–English
• English–French French–English
• English–German German–English
• English–Indonesian Indonesian–English
• English–Italian Italian–English
• English–Japanese Japanese–English
• English–Norwegian Norwegian–English
• English–Polish Polish–English
• English–Portuguese Portuguese–English
• English–Spanish Spanish–English
• English–Swedish Swedish–English
• Dictionary +Plus Word Lists
• English    Noun
• American    Noun
• Business    Noun
• Translations
• All translations

Add hypothesis to one of your lists below, or create a new one.

{{message}}

Something went wrong.

There was a problem sending your report.

• school Campus Bookshelves
• menu_book Bookshelves
• perm_media Learning Objects
• login Login
• how_to_reg Request Instructor Account
• hub Instructor Commons
• Download Page (PDF)
• Download Full Book (PDF)
• Periodic Table
• Physics Constants
• Scientific Calculator
• Reference & Cite
• Tools expand_more
• Readability

selected template will load here

This action is not available.

9.1: Null and Alternative Hypotheses

• Last updated
• Save as PDF
• Page ID 23459

The actual test begins by considering two hypotheses . They are called the null hypothesis and the alternative hypothesis . These hypotheses contain opposing viewpoints.

\(H_0\): The null hypothesis: It is a statement of no difference between the variables—they are not related. This can often be considered the status quo and as a result if you cannot accept the null it requires some action.

\(H_a\): The alternative hypothesis: It is a claim about the population that is contradictory to \(H_0\) and what we conclude when we reject \(H_0\). This is usually what the researcher is trying to prove.

Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data.

After you have determined which hypothesis the sample supports, you make a decision. There are two options for a decision. They are "reject \(H_0\)" if the sample information favors the alternative hypothesis or "do not reject \(H_0\)" or "decline to reject \(H_0\)" if the sample information is insufficient to reject the null hypothesis.

\(H_{0}\) always has a symbol with an equal in it. \(H_{a}\) never has a symbol with an equal in it. The choice of symbol depends on the wording of the hypothesis test. However, be aware that many researchers (including one of the co-authors in research work) use = in the null hypothesis, even with > or < as the symbol in the alternative hypothesis. This practice is acceptable because we only make the decision to reject or not reject the null hypothesis.

Example \(\PageIndex{1}\)

• \(H_{0}\): No more than 30% of the registered voters in Santa Clara County voted in the primary election. \(p \leq 30\)
• \(H_{a}\): More than 30% of the registered voters in Santa Clara County voted in the primary election. \(p > 30\)

Exercise \(\PageIndex{1}\)

A medical trial is conducted to test whether or not a new medicine reduces cholesterol by 25%. State the null and alternative hypotheses.

• \(H_{0}\): The drug reduces cholesterol by 25%. \(p = 0.25\)
• \(H_{a}\): The drug does not reduce cholesterol by 25%. \(p \neq 0.25\)

Example \(\PageIndex{2}\)

We want to test whether the mean GPA of students in American colleges is different from 2.0 (out of 4.0). The null and alternative hypotheses are:

• \(H_{0}: \mu = 2.0\)
• \(H_{a}: \mu \neq 2.0\)

Exercise \(\PageIndex{2}\)

We want to test whether the mean height of eighth graders is 66 inches. State the null and alternative hypotheses. Fill in the correct symbol \((=, \neq, \geq, <, \leq, >)\) for the null and alternative hypotheses.

• \(H_{0}: \mu \_ 66\)
• \(H_{a}: \mu \_ 66\)
• \(H_{0}: \mu = 66\)
• \(H_{a}: \mu \neq 66\)

Example \(\PageIndex{3}\)

We want to test if college students take less than five years to graduate from college, on the average. The null and alternative hypotheses are:

• \(H_{0}: \mu \geq 5\)
• \(H_{a}: \mu < 5\)

Exercise \(\PageIndex{3}\)

We want to test if it takes fewer than 45 minutes to teach a lesson plan. State the null and alternative hypotheses. Fill in the correct symbol ( =, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

• \(H_{0}: \mu \_ 45\)
• \(H_{a}: \mu \_ 45\)
• \(H_{0}: \mu \geq 45\)
• \(H_{a}: \mu < 45\)

Example \(\PageIndex{4}\)

In an issue of U. S. News and World Report , an article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third pass. The same article stated that 6.6% of U.S. students take advanced placement exams and 4.4% pass. Test if the percentage of U.S. students who take advanced placement exams is more than 6.6%. State the null and alternative hypotheses.

