what is mathematics hypothesis

A hypothesis is a proposition that is consistent with known data, but has been neither verified nor shown to be false.

In statistics, a hypothesis (sometimes called a statistical hypothesis) refers to a statement on which hypothesis testing will be based. Particularly important statistical hypotheses include the null hypothesis and alternative hypothesis .

In symbolic logic , a hypothesis is the first part of an implication (with the second part being known as the predicate ).

In general mathematical usage, "hypothesis" is roughly synonymous with " conjecture ."

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A statement that could be true, which might then be tested.

Example: Sam has a hypothesis that "large dogs are better at catching tennis balls than small dogs". We can test that hypothesis by having hundreds of different sized dogs try to catch tennis balls.

Sometimes the hypothesis won't be tested, it is simply a good explanation (which could be wrong). Conjecture is a better word for this.

Example: you notice the temperature drops just as the sun rises. Your hypothesis is that the sun warms the air high above you, which rises up and then cooler air comes from the sides.

Note: when someone says "I have a theory" they should say "I have a hypothesis", because in mathematics a theory is actually well proven.

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Hypothesis | Definition & Meaning

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Explanation of Hypothesis

Contradiction, simple hypothesis, complex hypothesis, null hypothesis, alternative hypothesis, empirical hypothesis, statistical hypothesis, special example of hypothesis, solution part (a), solution part (b), hypothesis|definition & meaning.

A hypothesis is a claim or statement that makes sense in the context of some information or data at hand but hasn’t been established as true or false through experimentation or proof.

In mathematics, any statement or equation that describes some relationship between certain variables can be termed as hypothesis if it is consistent with some initial supporting data or information, however, its yet to be proven true or false by some definite and trustworthy experiment or mathematical law.

Following example illustrates one such hypothesis to shed some light on this very fundamental concept which is often used in different areas of mathematics.

Figure 1: Example of Hypothesis

Here we have considered an example of a young startup company that manufactures state of the art batteries. The hypothesis or the claim of the company is that their batteries have a mean life of more than 1000 hours. Now its very easy to understand that they can prove their claim on some testing experiment in their lab.

However, the statement can only be proven if and only if at least one batch of their production batteries have actually been deployed in the real world for more than 1000 hours . After 1000 hours, data needs to be collected and it needs to be seen what is the probability of this statement being true .

The following paragraphs further explain this concept.

As explained with the help of an example earlier, a hypothesis in mathematics is an untested claim that is backed up by all the known data or some other discoveries or some weak experiments.

In any mathematical discovery, we first start by assuming something or some relationship . This supposed statement is called a supposition. A supposition, however, becomes a hypothesis when it is supported by all available data and a large number of contradictory findings.

The hypothesis is an important part of the scientific method that is widely known today for making new discoveries. The field of mathematics inherited this process. Following figure shows this cycle as a graphic:

Figure 2: Role of Hypothesis in the Scientific Method

The above figure shows a simplified version of the scientific method. It shows that whenever a supposition is supported by some data, its termed as hypothesis. Once a hypothesis is proven by some well known and widely acceptable experiment or proof, its becomes a law. If the hypothesis is rejected by some contradictory results then the supposition is changed and the cycle continues.

Lets try to understand the scientific method and the hypothesis concept with the help of an example. Lets say that a teacher wanted to analyze the relationship between the students performance in a certain subject, lets call it A, based on whether or not they studied a minor course, lets call it B.

Now the teacher puts forth a supposition that the students taking the course B prior to course A must perform better in the latter due to the obvious linkages in the key concepts. Due to this linkage, this supposition can be termed as a hypothesis.

However to test the hypothesis, the teacher has to collect data from all of his/her students such that he/she knows which students have taken course B and which ones haven’t. Then at the end of the semester, the performance of the students must be measured and compared with their course B enrollments.

If the students that took course B prior to course A perform better, then the hypothesis concludes successful . Otherwise, the supposition may need revision.

The following figure explains this problem graphically.

Figure 3: Teacher and Course Example of Hypothesis

Important Terms Related to Hypothesis

To further elaborate the concept of hypothesis, we first need to understand a few key terms that are widely used in this area such as conjecture, contradiction and some special types of hypothesis (simple, complex, null, alternative, empirical, statistical). These terms are briefly explained below:

A conjecture is a term used to describe a mathematical assertion that has notbeenproved. While testing may occasionally turn up millions of examples in favour of a conjecture, most experts in the area will typically only accept a proof . In mathematics, this term is synonymous to the term hypothesis.

In mathematics, a contradiction occurs if the results of an experiment or proof are against some hypothesis. In other words, a contradiction discredits a hypothesis.

A simple hypothesis is such a type of hypothesis that claims there is a correlation between two variables. The first is known as a dependent variable while the second is known as an independent variable.

A complex hypothesis is such a type of hypothesis that claims there is a correlation between more than two variables. Both the dependent and independent variables in this hypothesis may be more than one in numbers.

A null hypothesis, usually denoted by H0, is such a type of hypothesis that claims there is no statistical relationship and significance between two sets of observed data and measured occurrences for each set of defined, single observable variables. In short the variables are independent.

An alternative hypothesis, usually denoted by H1 or Ha, is such a type of hypothesis where the variables may be statistically influenced by some unknown factors or variables. In short the variables are dependent on some unknown phenomena .

An Empirical hypothesis is such a type of hypothesis that is built on top of some empirical data or experiment or formulation.

A statistical hypothesis is such a type of hypothesis that is built on top of some statistical data or experiment or formulation. It may be logical or illogical in nature.

According to the Riemann hypothesis, only negative even integers and complex numbers with real part 1/2 have zeros in the Riemann zeta function . It is regarded by many as the most significant open issue in pure mathematics.

Figure 4: Riemann Hypothesis

The Riemann hypothesis is the most well-known mathematical conjecture, and it has been the subject of innumerable proof efforts.

Numerical Examples

Identify the conclusions and hypothesis in the following given statements. Also state if the conclusion supports the hypothesis or not.

Part (a): If 30x = 30, then x = 1

Part (b): if 10x + 2 = 50, then x = 24

Hypothesis: 30x = 30

Conclusion: x = 10

Supports Hypothesis: Yes

Hypothesis: 10x + 2 = 50

Conclusion: x = 24

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Understanding Hypotheses

'What happens if ... ?' to ' This will happen if'

The experimentation of children continually moves on to the exploration of new ideas and the refinement of their world view of previously understood situations. This description of the playtime patterns of young children very nicely models the concept of 'making and testing hypotheses'. It follows this pattern:

Make some observations. Collect some data based on the observations.
Draw a conclusion (called a 'hypothesis') which will explain the pattern of the observations.
Test out your hypothesis by making some more targeted observations.

So, we have

A hypothesis is a statement or idea which gives an explanation to a series of observations.

Sometimes, following observation, a hypothesis will clearly need to be refined or rejected. This happens if a single contradictory observation occurs. For example, suppose that a child is trying to understand the concept of a dog. He reads about several dogs in children's books and sees that they are always friendly and fun. He makes the natural hypothesis in his mind that dogs are friendly and fun . He then meets his first real dog: his neighbour's puppy who is great fun to play with. This reinforces his hypothesis. His cousin's dog is also very friendly and great fun. He meets some of his friends' dogs on various walks to playgroup. They are also friendly and fun. He is now confident that his hypothesis is sound. Suddenly, one day, he sees a dog, tries to stroke it and is bitten. This experience contradicts his hypothesis. He will need to amend the hypothesis. We see that

Gathering more evidence/data can strengthen a hypothesis if it is in agreement with the hypothesis.
If the data contradicts the hypothesis then the hypothesis must be rejected or amended to take into account the contradictory situation.

A contradictory observation can cause us to know for certain that a hypothesis is incorrect.
Accumulation of supporting experimental evidence will strengthen a hypothesis but will never let us know for certain that the hypothesis is true.

In short, it is possible to show that a hypothesis is false, but impossible to prove that it is true!

Whilst we can never prove a scientific hypothesis to be true, there will be a certain stage at which we decide that there is sufficient supporting experimental data for us to accept the hypothesis. The point at which we make the choice to accept a hypothesis depends on many factors. In practice, the key issues are

What are the implications of mistakenly accepting a hypothesis which is false?
What are the cost / time implications of gathering more data?
What are the implications of not accepting in a timely fashion a true hypothesis?

For example, suppose that a drug company is testing a new cancer drug. They hypothesise that the drug is safe with no side effects. If they are mistaken in this belief and release the drug then the results could have a disastrous effect on public health. However, running extended clinical trials might be very costly and time consuming. Furthermore, a delay in accepting the hypothesis and releasing the drug might also have a negative effect on the health of many people.

In short, whilst we can never achieve absolute certainty with the testing of hypotheses, in order to make progress in science or industry decisions need to be made. There is a fine balance to be made between action and inaction.

Hypotheses and mathematics So where does mathematics enter into this picture? In many ways, both obvious and subtle:

A good hypothesis needs to be clear, precisely stated and testable in some way. Creation of these clear hypotheses requires clear general mathematical thinking.
The data from experiments must be carefully analysed in relation to the original hypothesis. This requires the data to be structured, operated upon, prepared and displayed in appropriate ways. The levels of this process can range from simple to exceedingly complex.

Very often, the situation under analysis will appear to be complicated and unclear. Part of the mathematics of the task will be to impose a clear structure on the problem. The clarity of thought required will actively be developed through more abstract mathematical study. Those without sufficient general mathematical skill will be unable to perform an appropriate logical analysis.

Using deductive reasoning in hypothesis testing

There is often confusion between the ideas surrounding proof, which is mathematics, and making and testing an experimental hypothesis, which is science. The difference is rather simple:

Mathematics is based on deductive reasoning : a proof is a logical deduction from a set of clear inputs.
Science is based on inductive reasoning : hypotheses are strengthened or rejected based on an accumulation of experimental evidence.

Of course, to be good at science, you need to be good at deductive reasoning, although experts at deductive reasoning need not be mathematicians. Detectives, such as Sherlock Holmes and Hercule Poirot, are such experts: they collect evidence from a crime scene and then draw logical conclusions from the evidence to support the hypothesis that, for example, Person M. committed the crime. They use this evidence to create sufficiently compelling deductions to support their hypotheses beyond reasonable doubt . The key word here is 'reasonable'. There is always the possibility of creating an exceedingly outlandish scenario to explain away any hypothesis of a detective or prosecution lawyer, but judges and juries in courts eventually make the decision that the probability of such eventualities are 'small' and the chance of the hypothesis being correct 'high'.

If a set of data is normally distributed with mean 0 and standard deviation 0.5 then there is a 97.7% certainty that a measurement will not exceed 1.0.
If the mean of a sample of data is 12, how confident can we be that the true mean of the population lies between 11 and 13?

It is at this point that making and testing hypotheses becomes a true branch of mathematics. This mathematics is difficult, but fascinating and highly relevant in the information-rich world of today.

To read more about the technical side of hypothesis testing, take a look at What is a Hypothesis Test?

You might also enjoy reading the articles on statistics on the Understanding Uncertainty website

This resource is part of the collection Statistics - Maths of Real Life

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9.1: Introduction to Hypothesis Testing

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Kyle Siegrist
University of Alabama in Huntsville via Random Services

Basic Theory

Preliminaries.