• \(H_{0}: p \leq 0.066\)
• \(H_{a}: p > 0.066\)

Exercise \(\PageIndex{4}\)

On a state driver’s test, about 40% pass the test on the first try. We want to test if more than 40% pass on the first try. Fill in the correct symbol (\(=, \neq, \geq, <, \leq, >\)) for the null and alternative hypotheses.

• \(H_{0}: p \_ 0.40\)
• \(H_{a}: p \_ 0.40\)
• \(H_{0}: p = 0.40\)
• \(H_{a}: p > 0.40\)

COLLABORATIVE EXERCISE

Bring to class a newspaper, some news magazines, and some Internet articles . In groups, find articles from which your group can write null and alternative hypotheses. Discuss your hypotheses with the rest of the class.

In a hypothesis test , sample data is evaluated in order to arrive at a decision about some type of claim. If certain conditions about the sample are satisfied, then the claim can be evaluated for a population. In a hypothesis test, we:

• Evaluate the null hypothesis , typically denoted with \(H_{0}\). The null is not rejected unless the hypothesis test shows otherwise. The null statement must always contain some form of equality \((=, \leq \text{or} \geq)\)
• Always write the alternative hypothesis , typically denoted with \(H_{a}\) or \(H_{1}\), using less than, greater than, or not equals symbols, i.e., \((\neq, >, \text{or} <)\).
• If we reject the null hypothesis, then we can assume there is enough evidence to support the alternative hypothesis.
• Never state that a claim is proven true or false. Keep in mind the underlying fact that hypothesis testing is based on probability laws; therefore, we can talk only in terms of non-absolute certainties.

Formula Review

\(H_{0}\) and \(H_{a}\) are contradictory.

• If \(\alpha \leq p\)-value, then do not reject \(H_{0}\).
• If\(\alpha > p\)-value, then reject \(H_{0}\).

\(\alpha\) is preconceived. Its value is set before the hypothesis test starts. The \(p\)-value is calculated from the data.References

Data from the National Institute of Mental Health. Available online at http://www.nimh.nih.gov/publicat/depression.cfm .

• school Campus Bookshelves
• menu_book Bookshelves
• perm_media Learning Objects
• login Login
• how_to_reg Request Instructor Account
• hub Instructor Commons
• Download Page (PDF)
• Download Full Book (PDF)
• Periodic Table
• Physics Constants
• Scientific Calculator
• Reference & Cite
• Tools expand_more
• Readability

selected template will load here

This action is not available.

3.6: Mathematical Induction - An Introduction

• Last updated
• Save as PDF
• Page ID 24565

• Harris Kwong
• State University of New York at Fredonia via OpenSUNY

Mathematical induction can be used to prove that an identity is valid for all integers \(n\geq1\). Here is a typical example of such an identity: \[1+2+3+\cdots+n = \frac{n(n+1)}{2}.\] More generally, we can use mathematical induction to prove that a propositional function \(P(n)\) is true for all integers \(n\geq a\).

Principal of Mathematical Induction (PMI)

Given a propositional function \(P(n)\) defined for integers \(n\), and a fixed integer \(a.\)

Then, if these two conditions are true

• \(P(a)\) is true.
• if \(P(k)\) is true for some integer \(k\geq a\), then \(P(k+1)\) is also true.

then the  \(P(n)\) is true for all integers \(n\geq a\).

Outline for Mathematical Induction

To show that a propositional function \(P(n)\) is true for all integers \(n\geq a\), follow these steps:

• Base Step : Verify that \(P(a)\) is true.
• Assume \(P(n)\) is true for an arbitrary integer, \(k\) with  \(k\geq a\).  This is the inductive hypothesis .
• With this assumption (the inductive hypothesis) , show \(P(k+1)\) is true.
• Conclude, by the Principle of Mathematical Induction (PMI) that \(P(n)\) is true for all integers \(n\geq a\).

The base step is also called the basis step or the  anchor step or the initial step . 

The base step and the inductive step, together, prove that \[P(a) \Rightarrow P(a+1) \Rightarrow P(a+2) \Rightarrow \cdots\,.\] Therefore, \(P(n)\) is true for all integers \(n\geq a\). Compare induction to falling dominoes. When the first domino falls, it knocks down the next domino. The second domino in turn knocks down the third domino. Eventually, all the dominoes will be knocked down. But it will not happen unless these conditions are met:

• The first domino must fall to start the motion. If it does not fall, no chain reaction will occur. This is the base step.
• The distance between adjacent dominoes must be set up correctly. Otherwise, a certain domino may fall down without knocking over the next. Then the chain reaction will stop, and will never be completed. Maintaining the right inter-domino distance ensures that \(P(k)\Rightarrow P(k+1)\) for each integer \(k\geq a\).

To prove the implication \[P(k) \Rightarrow P(k+1)\] in the inductive step, we need to carry out two steps: assuming that \(P(k)\) is true, then using it to prove \(P(k+1)\) is also true. So we can refine an induction proof into a 3-step procedure:

• Verify that \(P(a)\) is true.
• Assume that \(P(k)\) is true for some integer \(k\geq a\).
• Show that \(P(k+1)\) is also true.

The second step, the assumption that \(P(k)\) is true, is referred to as the inductive hypothesis.  This is how a mathematical induction proof may look:

The idea behind mathematical induction is rather simple. However, it must be delivered with precision.

• Be sure to say “Assume \(P(n)\) holds for some integer \(k\geq a\).” Do not say “Assume it holds for all integers \(k\geq a\).” If we already know the result holds for all \(k\geq a\), then there is no need to prove anything at all.
• Be sure to specify the requirement \(k\geq a\). This ensures that the chain reaction of the falling dominoes starts with the first one.
• Do not say “let \(n=k\)” or “let \(n=k+1\).” The point is, you are not assigning the value of \(k\) and \(k+1\) to \(n\). Rather, you are assuming that the statement is true when \(n\) equals \(k\), and using it to show that the statement also holds when \(n\) equals \(k+1\).

Some proofs by induction 

\(1+2+3+\cdots+n\).

Use mathematical induction to show proposition \(P(n)\) :   \[1+2+3+\cdots+n = \frac{n(n+1)}{2}\] for all integers \(n\geq1\).

Base Step:  consider n = 1

On the Left-Hand Side (LHS) we get 1.  On the Right-Hand Side ( RHS) we get \(\frac{1(1+1)}{2}=\frac{2}{2}=1.\)  Thus \(P(n)\) is true for \(n =1.\)

Inductive step: Assume  \(P(n)\) is true for \(n =k, k \geq 1.\)  In other words, \(P(k)\) is true so our  inductive hypothesis is  \[1+2+3+\cdots+k = \frac{k(k+1)}{2}.\]   

Consider the left-hand side of \(P(k+1)\).   \[1+2+3+\cdots+(k+1) = 1+2+\cdots+k+(k+1),\]

we can regroup this as

\[1+2+3+\cdots+(k+1) = [1+2+\cdots+k]+(k+1),\]

so that \(1+2+\cdots+k\) can be replaced by \(\frac{k(k+1)}{2}\), by the inductive hypothesis.

Using the inductive hypothesis, we find

\[\begin{aligned} 1+2+3+\cdots+(k+1) &=& 1+2+3+\cdots+k+(k+1) \\ &=& \frac{k(k+1)}{2}+(k+1) \\ &=& (k+1)\left(\frac{k}{2}+1\right) \\ &=& (k+1)\cdot\frac{k+2}{2}\\ &=& \frac{(k+1)(k+2)}{2}. \end{aligned}\]

Therefore, the identity also holds when \(n=k+1\).