As usual, our starting point is a random experiment with an underlying sample space and a probability measure $\P$. In the basic statistical model, we have an observable random variable $\bs{X}$ taking values in a set $S$. In general, $\bs{X}$ can have quite a complicated structure. For example, if the experiment is to sample $n$ objects from a population and record various measurements of interest, then \[ \bs{X} = (X_1, X_2, \ldots, X_n) \] where $X_i$ is the vector of measurements for the $i$th object. The most important special case occurs when $(X_1, X_2, \ldots, X_n)$ are independent and identically distributed. In this case, we have a random sample of size $n$ from the common distribution.

The purpose of this section is to define and discuss the basic concepts of statistical hypothesis testing . Collectively, these concepts are sometimes referred to as the Neyman-Pearson framework, in honor of Jerzy Neyman and Egon Pearson, who first formalized them.

A statistical hypothesis is a statement about the distribution of $\bs{X}$. Equivalently, a statistical hypothesis specifies a set of possible distributions of $\bs{X}$: the set of distributions for which the statement is true. A hypothesis that specifies a single distribution for $\bs{X}$ is called simple ; a hypothesis that specifies more than one distribution for $\bs{X}$ is called composite .

In hypothesis testing , the goal is to see if there is sufficient statistical evidence to reject a presumed null hypothesis in favor of a conjectured alternative hypothesis . The null hypothesis is usually denoted $H_0$ while the alternative hypothesis is usually denoted $H_1$.

An hypothesis test is a statistical decision ; the conclusion will either be to reject the null hypothesis in favor of the alternative, or to fail to reject the null hypothesis. The decision that we make must, of course, be based on the observed value $\bs{x}$ of the data vector $\bs{X}$. Thus, we will find an appropriate subset $R$ of the sample space $S$ and reject $H_0$ if and only if $\bs{x} \in R$. The set $R$ is known as the rejection region or the critical region . Note the asymmetry between the null and alternative hypotheses. This asymmetry is due to the fact that we assume the null hypothesis, in a sense, and then see if there is sufficient evidence in $\bs{x}$ to overturn this assumption in favor of the alternative.

An hypothesis test is a statistical analogy to proof by contradiction, in a sense. Suppose for a moment that $H_1$ is a statement in a mathematical theory and that $H_0$ is its negation. One way that we can prove $H_1$ is to assume $H_0$ and work our way logically to a contradiction. In an hypothesis test, we don't prove anything of course, but there are similarities. We assume $H_0$ and then see if the data $\bs{x}$ are sufficiently at odds with that assumption that we feel justified in rejecting $H_0$ in favor of $H_1$.

Often, the critical region is defined in terms of a statistic $w(\bs{X})$, known as a test statistic , where $w$ is a function from $S$ into another set $T$. We find an appropriate rejection region $R_T \subseteq T$ and reject $H_0$ when the observed value $w(\bs{x}) \in R_T$. Thus, the rejection region in $S$ is then $R = w^{-1}(R_T) = \left\{\bs{x} \in S: w(\bs{x}) \in R_T\right\}$. As usual, the use of a statistic often allows significant data reduction when the dimension of the test statistic is much smaller than the dimension of the data vector.

The ultimate decision may be correct or may be in error. There are two types of errors, depending on which of the hypotheses is actually true.

Types of errors:

A type 1 error is rejecting the null hypothesis $H_0$ when $H_0$ is true.
A type 2 error is failing to reject the null hypothesis $H_0$ when the alternative hypothesis $H_1$ is true.

Similarly, there are two ways to make a correct decision: we could reject $H_0$ when $H_1$ is true or we could fail to reject $H_0$ when $H_0$ is true. The possibilities are summarized in the following table:

Of course, when we observe $\bs{X} = \bs{x}$ and make our decision, either we will have made the correct decision or we will have committed an error, and usually we will never know which of these events has occurred. Prior to gathering the data, however, we can consider the probabilities of the various errors.

If $H_0$ is true (that is, the distribution of $\bs{X}$ is specified by $H_0$), then $\P(\bs{X} \in R)$ is the probability of a type 1 error for this distribution. If $H_0$ is composite, then $H_0$ specifies a variety of different distributions for $\bs{X}$ and thus there is a set of type 1 error probabilities.

The maximum probability of a type 1 error, over the set of distributions specified by $ H_0 $, is the significance level of the test or the size of the critical region.

The significance level is often denoted by $\alpha$. Usually, the rejection region is constructed so that the significance level is a prescribed, small value (typically 0.1, 0.05, 0.01).

If $H_1$ is true (that is, the distribution of $\bs{X}$ is specified by $H_1$), then $\P(\bs{X} \notin R)$ is the probability of a type 2 error for this distribution. Again, if $H_1$ is composite then $H_1$ specifies a variety of different distributions for $\bs{X}$, and thus there will be a set of type 2 error probabilities. Generally, there is a tradeoff between the type 1 and type 2 error probabilities. If we reduce the probability of a type 1 error, by making the rejection region $R$ smaller, we necessarily increase the probability of a type 2 error because the complementary region $S \setminus R$ is larger.

The extreme cases can give us some insight. First consider the decision rule in which we never reject $H_0$, regardless of the evidence $\bs{x}$. This corresponds to the rejection region $R = \emptyset$. A type 1 error is impossible, so the significance level is 0. On the other hand, the probability of a type 2 error is 1 for any distribution defined by $H_1$. At the other extreme, consider the decision rule in which we always rejects $H_0$ regardless of the evidence $\bs{x}$. This corresponds to the rejection region $R = S$. A type 2 error is impossible, but now the probability of a type 1 error is 1 for any distribution defined by $H_0$. In between these two worthless tests are meaningful tests that take the evidence $\bs{x}$ into account.

If $H_1$ is true, so that the distribution of $\bs{X}$ is specified by $H_1$, then $\P(\bs{X} \in R)$, the probability of rejecting $H_0$ is the power of the test for that distribution.

Thus the power of the test for a distribution specified by $ H_1 $ is the probability of making the correct decision.

Suppose that we have two tests, corresponding to rejection regions $R_1$ and $R_2$, respectively, each having significance level $\alpha$. The test with region $R_1$ is uniformly more powerful than the test with region $R_2$ if \[ \P(\bs{X} \in R_1) \ge \P(\bs{X} \in R_2) \text{ for every distribution of } \bs{X} \text{ specified by } H_1 \]

Naturally, in this case, we would prefer the first test. Often, however, two tests will not be uniformly ordered; one test will be more powerful for some distributions specified by $H_1$ while the other test will be more powerful for other distributions specified by $H_1$.

If a test has significance level $\alpha$ and is uniformly more powerful than any other test with significance level $\alpha$, then the test is said to be a uniformly most powerful test at level $\alpha$.

Clearly a uniformly most powerful test is the best we can do.

$P$-value

In most cases, we have a general procedure that allows us to construct a test (that is, a rejection region $R_\alpha$) for any given significance level $\alpha \in (0, 1)$. Typically, $R_\alpha$ decreases (in the subset sense) as $\alpha$ decreases.

The $P$-value of the observed value $\bs{x}$ of $\bs{X}$, denoted $P(\bs{x})$, is defined to be the smallest $\alpha$ for which $\bs{x} \in R_\alpha$; that is, the smallest significance level for which $H_0$ is rejected, given $\bs{X} = \bs{x}$.

Knowing $P(\bs{x})$ allows us to test $H_0$ at any significance level for the given data $\bs{x}$: If $P(\bs{x}) \le \alpha$ then we would reject $H_0$ at significance level $\alpha$; if $P(\bs{x}) \gt \alpha$ then we fail to reject $H_0$ at significance level $\alpha$. Note that $P(\bs{X})$ is a statistic . Informally, $P(\bs{x})$ can often be thought of as the probability of an outcome as or more extreme than the observed value $\bs{x}$, where extreme is interpreted relative to the null hypothesis $H_0$.

Analogy with Justice Systems

There is a helpful analogy between statistical hypothesis testing and the criminal justice system in the US and various other countries. Consider a person charged with a crime. The presumed null hypothesis is that the person is innocent of the crime; the conjectured alternative hypothesis is that the person is guilty of the crime. The test of the hypotheses is a trial with evidence presented by both sides playing the role of the data. After considering the evidence, the jury delivers the decision as either not guilty or guilty . Note that innocent is not a possible verdict of the jury, because it is not the point of the trial to prove the person innocent. Rather, the point of the trial is to see whether there is sufficient evidence to overturn the null hypothesis that the person is innocent in favor of the alternative hypothesis of that the person is guilty. A type 1 error is convicting a person who is innocent; a type 2 error is acquitting a person who is guilty. Generally, a type 1 error is considered the more serious of the two possible errors, so in an attempt to hold the chance of a type 1 error to a very low level, the standard for conviction in serious criminal cases is beyond a reasonable doubt .

Tests of an Unknown Parameter

Hypothesis testing is a very general concept, but an important special class occurs when the distribution of the data variable $\bs{X}$ depends on a parameter $\theta$ taking values in a parameter space $\Theta$. The parameter may be vector-valued, so that $\bs{\theta} = (\theta_1, \theta_2, \ldots, \theta_n)$ and $\Theta \subseteq \R^k$ for some $k \in \N_+$. The hypotheses generally take the form \[ H_0: \theta \in \Theta_0 \text{ versus } H_1: \theta \notin \Theta_0 \] where $\Theta_0$ is a prescribed subset of the parameter space $\Theta$. In this setting, the probabilities of making an error or a correct decision depend on the true value of $\theta$. If $R$ is the rejection region, then the power function $ Q $ is given by \[ Q(\theta) = \P_\theta(\bs{X} \in R), \quad \theta \in \Theta \] The power function gives a lot of information about the test.

The power function satisfies the following properties:

$Q(\theta)$ is the probability of a type 1 error when $\theta \in \Theta_0$.
$\max\left\{Q(\theta): \theta \in \Theta_0\right\}$ is the significance level of the test.
$1 - Q(\theta)$ is the probability of a type 2 error when $\theta \notin \Theta_0$.
$Q(\theta)$ is the power of the test when $\theta \notin \Theta_0$.

If we have two tests, we can compare them by means of their power functions.

Suppose that we have two tests, corresponding to rejection regions $R_1$ and $R_2$, respectively, each having significance level $\alpha$. The test with rejection region $R_1$ is uniformly more powerful than the test with rejection region $R_2$ if $ Q_1(\theta) \ge Q_2(\theta)$ for all $ \theta \notin \Theta_0 $.

Most hypothesis tests of an unknown real parameter $\theta$ fall into three special cases:

Suppose that $ \theta $ is a real parameter and $ \theta_0 \in \Theta $ a specified value. The tests below are respectively the two-sided test , the left-tailed test , and the right-tailed test .

$H_0: \theta = \theta_0$ versus $H_1: \theta \ne \theta_0$
$H_0: \theta \ge \theta_0$ versus $H_1: \theta \lt \theta_0$
$H_0: \theta \le \theta_0$ versus $H_1: \theta \gt \theta_0$

Thus the tests are named after the conjectured alternative. Of course, there may be other unknown parameters besides $\theta$ (known as nuisance parameters ).

Equivalence Between Hypothesis Test and Confidence Sets

There is an equivalence between hypothesis tests and confidence sets for a parameter $\theta$.