Thus, by the Principle of Mathematical Induction (PMI),  \[1+2+3+\cdots+n = \frac{n(n+1)}{2}\] for all integers \(n\geq1\).

We can use the summation notation (also called the sigma notation ) to abbreviate a sum. For example, the sum in the last example can be written as

\[\sum_{i=1}^n i.\]

The letter \(i\) is the index of summation . By putting \(i=1\) under \(\sum\) and \(n\) above, we declare that the sum starts with \(i=1\), and ranges through \(i=2\), \(i=3\), and so on, until \(i=n\). The quantity that follows \(\sum\) describes the pattern of the terms that we are adding in the summation. Accordingly,

\[\sum_{i=1}^{10} i^2 = 1^2+2^2+3^2+\cdots+10^2.\]

In general, the sum of the first \(n\) terms in a sequence \(\{a_1,a_2,a_3,\ldots\,\}\) is denoted \(\sum_{i=1}^n a_i\). Observe that

\[\sum_{i=1}^{k+1} a_i = \left(\sum_{i=1}^k a_i\right) + a_{k+1},\]

which provides the link between \(P(k+1)\) and \(P(k)\) in an induction proof.

\(\sum_{i=1}^n i^2\)

Example \(\PageIndex{2}\)

Use mathematical induction to show that, for all integers \(n\geq1\), \[\sum_{i=1}^n i^2 = 1^2+2^2+3^2+\cdots+n^2 = \frac{n(n+1)(2n+1)}{6}.\]

Base Step: When \(n=1\), the left-hand side reduces to \(1^2=1\), and the right-hand side becomes \(\frac{1\cdot2\cdot3}{6}=1\); hence, the identity holds when \(n=1\). Inductive Step: Assume it holds when \(n=k\) for some integer \(k\geq1\); that is, assume for some integer \(k\geq1\) that \[\sum_{i=1}^k i^2 = \frac{k(k+1)(2k+1)}{6}\] . Consider \(n=k+1\).   \[\sum_{i=1}^{k+1} i^2 =1^2+2^2+3^2+\cdots+k^2+(k+1)^2. \] From the inductive hypothesis, we find \[\sum_{i=1}^{k+1} i^2 = \sum_{i=1}^k i^2 + (k+1)^2\] \[=\frac{k(k+1)(2k+1)}{6}+(k+1)^2\]  \[=\frac{k(k+1)(2k+1)+6(k+1)^2}{6}\] \[\frac{(k+1)[k(2k+1)+6(k+1)]}{6}\] \[\frac{(k+1)(2k^2+7k+6)}{6}\] \[\frac{(k+1)(k+2)(2k+3)}{6}\] \[\frac{(k+1)(k+2)(2(k+1)+1)}{6}.\] Therefore, the identity also holds when \(n=k+1\).  Thus, by PMI for all integers \(n\geq1\), \[\sum_{i=1}^n i^2 = 1^2+2^2+3^2+\cdots+n^2 = \frac{n(n+1)(2n+1)}{6}.\]

hands-on exercise \(\PageIndex{1}\label{he:induct1-01}\)

It is time for you to write your own induction proof. Prove that \[1\cdot2 + 2\cdot3 + 3\cdot4 + \cdots + n(n+1) = \frac{n(n+1)(n+2)}{3}\] for all integers \(n\geq1\).

hands-on exercise \(\PageIndex{2}\label{he:induct1-02}\)

Use induction to prove that, for all positive integers \(n\), \[1\cdot2\cdot3 + 2\cdot3\cdot4 + \cdots + n(n+1)(n+2) = \frac{n(n+1)(n+2)(n+3)}{4}.\]

hands-on exercise \(\PageIndex{3}\label{he:sumfourn}\)

Use induction to prove that, for all positive integers \(n\), \[1+4^1+4^2+\cdots+4^n = \frac{4^{n+1}-1}{3}.\]

All three steps in an induction proof must be completed; otherwise, the proof may not be correct.

Example \(\PageIndex{3}\label{eg:induct1-03}\)

Can we just use examples?

Never attempt to prove \(P(k)\Rightarrow P(k+1)\) by examples alone . Consider \[P(n): \qquad n^2+n+11 \mbox{ is prime}.\] In the inductive step, we want to prove that \[P(k) \Rightarrow P(k+1) \qquad\mbox{ for ANY } k\geq1.\] The following table verifies that it is true for \(1\leq k\leq 9\): \[\begin{array}{|*{10}{c|}} \hline n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline n^2+n+11 & 13 & 17 & 23 & 31 & 41 & 53 & 67 & 83 & 101 \\ \hline \end{array}\] Nonetheless, when \(n=10\), \(n^2+n+11=121\) is composite. So \(P(9) \Rightarrow P(10)\) is false. The inductive step breaks down when \(k=9\).

Example \(\PageIndex{4}\label{eg:induct1-04}\)

The base step is equally important . Consider proving \[P(n): \qquad 3n+2 = 3q \mbox{ for some integer $q$}\] for all \(n\in\mathbb{N}\). Assume \(P(k)\) is true for some integer \(k\geq1\); that is, assume \(3k+2=3q\) for some integer \(q\). Then \[3(k+1)+2 = 3k+3+2 = 3+3q = 3(1+q).\] Therefore, \(3(k+1)+2\) can be written in the same form. This proves that \(P(k+1)\) is also true. Does it follow that \(P(n)\) is true for all integers \(n\geq1\)? We know that \(3n+2\) cannot be written as a multiple of 3. What is the problem?

The problem is: we need \(P(k)\) to be true for at least one value of \(k\) so as to start the sequence of implications \[P(1) \Rightarrow P(2), \qquad P(2) \Rightarrow P(3), \qquad P(3) \Rightarrow P(4), \qquad\ldots\] The induction fails because we have not established the basis step. In fact, \(P(1)\) is false. Since the first domino does not fall, we cannot even start the chain reaction.

Thus far, we have learned how to use mathematical induction to prove identities. In general, we can use mathematical induction to prove a statement about \(n\). This statement can take the form of an identity, an inequality, or simply a verbal statement about \(n\). We shall learn more about mathematical induction in the next few sections.

Summary and Review

• Mathematical induction can be used to prove that a statement about \(n\) is true for all integers \(n\geq a\).
• We have to complete three steps.
• In the base step, verify the statement for \(n=a\).
• In the inductive hypothesis, assume that the statement holds when \(n=k\) for some integer \(k\geq a\).
• In the inductive step, use the information gathered from the inductive hypothesis to prove that the statement also holds when \(n=k+1\).
• Be sure to complete all three steps.
• Pay attention to the wording. At the beginning, follow the template closely. When you feel comfortable with the whole process, you can start venturing out on your own.

Exercises 

Exercise \(\PageIndex{1}\label{ex:induct1-01}\)

Use induction to prove that \[1^3+2^3+3^3+\cdots+n^3 = \frac{n^2(n+1)^2}{4}\] for all integers \(n\geq1\).

Exercise \(\PageIndex{2}\)

Use induction to prove that the following identity holds for all integers \(n\geq1\): \[1+3+5+\cdots+(2n-1) = n^2.\]

Base Case: consider \(n=1\).  \(2(1)-1=1\) and \(1^2=1\) so the LHS & RHS are both 1. This works for  \(n=1\).

Inductive Step: Assume this works for some integer, \(k \geq 1.\) In other words,  \(1+3+5+\cdots+(2k-1) = k^2.\)  ( Inductive Hypothesis )

Consider the case of  \(n=k+1.\)   \(1+3+5+\cdots +(2k-1)+(2(k+1)-1)\)

 \[=k^2+(2(k+1)-1) \text{   by inductive hypothesis}\] \[=k^2+2k+2-1=k^2+2k+1=(k+1)^2 \text{   by algebra} \]

\(1+3+5+\cdots+(2(k+1)-1)=(k+1)^2\); assuming our proposition works for \(k\) it will also work for \(k+1.\)

By PMI, \(1+3+5+\cdots+(2n-1) = n^2\)  for all integers,  \(n\geq1\).