Suppose that $C(\bs{x})$ is a $1 - \alpha$ level confidence set for $\theta$. The following test has significance level $\alpha$ for the hypothesis $ H_0: \theta = \theta_0 $ versus $ H_1: \theta \ne \theta_0 $: Reject $H_0$ if and only if $\theta_0 \notin C(\bs{x})$

By definition, $\P[\theta \in C(\bs{X})] = 1 - \alpha$. Hence if $H_0$ is true so that $\theta = \theta_0$, then the probability of a type 1 error is $P[\theta \notin C(\bs{X})] = \alpha$.

Equivalently, we fail to reject $H_0$ at significance level $\alpha$ if and only if $\theta_0$ is in the corresponding $1 - \alpha$ level confidence set. In particular, this equivalence applies to interval estimates of a real parameter $\theta$ and the common tests for $\theta$ given above .

In each case below, the confidence interval has confidence level $1 - \alpha$ and the test has significance level $\alpha$.

Suppose that $\left[L(\bs{X}, U(\bs{X})\right]$ is a two-sided confidence interval for $\theta$. Reject $H_0: \theta = \theta_0$ versus $H_1: \theta \ne \theta_0$ if and only if $\theta_0 \lt L(\bs{X})$ or $\theta_0 \gt U(\bs{X})$.
Suppose that $L(\bs{X})$ is a confidence lower bound for $\theta$. Reject $H_0: \theta \le \theta_0$ versus $H_1: \theta \gt \theta_0$ if and only if $\theta_0 \lt L(\bs{X})$.
Suppose that $U(\bs{X})$ is a confidence upper bound for $\theta$. Reject $H_0: \theta \ge \theta_0$ versus $H_1: \theta \lt \theta_0$ if and only if $\theta_0 \gt U(\bs{X})$.

Pivot Variables and Test Statistics

Recall that confidence sets of an unknown parameter $\theta$ are often constructed through a pivot variable , that is, a random variable $W(\bs{X}, \theta)$ that depends on the data vector $\bs{X}$ and the parameter $\theta$, but whose distribution does not depend on $\theta$ and is known. In this case, a natural test statistic for the basic tests given above is $W(\bs{X}, \theta_0)$.

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Here’s why we care about attempts to prove the riemann hypothesis.

The latest effort shines a spotlight on an enduring prime numbers mystery

LINED UP The Riemann zeta function has an infinite number of points where the function’s value is zero, located at the whirls of color in this plot. The Riemann hypothesis predicts that certain zeros lie along a single line, which is horizontal in this image, where the colorful bands meet the red.

Empetrisor/Wikimedia Commons ( CC BY-SA 4.0 )

What is the Riemann hypothesis?

The Riemann hypothesis is a statement about a mathematical curiosity known as the Riemann zeta function. That function is closely entwined with prime numbers — whole numbers that are evenly divisible only by 1 and themselves. Prime numbers are mysterious: They are scattered in an inscrutable pattern across the number line, making it difficult to predict where each prime number will fall ( SN Online: 4/2/08 ).

But if the Riemann zeta function meets a certain condition, Riemann realized, it would reveal secrets of the prime numbers, such as how many primes exist below a given number. That required condition is the Riemann hypothesis. It conjectures that certain zeros of the function — the points where the function’s value equals zero — all lie along a particular line when plotted ( SN: 9/27/08, p. 14 ). If the hypothesis is confirmed, it could help expose a method to the primes’ madness.

Why is it so important?

Prime numbers are mathematical VIPs: Like atoms of the periodic table, they are the building blocks for larger numbers. Primes matter for practical purposes, too, as they are important for securing encrypted transmissions sent over the internet. And importantly, a multitude of mathematical papers take the Riemann hypothesis as a given. If this foundational assumption were proved correct, “many results that are believed to be true will be known to be true,” says mathematician Ken Ono of Emory University in Atlanta. “It’s a kind of mathematical oracle.”

Haven’t people tried to prove this before?

Yep. It’s difficult to count the number of attempts, but probably hundreds of researchers have tried their hands at a proof. So far none of the proofs have stood up to scrutiny. The problem is so stubborn that it now has a bounty on its head : The Clay Mathematics Institute has offered up $1 million to anyone who can prove the Riemann hypothesis.

Why is it so difficult to prove?

The Riemann zeta function is a difficult beast to work with. Even defining it is a challenge, Ono says. Furthermore, the function has an infinite number of zeros. If any one of those zeros is not on its expected line, the Riemann hypothesis is wrong. And since there are infinite zeros, manually checking each one won’t work. Instead, a proof must show without a doubt that no zero can be an outlier. For difficult mathematical quandaries like the Riemann hypothesis, the bar for acceptance of a proof is extremely high. Verification of such a proof typically requires months or even years of double-checking by other mathematicians before either everyone is convinced, or the proof is deemed flawed.

What will it take to prove the Riemann hypothesis?

Various mathematicians have made some amount of headway toward a proof. Ono likens it to attempting to climb Mount Everest and making it to base camp. While some clever mathematician may eventually be able to finish that climb, Ono says, “there is this belief that the ultimate proof … if one ever is made, will require a different level of mathematics.”

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AP®︎/College Statistics

Course: ap®︎/college statistics > unit 10.

Idea behind hypothesis testing

Examples of null and alternative hypotheses

Writing null and alternative hypotheses
P-values and significance tests
Comparing P-values to different significance levels
Estimating a P-value from a simulation
Estimating P-values from simulations
Using P-values to make conclusions

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Video transcript

Module 9: Hypothesis Testing With One Sample

Null and alternative hypotheses, learning outcomes.

Describe hypothesis testing in general and in practice

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

H 0 : The null hypothesis: It is a statement about the population that either is believed to be true or is used to put forth an argument unless it can be shown to be incorrect beyond a reasonable doubt.

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 .

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 adecision. There are two options for a decision . They are “reject H 0 ” if the sample information favors the alternative hypothesis or “do not reject H 0 ” or “decline to reject H 0 ” if the sample information is insufficient to reject the null hypothesis.

Mathematical Symbols Used in H 0 and H a :

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

H 0 : No more than 30% of the registered voters in Santa Clara County voted in the primary election. p ≤ 30

H a : More than 30% of the registered voters in Santa Clara County voted in the primary election. p > 30

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

H 0 : The drug reduces cholesterol by 25%. p = 0.25

H a : The drug does not reduce cholesterol by 25%. p ≠ 0.25

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

H 0 : μ = 2.0

H a : μ ≠ 2.0

We want to test whether the mean height of eighth graders is 66 inches. State the null and alternative hypotheses. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses. H 0 : μ __ 66 H a : μ __ 66

H 0 : μ = 66
H a : μ ≠ 66

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

H 0 : μ ≥ 5

H a : μ < 5

We want to test if it takes fewer than 45 minutes to teach a lesson plan. State the null and alternative hypotheses. Fill in the correct symbol ( =, ≠, ≥, <, ≤, >) for the null and alternative hypotheses. H 0 : μ __ 45 H a : μ __ 45

H 0 : μ ≥ 45
H a : μ < 45

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

H 0 : p ≤ 0.066

H a : p > 0.066

On a state driver’s test, about 40% pass the test on the first try. We want to test if more than 40% pass on the first try. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses. H 0 : p __ 0.40 H a : p __ 0.40

H 0 : p = 0.40
H a : p > 0.40

Concept Review

In a hypothesis test , sample data is evaluated in order to arrive at a decision about some type of claim. If certain conditions about the sample are satisfied, then the claim can be evaluated for a population. In a hypothesis test, we: Evaluate the null hypothesis , typically denoted with H 0 . The null is not rejected unless the hypothesis test shows otherwise. The null statement must always contain some form of equality (=, ≤ or ≥) Always write the alternative hypothesis , typically denoted with H a or H 1 , using less than, greater than, or not equals symbols, i.e., (≠, >, or <). If we reject the null hypothesis, then we can assume there is enough evidence to support the alternative hypothesis. Never state that a claim is proven true or false. Keep in mind the underlying fact that hypothesis testing is based on probability laws; therefore, we can talk only in terms of non-absolute certainties.

Formula Review

H 0 and H a are contradictory.

OpenStax, Statistics, Null and Alternative Hypotheses. Provided by : OpenStax. Located at : http://cnx.org/contents/[email protected]:58/Introductory_Statistics . License : CC BY: Attribution
Introductory Statistics . Authored by : Barbara Illowski, Susan Dean. Provided by : Open Stax. Located at : http://cnx.org/contents/[email protected] . License : CC BY: Attribution . License Terms : Download for free at http://cnx.org/contents/[email protected]
Simple hypothesis testing | Probability and Statistics | Khan Academy. Authored by : Khan Academy. Located at : https://youtu.be/5D1gV37bKXY . License : All Rights Reserved . License Terms : Standard YouTube License

What Is a Hypothesis? (Science)

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A hypothesis (plural hypotheses) is a proposed explanation for an observation. The definition depends on the subject.

In science, a hypothesis is part of the scientific method. It is a prediction or explanation that is tested by an experiment. Observations and experiments may disprove a scientific hypothesis, but can never entirely prove one.

In the study of logic, a hypothesis is an if-then proposition, typically written in the form, "If X , then Y ."

In common usage, a hypothesis is simply a proposed explanation or prediction, which may or may not be tested.

Writing a Hypothesis

Most scientific hypotheses are proposed in the if-then format because it's easy to design an experiment to see whether or not a cause and effect relationship exists between the independent variable and the dependent variable . The hypothesis is written as a prediction of the outcome of the experiment.

Null Hypothesis and Alternative Hypothesis

Statistically, it's easier to show there is no relationship between two variables than to support their connection. So, scientists often propose the null hypothesis . The null hypothesis assumes changing the independent variable will have no effect on the dependent variable.

In contrast, the alternative hypothesis suggests changing the independent variable will have an effect on the dependent variable. Designing an experiment to test this hypothesis can be trickier because there are many ways to state an alternative hypothesis.

For example, consider a possible relationship between getting a good night's sleep and getting good grades. The null hypothesis might be stated: "The number of hours of sleep students get is unrelated to their grades" or "There is no correlation between hours of sleep and grades."

An experiment to test this hypothesis might involve collecting data, recording average hours of sleep for each student and grades. If a student who gets eight hours of sleep generally does better than students who get four hours of sleep or 10 hours of sleep, the hypothesis might be rejected.

But the alternative hypothesis is harder to propose and test. The most general statement would be: "The amount of sleep students get affects their grades." The hypothesis might also be stated as "If you get more sleep, your grades will improve" or "Students who get nine hours of sleep have better grades than those who get more or less sleep."

In an experiment, you can collect the same data, but the statistical analysis is less likely to give you a high confidence limit.

Usually, a scientist starts out with the null hypothesis. From there, it may be possible to propose and test an alternative hypothesis, to narrow down the relationship between the variables.