Exercise \(\PageIndex{3}\label{ex:induct1-03}\)

Use induction to show that \[1+\frac{1}{3}+\frac{1}{3^2}+\cdots+\frac{1}{3^n} = \frac{3}{2}\left(1-\frac{1}{3^{n+1}}\right)\] for all positive integers \(n\).

Exercise \(\PageIndex{4}\label{ex:induct1-04}\)

Use induction to establish the following identity for any integer \(n\geq1\): \[1-3+9-\cdots+(-3)^n = \frac{1-(-3)^{n+1}}{4}.\]

Exercise \(\PageIndex{5}\label{ex:induct1-05}\)

Use induction to show that, for any integer \(n\geq1\): \[\sum_{i=1}^n i\cdot i! = (n+1)!-1.\]

Exercise \(\PageIndex{6}\label{ex:induct1-06}\)

Use induction to prove the following identity for integers \(n\geq1\): \[\sum_{i=1}^n \frac{1}{(2i-1)(2i+1)} = \frac{n}{2n+1}.\]

Exercise \(\PageIndex{7}\)

Prove \(2^{2n}-1\) is divisible by 3, for all integers \(n\geq0.\)

Base Case: consider \(n=0\).  \(2^{2(0)}-1=1-1=0.\)  \(0\) is divisible by 3 because 0 = 0(3).

Inductive Step: Assume this works for some integer, \(k \geq 0.\) In other words, \(2^{2k}-1\) is divisible by 3. ( Inductive Hypothesis )

Since \(2^{2k}-1\) is divisible by 3, there exists some integer, m such that \(2^{2k}-1=3m,\)  by definition of divides.

Consider the case of  \(n=k+1.\)  By algebra: \[2^{2(k+1)}-1=2^{2k+2}-1=2^{2k}\cdot 2^2-1=2^{2k}\cdot 4 -1=2^{2k}\cdot (3+1)-1=3 \cdot 2^{2k}+2^{2k}-1\] \[=3 \cdot 2^{2k}+3m \text{   by inductive hypothesis}\]

\[=3(2^{2k}+m) \text{   by algebra}\]

\(2^{2(k+1)}-1=3(2^{2k}+m)\) and  \((2^{2k}+m)\in \mathbb{Z}\) since the integers are closed under addition and multiplication.  

So, \(2^{2(k+1)}-1\) is divisible by 3 by the definition of divisible.

Thus assuming our proposition works for \(k\) it will also work for \(k+1.\)

By PMI,  \(2^{2n}-1\) is divisible by 3, for all integers \(n\geq0.\)

Exercise \(\PageIndex{8}\label{ex:induct1-08}\)

Evaluate \(\sum_{i=1}^n \frac{1}{i(i+1)}\) for a few values of \(n\). What do you think the result should be? Use induction to prove your conjecture.

Exercise \(\PageIndex{9}\label{ex:induct1-09}\)

Use induction to prove that \[\sum_{i=1}^n (2i-1)^3 = n^2(2n^2-1)\] whenever \(n\) is a positive integer.

Exercise \(\PageIndex{10}\label{ex:induct1-10}\)

Use induction to show that, for any integer \(n\geq1\): \[1^2-2^2+3^2-\cdots+(-1)^{n-1}n^2 = (-1)^{n-1}\,\frac{n(n+1)}{2}.\]

Exercise \(\PageIndex{11}\label{ex:induct1-11}\)

Use mathematical induction to show that \[\sum_{i=1}^n \frac{i+4}{i(i+1)(i+2)} = \frac{n(3n+7)}{2(n+1)(n+2)}\] for all integers \(n\geq1\).

Exercise \(\PageIndex{12}\)

Use mathematical induction to show that \[3+\sum_{i=1}^n (3+5i) = \frac{(n+1)(5n+6)}{2}\] for all integers \(n\geq1\).

No answer here at this time.