Example of a Hypothesis

Examples of a hypothesis include:

If you drop a rock and a feather, (then) they will fall at the same rate.
Plants need sunlight in order to live. (if sunlight, then life)
Eating sugar gives you energy. (if sugar, then energy)
White, Jay D. Research in Public Administration . Conn., 1998.
Schick, Theodore, and Lewis Vaughn. How to Think about Weird Things: Critical Thinking for a New Age . McGraw-Hill Higher Education, 2002.
Null Hypothesis Definition and Examples
Definition of a Hypothesis
What Are the Elements of a Good Hypothesis?
Six Steps of the Scientific Method
What Are Examples of a Hypothesis?
Understanding Simple vs Controlled Experiments
Scientific Method Flow Chart
Scientific Method Vocabulary Terms
What Is a Testable Hypothesis?
Null Hypothesis Examples
What 'Fail to Reject' Means in a Hypothesis Test
How To Design a Science Fair Experiment
What Is an Experiment? Definition and Design
Hypothesis Test for the Difference of Two Population Proportions
How to Conduct a Hypothesis Test

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8.1: The Elements of Hypothesis Testing

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Learning Objectives

To understand the logical framework of tests of hypotheses.
To learn basic terminology connected with hypothesis testing.
To learn fundamental facts about hypothesis testing.

Types of Hypotheses

A hypothesis about the value of a population parameter is an assertion about its value. As in the introductory example we will be concerned with testing the truth of two competing hypotheses, only one of which can be true.

Definition: null hypothesis and alternative hypothesis

The null hypothesis , denoted $H_0$, is the statement about the population parameter that is assumed to be true unless there is convincing evidence to the contrary.
The alternative hypothesis , denoted $H_a$, is a statement about the population parameter that is contradictory to the null hypothesis, and is accepted as true only if there is convincing evidence in favor of it.

Definition: statistical procedure

Hypothesis testing is a statistical procedure in which a choice is made between a null hypothesis and an alternative hypothesis based on information in a sample.

The end result of a hypotheses testing procedure is a choice of one of the following two possible conclusions:

Reject $H_0$ (and therefore accept $H_a$), or
Fail to reject $H_0$ (and therefore fail to accept $H_a$).

The null hypothesis typically represents the status quo, or what has historically been true. In the example of the respirators, we would believe the claim of the manufacturer unless there is reason not to do so, so the null hypotheses is $H_0:\mu =75$. The alternative hypothesis in the example is the contradictory statement $H_a:\mu <75$. The null hypothesis will always be an assertion containing an equals sign, but depending on the situation the alternative hypothesis can have any one of three forms: with the symbol $<$, as in the example just discussed, with the symbol $>$, or with the symbol $\neq$. The following two examples illustrate the latter two cases.

Example $\PageIndex{1}$

A publisher of college textbooks claims that the average price of all hardbound college textbooks is $\$127.50$. A student group believes that the actual mean is higher and wishes to test their belief. State the relevant null and alternative hypotheses.

The default option is to accept the publisher’s claim unless there is compelling evidence to the contrary. Thus the null hypothesis is $H_0:\mu =127.50$. Since the student group thinks that the average textbook price is greater than the publisher’s figure, the alternative hypothesis in this situation is $H_a:\mu >127.50$.

Example $\PageIndex{2}$

The recipe for a bakery item is designed to result in a product that contains $8$ grams of fat per serving. The quality control department samples the product periodically to insure that the production process is working as designed. State the relevant null and alternative hypotheses.

The default option is to assume that the product contains the amount of fat it was formulated to contain unless there is compelling evidence to the contrary. Thus the null hypothesis is $H_0:\mu =8.0$. Since to contain either more fat than desired or to contain less fat than desired are both an indication of a faulty production process, the alternative hypothesis in this situation is that the mean is different from $8.0$, so $H_a:\mu \neq 8.0$.

In Example $\PageIndex{1}$, the textbook example, it might seem more natural that the publisher’s claim be that the average price is at most $\$127.50$, not exactly $\$127.50$. If the claim were made this way, then the null hypothesis would be $H_0:\mu \leq 127.50$, and the value $\$127.50$ given in the example would be the one that is least favorable to the publisher’s claim, the null hypothesis. It is always true that if the null hypothesis is retained for its least favorable value, then it is retained for every other value.

Thus in order to make the null and alternative hypotheses easy for the student to distinguish, in every example and problem in this text we will always present one of the two competing claims about the value of a parameter with an equality. The claim expressed with an equality is the null hypothesis. This is the same as always stating the null hypothesis in the least favorable light. So in the introductory example about the respirators, we stated the manufacturer’s claim as “the average is $75$ minutes” instead of the perhaps more natural “the average is at least $75$ minutes,” essentially reducing the presentation of the null hypothesis to its worst case.

The first step in hypothesis testing is to identify the null and alternative hypotheses.

The Logic of Hypothesis Testing

Although we will study hypothesis testing in situations other than for a single population mean (for example, for a population proportion instead of a mean or in comparing the means of two different populations), in this section the discussion will always be given in terms of a single population mean $\mu$.

The null hypothesis always has the form $H_0:\mu =\mu _0$ for a specific number $\mu _0$ (in the respirator example $\mu _0=75$, in the textbook example $\mu _0=127.50$, and in the baked goods example $\mu _0=8.0$). Since the null hypothesis is accepted unless there is strong evidence to the contrary, the test procedure is based on the initial assumption that $H_0$ is true. This point is so important that we will repeat it in a display:

The test procedure is based on the initial assumption that $H_0$ is true.

The criterion for judging between $H_0$ and $H_a$ based on the sample data is: if the value of $\overline{X}$ would be highly unlikely to occur if $H_0$ were true, but favors the truth of $H_a$, then we reject $H_0$ in favor of $H_a$. Otherwise we do not reject $H_0$.

Supposing for now that $\overline{X}$ follows a normal distribution, when the null hypothesis is true the density function for the sample mean $\overline{X}$ must be as in Figure $\PageIndex{1}$: a bell curve centered at $\mu _0$. Thus if $H_0$ is true then $\overline{X}$ is likely to take a value near $\mu _0$ and is unlikely to take values far away. Our decision procedure therefore reduces simply to:

if $H_a$ has the form $H_a:\mu <\mu _0$ then reject $H_0$ if $\bar{x}$ is far to the left of $\mu _0$;
if $H_a$ has the form $H_a:\mu >\mu _0$ then reject $H_0$ if $\bar{x}$ is far to the right of $\mu _0$;
if $H_a$ has the form $H_a:\mu \neq \mu _0$ then reject $H_0$ if $\bar{x}$ is far away from $\mu _0$ in either direction.

Think of the respirator example, for which the null hypothesis is $H_0:\mu =75$, the claim that the average time air is delivered for all respirators is $75$ minutes. If the sample mean is $75$ or greater then we certainly would not reject $H_0$ (since there is no issue with an emergency respirator delivering air even longer than claimed).

If the sample mean is slightly less than $75$ then we would logically attribute the difference to sampling error and also not reject $H_0$ either.

Values of the sample mean that are smaller and smaller are less and less likely to come from a population for which the population mean is $75$. Thus if the sample mean is far less than $75$, say around $60$ minutes or less, then we would certainly reject $H_0$, because we know that it is highly unlikely that the average of a sample would be so low if the population mean were $75$. This is the rare event criterion for rejection: what we actually observed $(\overline{X}<60)$ would be so rare an event if $\mu =75$ were true that we regard it as much more likely that the alternative hypothesis $\mu <75$ holds.

In summary, to decide between $H_0$ and $H_a$ in this example we would select a “rejection region” of values sufficiently far to the left of $75$, based on the rare event criterion, and reject $H_0$ if the sample mean $\overline{X}$ lies in the rejection region, but not reject $H_0$ if it does not.

The Rejection Region

Each different form of the alternative hypothesis Ha has its own kind of rejection region:

if (as in the respirator example) $H_a$ has the form $H_a:\mu <\mu _0$, we reject $H_0$ if $\bar{x}$ is far to the left of $\mu _0$, that is, to the left of some number $C$, so the rejection region has the form of an interval $(-\infty ,C]$;
if (as in the textbook example) $H_a$ has the form $H_a:\mu >\mu _0$, we reject $H_0$ if $\bar{x}$ is far to the right of $\mu _0$, that is, to the right of some number $C$, so the rejection region has the form of an interval $[C,\infty )$;
if (as in the baked good example) $H_a$ has the form $H_a:\mu \neq \mu _0$, we reject $H_0$ if $\bar{x}$ is far away from $\mu _0$ in either direction, that is, either to the left of some number $C$ or to the right of some other number $C′$, so the rejection region has the form of the union of two intervals $(-\infty ,C]\cup [C',\infty )$.

The key issue in our line of reasoning is the question of how to determine the number $C$ or numbers $C$ and $C′$, called the critical value or critical values of the statistic, that determine the rejection region.

Definition: critical values

The critical value or critical values of a test of hypotheses are the number or numbers that determine the rejection region.

Suppose the rejection region is a single interval, so we need to select a single number $C$. Here is the procedure for doing so. We select a small probability, denoted $\alpha$, say $1\%$, which we take as our definition of “rare event:” an event is “rare” if its probability of occurrence is less than $\alpha$. (In all the examples and problems in this text the value of $\alpha$ will be given already.) The probability that $\overline{X}$ takes a value in an interval is the area under its density curve and above that interval, so as shown in Figure $\PageIndex{2}$ (drawn under the assumption that $H_0$ is true, so that the curve centers at $\mu _0$) the critical value $C$ is the value of $\overline{X}$ that cuts off a tail area $\alpha$ in the probability density curve of $\overline{X}$. When the rejection region is in two pieces, that is, composed of two intervals, the total area above both of them must be $\alpha$, so the area above each one is $\alpha /2$, as also shown in Figure $\PageIndex{2}$.

The number $\alpha$ is the total area of a tail or a pair of tails.

Example $\PageIndex{3}$

In the context of Example $\PageIndex{2}$, suppose that it is known that the population is normally distributed with standard deviation $\alpha =0.15$ gram, and suppose that the test of hypotheses $H_0:\mu =8.0$ versus $H_a:\mu \neq 8.0$ will be performed with a sample of size $5$. Construct the rejection region for the test for the choice $\alpha =0.10$. Explain the decision procedure and interpret it.

If $H_0$ is true then the sample mean $\overline{X}$ is normally distributed with mean and standard deviation

\[\begin{align} \mu _{\overline{X}} &=\mu \nonumber \\[5pt] &=8.0 \nonumber \end{align} \nonumber \]

\[\begin{align} \sigma _{\overline{X}}&=\dfrac{\sigma}{\sqrt{n}} \nonumber \\[5pt] &= \dfrac{0.15}{\sqrt{5}} \nonumber\\[5pt] &=0.067 \nonumber \end{align} \nonumber \]

Since $H_a$ contains the $\neq$ symbol the rejection region will be in two pieces, each one corresponding to a tail of area $\alpha /2=0.10/2=0.05$. From Figure 7.1.6, $z_{0.05}=1.645$, so $C$ and $C′$ are $1.645$ standard deviations of $\overline{X}$ to the right and left of its mean $8.0$:

\[C=8.0-(1.645)(0.067) = 7.89 \; \; \text{and}\; \; C'=8.0 + (1.645)(0.067) = 8.11 \nonumber \]

The result is shown in Figure $\PageIndex{3}$. α = 0.1

The decision procedure is: take a sample of size $5$ and compute the sample mean $\bar{x}$. If $\bar{x}$ is either $7.89$ grams or less or $8.11$ grams or more then reject the hypothesis that the average amount of fat in all servings of the product is $8.0$ grams in favor of the alternative that it is different from $8.0$ grams. Otherwise do not reject the hypothesis that the average amount is $8.0$ grams.

The reasoning is that if the true average amount of fat per serving were $8.0$ grams then there would be less than a $10\%$ chance that a sample of size $5$ would produce a mean of either $7.89$ grams or less or $8.11$ grams or more. Hence if that happened it would be more likely that the value $8.0$ is incorrect (always assuming that the population standard deviation is $0.15$ gram).

Because the rejection regions are computed based on areas in tails of distributions, as shown in Figure $\PageIndex{2}$, hypothesis tests are classified according to the form of the alternative hypothesis in the following way.

Definitions: Test classifications

If $H_a$ has the form $\mu \neq \mu _0$ the test is called a two-tailed test .
If $H_a$ has the form $\mu < \mu _0$ the test is called a left-tailed test .
If $H_a$ has the form $\mu > \mu _0$the test is called a right-tailed test .

Each of the last two forms is also called a one-tailed test .

Two Types of Errors

The format of the testing procedure in general terms is to take a sample and use the information it contains to come to a decision about the two hypotheses. As stated before our decision will always be either

reject the null hypothesis $H_0$ in favor of the alternative $H_a$ presented, or
do not reject the null hypothesis $H_0$ in favor of the alternative $H_0$ presented.

There are four possible outcomes of hypothesis testing procedure, as shown in the following table:

As the table shows, there are two ways to be right and two ways to be wrong. Typically to reject $H_0$ when it is actually true is a more serious error than to fail to reject it when it is false, so the former error is labeled “ Type I ” and the latter error “ Type II ”.

Definition: Type I and Type II errors

In a test of hypotheses:

A Type I error is the decision to reject $H_0$ when it is in fact true.
A Type II error is the decision not to reject $H_0$ when it is in fact not true.

Unless we perform a census we do not have certain knowledge, so we do not know whether our decision matches the true state of nature or if we have made an error. We reject $H_0$ if what we observe would be a “rare” event if $H_0$ were true. But rare events are not impossible: they occur with probability $\alpha$. Thus when $H_0$ is true, a rare event will be observed in the proportion $\alpha$ of repeated similar tests, and $H_0$ will be erroneously rejected in those tests. Thus $\alpha$ is the probability that in following the testing procedure to decide between $H_0$ and $H_a$ we will make a Type I error.

Definition: level of significance

The number $\alpha$ that is used to determine the rejection region is called the level of significance of the test. It is the probability that the test procedure will result in a Type I error .

The probability of making a Type II error is too complicated to discuss in a beginning text, so we will say no more about it than this: for a fixed sample size, choosing $alpha$ smaller in order to reduce the chance of making a Type I error has the effect of increasing the chance of making a Type II error . The only way to simultaneously reduce the chances of making either kind of error is to increase the sample size.

Standardizing the Test Statistic

Hypotheses testing will be considered in a number of contexts, and great unification as well as simplification results when the relevant sample statistic is standardized by subtracting its mean from it and then dividing by its standard deviation. The resulting statistic is called a standardized test statistic . In every situation treated in this and the following two chapters the standardized test statistic will have either the standard normal distribution or Student’s $t$-distribution.

Definition: hypothesis test

A standardized test statistic for a hypothesis test is the statistic that is formed by subtracting from the statistic of interest its mean and dividing by its standard deviation.

For example, reviewing Example $\PageIndex{3}$, if instead of working with the sample mean $\overline{X}$ we instead work with the test statistic

\[\frac{\overline{X}-8.0}{0.067} \nonumber \]

then the distribution involved is standard normal and the critical values are just $\pm z_{0.05}$. The extra work that was done to find that $C=7.89$ and $C′=8.11$ is eliminated. In every hypothesis test in this book the standardized test statistic will be governed by either the standard normal distribution or Student’s $t$-distribution. Information about rejection regions is summarized in the following tables:

Every instance of hypothesis testing discussed in this and the following two chapters will have a rejection region like one of the six forms tabulated in the tables above.

No matter what the context a test of hypotheses can always be performed by applying the following systematic procedure, which will be illustrated in the examples in the succeeding sections.

Systematic Hypothesis Testing Procedure: Critical Value Approach

Identify the null and alternative hypotheses.
Identify the relevant test statistic and its distribution.
Compute from the data the value of the test statistic.
Construct the rejection region.
Compare the value computed in Step 3 to the rejection region constructed in Step 4 and make a decision. Formulate the decision in the context of the problem, if applicable.

The procedure that we have outlined in this section is called the “Critical Value Approach” to hypothesis testing to distinguish it from an alternative but equivalent approach that will be introduced at the end of Section 8.3.

Key Takeaway

A test of hypotheses is a statistical process for deciding between two competing assertions about a population parameter.
The testing procedure is formalized in a five-step procedure.

Hypothesis Testing

Hypothesis testing is a tool for making statistical inferences about the population data. It is an analysis tool that tests assumptions and determines how likely something is within a given standard of accuracy. Hypothesis testing provides a way to verify whether the results of an experiment are valid.

A null hypothesis and an alternative hypothesis are set up before performing the hypothesis testing. This helps to arrive at a conclusion regarding the sample obtained from the population. In this article, we will learn more about hypothesis testing, its types, steps to perform the testing, and associated examples.

What is Hypothesis Testing in Statistics?

Hypothesis testing uses sample data from the population to draw useful conclusions regarding the population probability distribution . It tests an assumption made about the data using different types of hypothesis testing methodologies. The hypothesis testing results in either rejecting or not rejecting the null hypothesis.

Hypothesis Testing Definition

Hypothesis testing can be defined as a statistical tool that is used to identify if the results of an experiment are meaningful or not. It involves setting up a null hypothesis and an alternative hypothesis. These two hypotheses will always be mutually exclusive. This means that if the null hypothesis is true then the alternative hypothesis is false and vice versa. An example of hypothesis testing is setting up a test to check if a new medicine works on a disease in a more efficient manner.

Null Hypothesis

The null hypothesis is a concise mathematical statement that is used to indicate that there is no difference between two possibilities. In other words, there is no difference between certain characteristics of data. This hypothesis assumes that the outcomes of an experiment are based on chance alone. It is denoted as $H_{0}$. Hypothesis testing is used to conclude if the null hypothesis can be rejected or not. Suppose an experiment is conducted to check if girls are shorter than boys at the age of 5. The null hypothesis will say that they are the same height.

Alternative Hypothesis

The alternative hypothesis is an alternative to the null hypothesis. It is used to show that the observations of an experiment are due to some real effect. It indicates that there is a statistical significance between two possible outcomes and can be denoted as $H_{1}$ or $H_{a}$. For the above-mentioned example, the alternative hypothesis would be that girls are shorter than boys at the age of 5.

Hypothesis Testing P Value

In hypothesis testing, the p value is used to indicate whether the results obtained after conducting a test are statistically significant or not. It also indicates the probability of making an error in rejecting or not rejecting the null hypothesis.This value is always a number between 0 and 1. The p value is compared to an alpha level, $\alpha$ or significance level. The alpha level can be defined as the acceptable risk of incorrectly rejecting the null hypothesis. The alpha level is usually chosen between 1% to 5%.

Hypothesis Testing Critical region

All sets of values that lead to rejecting the null hypothesis lie in the critical region. Furthermore, the value that separates the critical region from the non-critical region is known as the critical value.

Hypothesis Testing Formula

Depending upon the type of data available and the size, different types of hypothesis testing are used to determine whether the null hypothesis can be rejected or not. The hypothesis testing formula for some important test statistics are given below:

z = $\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}$. $\overline{x}$ is the sample mean, $\mu$ is the population mean, $\sigma$ is the population standard deviation and n is the size of the sample.
t = $\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}$. s is the sample standard deviation.
$\chi ^{2} = \sum \frac{(O_{i}-E_{i})^{2}}{E_{i}}$. $O_{i}$ is the observed value and $E_{i}$ is the expected value.

We will learn more about these test statistics in the upcoming section.

Types of Hypothesis Testing

Selecting the correct test for performing hypothesis testing can be confusing. These tests are used to determine a test statistic on the basis of which the null hypothesis can either be rejected or not rejected. Some of the important tests used for hypothesis testing are given below.

Hypothesis Testing Z Test

A z test is a way of hypothesis testing that is used for a large sample size (n ≥ 30). It is used to determine whether there is a difference between the population mean and the sample mean when the population standard deviation is known. It can also be used to compare the mean of two samples. It is used to compute the z test statistic. The formulas are given as follows:

One sample: z = $\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}$.
Two samples: z = $\frac{(\overline{x_{1}}-\overline{x_{2}})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}}$.

Hypothesis Testing t Test

The t test is another method of hypothesis testing that is used for a small sample size (n < 30). It is also used to compare the sample mean and population mean. However, the population standard deviation is not known. Instead, the sample standard deviation is known. The mean of two samples can also be compared using the t test.

One sample: t = $\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}$.
Two samples: t = $\frac{(\overline{x_{1}}-\overline{x_{2}})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}}}$.

Hypothesis Testing Chi Square

The Chi square test is a hypothesis testing method that is used to check whether the variables in a population are independent or not. It is used when the test statistic is chi-squared distributed.

One Tailed Hypothesis Testing

One tailed hypothesis testing is done when the rejection region is only in one direction. It can also be known as directional hypothesis testing because the effects can be tested in one direction only. This type of testing is further classified into the right tailed test and left tailed test.

Right Tailed Hypothesis Testing

The right tail test is also known as the upper tail test. This test is used to check whether the population parameter is greater than some value. The null and alternative hypotheses for this test are given as follows:

$H_{0}$: The population parameter is ≤ some value

$H_{1}$: The population parameter is > some value.

If the test statistic has a greater value than the critical value then the null hypothesis is rejected

Left Tailed Hypothesis Testing

The left tail test is also known as the lower tail test. It is used to check whether the population parameter is less than some value. The hypotheses for this hypothesis testing can be written as follows:

$H_{0}$: The population parameter is ≥ some value

$H_{1}$: The population parameter is < some value.

The null hypothesis is rejected if the test statistic has a value lesser than the critical value.

Two Tailed Hypothesis Testing

In this hypothesis testing method, the critical region lies on both sides of the sampling distribution. It is also known as a non - directional hypothesis testing method. The two-tailed test is used when it needs to be determined if the population parameter is assumed to be different than some value. The hypotheses can be set up as follows:

$H_{0}$: the population parameter = some value

$H_{1}$: the population parameter ≠ some value

The null hypothesis is rejected if the test statistic has a value that is not equal to the critical value.

Hypothesis Testing Steps

Hypothesis testing can be easily performed in five simple steps. The most important step is to correctly set up the hypotheses and identify the right method for hypothesis testing. The basic steps to perform hypothesis testing are as follows:

Step 1: Set up the null hypothesis by correctly identifying whether it is the left-tailed, right-tailed, or two-tailed hypothesis testing.
Step 2: Set up the alternative hypothesis.
Step 3: Choose the correct significance level, $\alpha$, and find the critical value.
Step 4: Calculate the correct test statistic (z, t or $\chi$) and p-value.
Step 5: Compare the test statistic with the critical value or compare the p-value with $\alpha$ to arrive at a conclusion. In other words, decide if the null hypothesis is to be rejected or not.

Hypothesis Testing Example

The best way to solve a problem on hypothesis testing is by applying the 5 steps mentioned in the previous section. Suppose a researcher claims that the mean average weight of men is greater than 100kgs with a standard deviation of 15kgs. 30 men are chosen with an average weight of 112.5 Kgs. Using hypothesis testing, check if there is enough evidence to support the researcher's claim. The confidence interval is given as 95%.

Step 1: This is an example of a right-tailed test. Set up the null hypothesis as $H_{0}$: $\mu$ = 100.

Step 2: The alternative hypothesis is given by $H_{1}$: $\mu$ > 100.

Step 3: As this is a one-tailed test, $\alpha$ = 100% - 95% = 5%. This can be used to determine the critical value.

1 - $\alpha$ = 1 - 0.05 = 0.95

0.95 gives the required area under the curve. Now using a normal distribution table, the area 0.95 is at z = 1.645. A similar process can be followed for a t-test. The only additional requirement is to calculate the degrees of freedom given by n - 1.

Step 4: Calculate the z test statistic. This is because the sample size is 30. Furthermore, the sample and population means are known along with the standard deviation.

z = $\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}$.

$\mu$ = 100, $\overline{x}$ = 112.5, n = 30, $\sigma$ = 15

z = $\frac{112.5-100}{\frac{15}{\sqrt{30}}}$ = 4.56

Step 5: Conclusion. As 4.56 > 1.645 thus, the null hypothesis can be rejected.

Hypothesis Testing and Confidence Intervals

Confidence intervals form an important part of hypothesis testing. This is because the alpha level can be determined from a given confidence interval. Suppose a confidence interval is given as 95%. Subtract the confidence interval from 100%. This gives 100 - 95 = 5% or 0.05. This is the alpha value of a one-tailed hypothesis testing. To obtain the alpha value for a two-tailed hypothesis testing, divide this value by 2. This gives 0.05 / 2 = 0.025.

Probability and Statistics
Data Handling

Important Notes on Hypothesis Testing

Hypothesis testing is a technique that is used to verify whether the results of an experiment are statistically significant.
It involves the setting up of a null hypothesis and an alternate hypothesis.
There are three types of tests that can be conducted under hypothesis testing - z test, t test, and chi square test.
Hypothesis testing can be classified as right tail, left tail, and two tail tests.

Examples on Hypothesis Testing

Example 1: The average weight of a dumbbell in a gym is 90lbs. However, a physical trainer believes that the average weight might be higher. A random sample of 5 dumbbells with an average weight of 110lbs and a standard deviation of 18lbs. Using hypothesis testing check if the physical trainer's claim can be supported for a 95% confidence level. Solution: As the sample size is lesser than 30, the t-test is used. $H_{0}$: $\mu$ = 90, $H_{1}$: $\mu$ > 90 $\overline{x}$ = 110, $\mu$ = 90, n = 5, s = 18. $\alpha$ = 0.05 Using the t-distribution table, the critical value is 2.132 t = $\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}$ t = 2.484 As 2.484 > 2.132, the null hypothesis is rejected. Answer: The average weight of the dumbbells may be greater than 90lbs
Example 2: The average score on a test is 80 with a standard deviation of 10. With a new teaching curriculum introduced it is believed that this score will change. On random testing, the score of 38 students, the mean was found to be 88. With a 0.05 significance level, is there any evidence to support this claim? Solution: This is an example of two-tail hypothesis testing. The z test will be used. $H_{0}$: $\mu$ = 80, $H_{1}$: $\mu$ ≠ 80 $\overline{x}$ = 88, $\mu$ = 80, n = 36, $\sigma$ = 10. $\alpha$ = 0.05 / 2 = 0.025 The critical value using the normal distribution table is 1.96 z = $\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}$ z = $\frac{88-80}{\frac{10}{\sqrt{36}}}$ = 4.8 As 4.8 > 1.96, the null hypothesis is rejected. Answer: There is a difference in the scores after the new curriculum was introduced.
Example 3: The average score of a class is 90. However, a teacher believes that the average score might be lower. The scores of 6 students were randomly measured. The mean was 82 with a standard deviation of 18. With a 0.05 significance level use hypothesis testing to check if this claim is true. Solution: The t test will be used. $H_{0}$: $\mu$ = 90, $H_{1}$: $\mu$ < 90 $\overline{x}$ = 110, $\mu$ = 90, n = 6, s = 18 The critical value from the t table is -2.015 t = $\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}$ t = $\frac{82-90}{\frac{18}{\sqrt{6}}}$ t = -1.088 As -1.088 > -2.015, we fail to reject the null hypothesis. Answer: There is not enough evidence to support the claim.

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FAQs on Hypothesis Testing

What is hypothesis testing.

Hypothesis testing in statistics is a tool that is used to make inferences about the population data. It is also used to check if the results of an experiment are valid.

What is the z Test in Hypothesis Testing?

The z test in hypothesis testing is used to find the z test statistic for normally distributed data . The z test is used when the standard deviation of the population is known and the sample size is greater than or equal to 30.

What is the t Test in Hypothesis Testing?

The t test in hypothesis testing is used when the data follows a student t distribution . It is used when the sample size is less than 30 and standard deviation of the population is not known.

What is the formula for z test in Hypothesis Testing?

The formula for a one sample z test in hypothesis testing is z = $\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}$ and for two samples is z = $\frac{(\overline{x_{1}}-\overline{x_{2}})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}}$.

What is the p Value in Hypothesis Testing?

The p value helps to determine if the test results are statistically significant or not. In hypothesis testing, the null hypothesis can either be rejected or not rejected based on the comparison between the p value and the alpha level.

What is One Tail Hypothesis Testing?

When the rejection region is only on one side of the distribution curve then it is known as one tail hypothesis testing. The right tail test and the left tail test are two types of directional hypothesis testing.

What is the Alpha Level in Two Tail Hypothesis Testing?

To get the alpha level in a two tail hypothesis testing divide $\alpha$ by 2. This is done as there are two rejection regions in the curve.

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The majority of statements in mathematics can be written in the form: "If A, then B." For example: "If a function is differentiable, then it is continuous". In this example, the "A" part is "a function is differentiable" and the "B" part is "a function is continuous." The "A" part of the statement is called the "hypothesis", and the "B" part of the statement is called the "conclusion". Thus the hypothesis is what we must assume in order to be positive that the conclusion will hold.

Whenever you are asked to state a theorem, be sure to include the hypothesis. In order to know when you may apply the theorem, you need to know what constraints you have. So in the example above, if we know that a function is differentiable, we may assume that it is continuous. However, if we do not know that a function is differentiable, continuity may not hold. Some theorems have MANY hypotheses, some of which are written in sentences before the ultimate "if, then" statement. For example, there might be a sentence that says: "Assume n is even." which is then followed by an if,then statement. Include all hypotheses and assumptions when asked to state theorems and definitions!

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Math Article

Hypothesis Definition

In Statistics, the determination of the variation between the group of data due to true variation is done by hypothesis testing. The sample data are taken from the population parameter based on the assumptions. The hypothesis can be classified into various types. In this article, let us discuss the hypothesis definition, various types of hypothesis and the significance of hypothesis testing, which are explained in detail.

Hypothesis Definition in Statistics

In Statistics, a hypothesis is defined as a formal statement, which gives the explanation about the relationship between the two or more variables of the specified population. It helps the researcher to translate the given problem to a clear explanation for the outcome of the study. It clearly explains and predicts the expected outcome. It indicates the types of experimental design and directs the study of the research process.

Types of Hypothesis

The hypothesis can be broadly classified into different types. They are:

Simple Hypothesis

A simple hypothesis is a hypothesis that there exists a relationship between two variables. One is called a dependent variable, and the other is called an independent variable.

Complex Hypothesis

A complex hypothesis is used when there is a relationship between the existing variables. In this hypothesis, the dependent and independent variables are more than two.

Null Hypothesis

In the null hypothesis, there is no significant difference between the populations specified in the experiments, due to any experimental or sampling error. The null hypothesis is denoted by H 0 .

Alternative Hypothesis

In an alternative hypothesis, the simple observations are easily influenced by some random cause. It is denoted by the H a or H 1 .

Empirical Hypothesis

An empirical hypothesis is formed by the experiments and based on the evidence.

Statistical Hypothesis

In a statistical hypothesis, the statement should be logical or illogical, and the hypothesis is verified statistically.

Apart from these types of hypothesis, some other hypotheses are directional and non-directional hypothesis, associated hypothesis, casual hypothesis.

Characteristics of Hypothesis

The important characteristics of the hypothesis are:

The hypothesis should be short and precise
It should be specific
A hypothesis must be related to the existing body of knowledge
It should be capable of verification

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Hypothesis is a testable statement that explains what is happening or observed. It proposes the relation between the various participating variables. Hypothesis is also called Theory, Thesis, Guess, Assumption, or Suggestion. Hypothesis creates a structure that guides the search for knowledge.

In this article, we will learn what is hypothesis, its characteristics, types, and examples. We will also learn how hypothesis helps in scientific research.

What is Hypothesis?

A hypothesis is a suggested idea or plan that has little proof, meant to lead to more study. It’s mainly a smart guess or suggested answer to a problem that can be checked through study and trial. In science work, we make guesses called hypotheses to try and figure out what will happen in tests or watching. These are not sure things but rather ideas that can be proved or disproved based on real-life proofs. A good theory is clear and can be tested and found wrong if the proof doesn’t support it.

Hypothesis Meaning

A hypothesis is a proposed statement that is testable and is given for something that happens or observed.

It is made using what we already know and have seen, and it’s the basis for scientific research.
A clear guess tells us what we think will happen in an experiment or study.
It’s a testable clue that can be proven true or wrong with real-life facts and checking it out carefully.
It usually looks like a “if-then” rule, showing the expected cause and effect relationship between what’s being studied.

Characteristics of Hypothesis

Here are some key characteristics of a hypothesis:

Testable: An idea (hypothesis) should be made so it can be tested and proven true through doing experiments or watching. It should show a clear connection between things.
Specific: It needs to be easy and on target, talking about a certain part or connection between things in a study.
Falsifiable: A good guess should be able to show it’s wrong. This means there must be a chance for proof or seeing something that goes against the guess.
Logical and Rational: It should be based on things we know now or have seen, giving a reasonable reason that fits with what we already know.
Predictive: A guess often tells what to expect from an experiment or observation. It gives a guide for what someone might see if the guess is right.
Concise: It should be short and clear, showing the suggested link or explanation simply without extra confusion.
Grounded in Research: A guess is usually made from before studies, ideas or watching things. It comes from a deep understanding of what is already known in that area.
Flexible: A guess helps in the research but it needs to change or fix when new information comes up.
Relevant: It should be related to the question or problem being studied, helping to direct what the research is about.
Empirical: Hypotheses come from observations and can be tested using methods based on real-world experiences.

Sources of Hypothesis

Hypotheses can come from different places based on what you’re studying and the kind of research. Here are some common sources from which hypotheses may originate:

Existing Theories: Often, guesses come from well-known science ideas. These ideas may show connections between things or occurrences that scientists can look into more.
Observation and Experience: Watching something happen or having personal experiences can lead to guesses. We notice odd things or repeat events in everyday life and experiments. This can make us think of guesses called hypotheses.
Previous Research: Using old studies or discoveries can help come up with new ideas. Scientists might try to expand or question current findings, making guesses that further study old results.
Literature Review: Looking at books and research in a subject can help make guesses. Noticing missing parts or mismatches in previous studies might make researchers think up guesses to deal with these spots.
Problem Statement or Research Question: Often, ideas come from questions or problems in the study. Making clear what needs to be looked into can help create ideas that tackle certain parts of the issue.
Analogies or Comparisons: Making comparisons between similar things or finding connections from related areas can lead to theories. Understanding from other fields could create new guesses in a different situation.
Hunches and Speculation: Sometimes, scientists might get a gut feeling or make guesses that help create ideas to test. Though these may not have proof at first, they can be a beginning for looking deeper.
Technology and Innovations: New technology or tools might make guesses by letting us look at things that were hard to study before.
Personal Interest and Curiosity: People’s curiosity and personal interests in a topic can help create guesses. Scientists could make guesses based on their own likes or love for a subject.

Types of Hypothesis

Here are some common types of hypotheses:

Simple Hypothesis

Complex hypothesis, directional hypothesis.

Non-directional Hypothesis

Null Hypothesis (H0)

Alternative hypothesis (h1 or ha), statistical hypothesis, research hypothesis, associative hypothesis, causal hypothesis.

Simple Hypothesis guesses a connection between two things. It says that there is a connection or difference between variables, but it doesn’t tell us which way the relationship goes.

Complex Hypothesis tells us what will happen when more than two things are connected. It looks at how different things interact and may be linked together.

Directional Hypothesis says how one thing is related to another. For example, it guesses that one thing will help or hurt another thing.

Non-Directional Hypothesis

Non-Directional Hypothesis are the one that don’t say how the relationship between things will be. They just say that there is a connection, without telling which way it goes.

Null hypothesis is a statement that says there’s no connection or difference between different things. It implies that any seen impacts are because of luck or random changes in the information.

Alternative Hypothesis is different from the null hypothesis and shows that there’s a big connection or gap between variables. Scientists want to say no to the null hypothesis and choose the alternative one.

Statistical Hypotheis are used in math testing and include making ideas about what groups or bits of them look like. You aim to get information or test certain things using these top-level, common words only.

Research Hypothesis comes from the research question and tells what link is expected between things or factors. It leads the study and chooses where to look more closely.

Associative Hypotheis guesses that there is a link or connection between things without really saying it caused them. It means that when one thing changes, it is connected to another thing changing.

Causal Hypothesis are different from other ideas because they say that one thing causes another. This means there’s a cause and effect relationship between variables involved in the situation. They say that when one thing changes, it directly makes another thing change.

Hypothesis Examples

Following are the examples of hypotheses based on their types:

Simple Hypothesis Example

Studying more can help you do better on tests.
Getting more sun makes people have higher amounts of vitamin D.

Complex Hypothesis Example

How rich you are, how easy it is to get education and healthcare greatly affects the number of years people live.
A new medicine’s success relies on the amount used, how old a person is who takes it and their genes.

Directional Hypothesis Example

Drinking more sweet drinks is linked to a higher body weight score.
Too much stress makes people less productive at work.

Non-directional Hypothesis Example

Drinking caffeine can affect how well you sleep.
People often like different kinds of music based on their gender.
The average test scores of Group A and Group B are not much different.
There is no connection between using a certain fertilizer and how much it helps crops grow.

Alternative Hypothesis (Ha)

Patients on Diet A have much different cholesterol levels than those following Diet B.
Exposure to a certain type of light can change how plants grow compared to normal sunlight.
The average smarts score of kids in a certain school area is 100.
The usual time it takes to finish a job using Method A is the same as with Method B.
Having more kids go to early learning classes helps them do better in school when they get older.
Using specific ways of talking affects how much customers get involved in marketing activities.
Regular exercise helps to lower the chances of heart disease.
Going to school more can help people make more money.
Playing violent video games makes teens more likely to act aggressively.
Less clean air directly impacts breathing health in city populations.

Functions of Hypothesis

Hypotheses have many important jobs in the process of scientific research. Here are the key functions of hypotheses:

Guiding Research: Hypotheses give a clear and exact way for research. They act like guides, showing the predicted connections or results that scientists want to study.
Formulating Research Questions: Research questions often create guesses. They assist in changing big questions into particular, checkable things. They guide what the study should be focused on.
Setting Clear Objectives: Hypotheses set the goals of a study by saying what connections between variables should be found. They set the targets that scientists try to reach with their studies.
Testing Predictions: Theories guess what will happen in experiments or observations. By doing tests in a planned way, scientists can check if what they see matches the guesses made by their ideas.
Providing Structure: Theories give structure to the study process by arranging thoughts and ideas. They aid scientists in thinking about connections between things and plan experiments to match.
Focusing Investigations: Hypotheses help scientists focus on certain parts of their study question by clearly saying what they expect links or results to be. This focus makes the study work better.
Facilitating Communication: Theories help scientists talk to each other effectively. Clearly made guesses help scientists to tell others what they plan, how they will do it and the results expected. This explains things well with colleagues in a wide range of audiences.
Generating Testable Statements: A good guess can be checked, which means it can be looked at carefully or tested by doing experiments. This feature makes sure that guesses add to the real information used in science knowledge.
Promoting Objectivity: Guesses give a clear reason for study that helps guide the process while reducing personal bias. They motivate scientists to use facts and data as proofs or disprovals for their proposed answers.
Driving Scientific Progress: Making, trying out and adjusting ideas is a cycle. Even if a guess is proven right or wrong, the information learned helps to grow knowledge in one specific area.

How Hypothesis help in Scientific Research?

Researchers use hypotheses to put down their thoughts directing how the experiment would take place. Following are the steps that are involved in the scientific method:

Initiating Investigations: Hypotheses are the beginning of science research. They come from watching, knowing what’s already known or asking questions. This makes scientists make certain explanations that need to be checked with tests.
Formulating Research Questions: Ideas usually come from bigger questions in study. They help scientists make these questions more exact and testable, guiding the study’s main point.
Setting Clear Objectives: Hypotheses set the goals of a study by stating what we think will happen between different things. They set the goals that scientists want to reach by doing their studies.
Designing Experiments and Studies: Assumptions help plan experiments and watchful studies. They assist scientists in knowing what factors to measure, the techniques they will use and gather data for a proposed reason.
Testing Predictions: Ideas guess what will happen in experiments or observations. By checking these guesses carefully, scientists can see if the seen results match up with what was predicted in each hypothesis.
Analysis and Interpretation of Data: Hypotheses give us a way to study and make sense of information. Researchers look at what they found and see if it matches the guesses made in their theories. They decide if the proof backs up or disagrees with these suggested reasons why things are happening as expected.
Encouraging Objectivity: Hypotheses help make things fair by making sure scientists use facts and information to either agree or disagree with their suggested reasons. They lessen personal preferences by needing proof from experience.
Iterative Process: People either agree or disagree with guesses, but they still help the ongoing process of science. Findings from testing ideas make us ask new questions, improve those ideas and do more tests. It keeps going on in the work of science to keep learning things.

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Summary – Hypothesis

A hypothesis is a testable statement serving as an initial explanation for phenomena, based on observations, theories, or existing knowledge. It acts as a guiding light for scientific research, proposing potential relationships between variables that can be empirically tested through experiments and observations. The hypothesis must be specific, testable, falsifiable, and grounded in prior research or observation, laying out a predictive, if-then scenario that details a cause-and-effect relationship. It originates from various sources including existing theories, observations, previous research, and even personal curiosity, leading to different types, such as simple, complex, directional, non-directional, null, and alternative hypotheses, each serving distinct roles in research methodology. The hypothesis not only guides the research process by shaping objectives and designing experiments but also facilitates objective analysis and interpretation of data, ultimately driving scientific progress through a cycle of testing, validation, and refinement.

FAQs on Hypothesis

What is a hypothesis.

A guess is a possible explanation or forecast that can be checked by doing research and experiments.

What are Components of a Hypothesis?

The components of a Hypothesis are Independent Variable, Dependent Variable, Relationship between Variables, Directionality etc.

What makes a Good Hypothesis?

Testability, Falsifiability, Clarity and Precision, Relevance are some parameters that makes a Good Hypothesis

Can a Hypothesis be Proven True?

You cannot prove conclusively that most hypotheses are true because it’s generally impossible to examine all possible cases for exceptions that would disprove them.

How are Hypotheses Tested?

Hypothesis testing is used to assess the plausibility of a hypothesis by using sample data

Can Hypotheses change during Research?

Yes, you can change or improve your ideas based on new information discovered during the research process.

What is the Role of a Hypothesis in Scientific Research?

Hypotheses are used to support scientific research and bring about advancements in knowledge.

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How to Write a Strong Hypothesis | Steps & Examples

How to Write a Strong Hypothesis | Steps & Examples

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

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

Example: Hypothesis

Daily apple consumption leads to fewer doctor’s visits.

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

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

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

Variables in hypotheses

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

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

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

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

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Step 1. Ask a question

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

Step 2. Do some preliminary research

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

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

Step 3. Formulate your hypothesis

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

4. Refine your hypothesis

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

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

5. Phrase your hypothesis in three ways

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

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

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

6. Write a null hypothesis

If your research involves statistical hypothesis testing , you will also have to write a null hypothesis . The null hypothesis is the default position that there is no association between the variables. The null hypothesis is written as H 0 , while the alternative hypothesis is H 1 or H a .

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

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

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

Statistics

Null hypothesis
Statistical power
Probability distribution
Effect size
Poisson distribution

Research bias

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

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

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

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

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Understanding the Role of Hypotheses and Conclusions in Mathematical Reasoning

Hypothesis and conclusion.

In the context of mathematics and logic, a hypothesis is a statement or proposition that is assumed to be true for the purpose of a logical argument or investigation. It is usually denoted by “H” or “P” and is the starting point for many mathematical proofs.

For example, let’s consider the hypothesis: “If it is raining outside, then the ground is wet.” This statement assumes that whenever it rains, the ground will be wet.

The conclusion, on the other hand, is the statement or proposition that is inferred or reached by logical reasoning, based on the hypothesis or given information. It is typically denoted by “C” or “Q”.

Using the same example, the conclusion derived from the hypothesis could be: “It is currently raining outside, so the ground is wet.” This conclusion is based on the assumption that the given condition of rain implies a wet ground.

In mathematics, hypotheses and conclusions are commonly used in proofs and logical arguments. By stating a hypothesis and then deducing a conclusion from it, mathematicians can demonstrate the validity of certain mathematical concepts, theorems, or formulas.

It’s important to note that in mathematics, a hypothesis is not the same as a guess or a prediction. It is a statement that is assumed to be true and serves as the basis for logical reasoning, while the conclusion is the logical consequence or outcome that is drawn from the hypothesis.

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Hypothesis test

A significance test, also referred to as a statistical hypothesis test, is a method of statistical inference in which observed data is compared to a claim (referred to as a hypothesis) in order to assess the truth of the claim. For example, one might wonder whether age affects the number of apples a person can eat, and may use a significance test to determine whether there is any evidence to suggest that it does.

Generally, the process of statistical hypothesis testing involves the following steps:

State the null hypothesis.
State the alternative hypothesis.
Select the appropriate test statistic and select a significance level.
Compute the observed value of the test statistic and its corresponding p-value.
Reject the null hypothesis in favor of the alternative hypothesis, or do not reject the null hypothesis.

The null hypothesis

The null hypothesis, H 0 , is the claim that is being tested in a statistical hypothesis test. It typically is a statement that there is no difference between the populations being studied, or that there is no evidence to support a claim being made. For example, "age has no effect on the number of apples a person can eat."

A significance test is designed to test the evidence against the null hypothesis. This is because it is easier to prove that a claim is false than to prove that it is true; demonstrating that the claim is false in one case is sufficient, while proving that it is true requires that the claim be true in all cases.

The alternative hypothesis

The alternative hypothesis is the opposite of the null hypothesis in that it is a statement that there is some difference between the populations being studied. For example, "younger people can eat more apples than older people."

The alternative hypothesis is typically the hypothesis that researchers are trying to prove. A significance test is meant to determine whether there is sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis. Note that the results of a significance test should either be to reject the null hypothesis in favor of the alternative hypothesis, or to not reject the null hypothesis. The result should not be to reject the alternative hypothesis or to accept the alternative hypothesis.

Test statistics and significance level

A test statistic is a statistic that is calculated as part of hypothesis testing that compares the distribution of observed data to the expected distribution, based on the null hypothesis. Examples of test statistics include the Z-score, T-statistic, F-statistic, and the Chi-square statistic. The test statistic used is dependent on the significance test used, which is dependent on the type of data collected and the type of relationship to be tested.

In many cases, the chosen significance level is 0.05, though 0.01 is also used. A significance level of 0.05 indicates that there is a 5% chance of rejecting the null hypothesis when the null hypothesis is actually true. Thus, a smaller selected significance level will require more evidence if the null hypothesis is to be rejected in favor of the alternative hypothesis.

After the test statistic is computed, the p-value can be determined based on the result of the test statistic. The p-value indicates the probability of obtaining test results that are at least as extreme as the observed results, under the assumption that the null hypothesis is correct. It tells us how likely it is to obtain a result based solely on chance. The smaller the p-value, the less likely a result can occur purely by chance, while a larger p-value makes it more likely. For example, a p-value of 0.01 means that there is a 1% chance that a result occurred solely by chance, given that the null hypothesis is true; a p-value of 0.90 means that there is a 90% chance.

A p-value is significantly affected by sample size. The larger the sample size, the smaller the p-value, even if the difference between populations may not be meaningful. On the other hand, if a sample size is too small, a meaningful difference may not be detected.

The last step in a significance test is to determine whether the p-value provides evidence that the null hypothesis should be rejected in favor of the alternative hypothesis. This is based on the selected significance level. If the p-value is less than or equal to the selected significance level, the null hypothesis is rejected in favor of the alternative hypothesis, and the result is deemed statistically significant. If the p-value is greater than the selected significance level, the null hypothesis is not rejected, and the result is deemed not statistically significant.

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Title: disproof of the riemann hypothesis.

Abstract: The Riemann Hypothesis is a conjecture that all non-trivial zeros of Riemann Zeta function are located on the critical line in the complex plane. Hundreds of propositions in function theory and analytic number theory rely on this hypothesis. However, the problem has been unresolved for over a century. Here we show that at least one set of quadruplet-zeros exists outside the critical line through expanding the infinite product of the Riemann Xi zero function. We found that assuming there are no zeros outside the critical line will result in a contradiction with the known result that the reciprocal sum of all zeros of the xi-function is a constant, thereby refuting the Riemann Hypothesis.

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Quantum impossibility is made possible using ‘exotic' mathematics

B erislav Buca has developed a new theory that enables the calculation of the dynamics, i.e., movements and interactions, of systems with enormous quantities of quantum particles.

This feat was previously thought to be impossible. The research, published in the journal Physical Review X , has reinvigorated an old and fundamental scientific question: Can we predict everything by calculating its smallest particles ?

"Many physics disciplines are ultimately about explaining and predicting the world by understanding the laws of physics and calculating the behavior of the smallest particles," says Buca, a researcher at the University of Copenhagen's Niels Bohr Institute .

"In principle, we would be able to answer any possible question about how all sorts of things behave if we were able to," Buca noted.

Complexity of quantum particles

Despite the potential of this research, Buca quickly appeals for caution. "Of course I can't do that," says the theorist.

The interactions and movements of quantum particles in their systems are so complex that even the world's most powerful supercomputer today is only able to perform calculations on a dozen of these particles at a time.

"So in practice, it isn't possible. Not currently. However, my theory is a significant step in the right direction," Buca explained.

"This is because it takes a kind of mathematical shortcut to understanding the dynamics of the whole, without computing power being lost in the details for a broad class of systems with many quantum particles. That is, without the need to calculate all of the individual particles in a system," he continued.

Proving a long-held hypothesis

Buca's theory has already made a name for itself by providing the first mathematical proof of a long-held hypothesis in theoretical physics.

The eigenstate-thermalization hypothesis, which concerns the ability of mathematics to describe the motions of quantum systems as wholes, had been an assumption -- an educated guess -- in physics that had yet to be explained mathematically.

Quantum particles as keys to revolutionary technologies

While the results mainly interest the bright minds of physics for now, the consequences could eventually be great for us all.

This knowledge could end up showing the way to sought for quantum materials with properties so unique that they could transform our world.

"We are looking for a material for quantum computers that can withstand entropy -- a law of nature that causes complex systems -- e.g., materials -- to decay into less complex forms. Entropy destroys the coherence needed for quantum computers to be stable and keep working," Buca explains.

Simplifying quantum systems with exotic math

The exotic math systems that initially inspired Buca and made his research breakthrough possible may be just what a quantum computer needs to be truly useful.

"The so-called qubits that a quantum computer theoretically works with must be in a state of superposition to function, meaning that they are simultaneously turned on and off -- in common phrasing," Buca says.

"This requires them to be in a stable quantum state. However, thermodynamics does not like the structures required by the current materials. My theory may be able to inform us whether these exotic systems can be a way of structuring things so this quantum state could be more permanent," he concluded.

Road map for quantum particle discoveries

Buca's method is a bit like a road map that can guide researchers across a vast landscape of possible materials by allowing for predictions of how these materials would behave under experimental conditions.

For the first time, this gives researchers a way to target their search for quantum materials equipped with special properties.

"Until now, the hunt for these materials has been governed by chance. But my results can, for the first time, provide a guiding principle to navigate by when searching for unique properties in materials," says Buca.

Dawn of a potential new era in quantum physics

In summary, Berislav Buca's mind-bending theory opens up new possibilities in the field of theoretical physics and quantum computing.

By providing a mathematical shortcut to calculate the dynamics of quantum systems, Buca has proven a long-held hypothesis and paved the way for targeted searches of quantum materials with unique properties.

This research could lead to the development of stable quantum computers and room-temperature superconductors, revolutionizing our understanding of the universe and transforming our world.

As scientists continue to explore the implications of this breakthrough, we stand at the precipice of a new era in quantum physics, eagerly anticipating the exciting discoveries and innovations that lie ahead.

The full study was published in the journal Physical Review X .

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Quantum impossibility is made possible using ‘exotic' mathematics

IMAGES

A Direct Proof of the Riemann Hypothesis
How to Write a Hypothesis
How to Write a Hypothesis: The Ultimate Guide with Examples
How to Write a Strong Hypothesis in 6 Simple Steps
Hypothesis
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VIDEO

Session 8- Hypothesis testing by Non Parametric Tests (7/12/23)
lecture 9 mathematics hypothesis testing ❇✅
HYPOTHESIS TESTING PROBLEM-1 USING Z TEST VIDEO-4
Hypothesis Testing, Difference of Means, Sigma Unknown, Critical Region(Traditional) Method
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What Is A Hypothesis?

COMMENTS

Hypothesis -- from Wolfram MathWorld
A hypothesis is a proposition that is consistent with known data, but has been neither verified nor shown to be false. In statistics, a hypothesis (sometimes called a statistical hypothesis) refers to a statement on which hypothesis testing will be based. Particularly important statistical hypotheses include the null hypothesis and alternative hypothesis. In symbolic logic, a hypothesis is the ...
Hypothesis Definition (Illustrated Mathematics Dictionary)
Hypothesis. A statement that could be true, which might then be tested. Example: Sam has a hypothesis that "large dogs are better at catching tennis balls than small dogs". We can test that hypothesis by having hundreds of different sized dogs try to catch tennis balls. Sometimes the hypothesis won't be tested, it is simply a good explanation ...
Hypothesis
A hypothesis is a claim or statement that makes sense in the context of some information or data at hand but hasn't been established as true or false through experimentation or proof. In mathematics, any statement or equation that describes some relationship between certain variables can be termed as hypothesis if it is consistent with some ...
Understanding Hypotheses
A hypothesis is a statement or idea which gives an explanation to a series of observations. Sometimes, following observation, a hypothesis will clearly need to be refined or rejected. This happens if a single contradictory observation occurs. For example, suppose that a child is trying to understand the concept of a dog.
9.1: Introduction to Hypothesis Testing
In hypothesis testing, the goal is to see if there is sufficient statistical evidence to reject a presumed null hypothesis in favor of a conjectured alternative hypothesis. The null hypothesis is usually denoted H0 while the alternative hypothesis is usually denoted H1. An hypothesis test is a statistical decision; the conclusion will either be ...
Riemann hypothesis
The Riemann hypothesis has long been considered the greatest unsolved problem in mathematics.It was one of 10 unsolved mathematical problems (23 in the printed address) presented as a challenge for 20th-century mathematicians by German mathematician David Hilbert at the Second International Congress of Mathematics in Paris on Aug. 8, 1900. In 2000 American mathematician Stephen Smale updated ...
Here's why we care about attempts to prove the Riemann hypothesis
The Riemann hypothesis is a statement about a mathematical curiosity known as the Riemann zeta function. That function is closely entwined with prime numbers — whole numbers that are evenly ...
Hypothesis
A hypothesis (pl.: hypotheses) is a proposed explanation for a phenomenon. For a hypothesis to be a scientific hypothesis, ... In this sense, 'hypothesis' refers to a clever idea or to a convenient mathematical approach that simplifies cumbersome calculations.
Examples of null and alternative hypotheses
The null hypothesis is what happens at baseline. It is the uninteresting hypothesis--the boring hypothesis. Usually, it is the hypothesis that assumes no difference. It is the opposite of your research hypothesis. The alternative hypothesis--that is, the research hypothesis--is the idea, phenomenon, observation that you want to prove.
Mathematical universe hypothesis
Mathematical universe hypothesis. In physics and cosmology, the mathematical universe hypothesis ( MUH ), also known as the ultimate ensemble theory, is a speculative "theory of everything" (TOE) proposed by cosmologist Max Tegmark. [1] [2] According to the hypothesis, the universe is a mathematical object in and of itself.
1.1: Statements and Conditional Statements
Mathematical Logic and Proofs Mathematical Reasoning - Writing and Proof (Sundstrom) 1: Introduction to Writing Proofs in Mathematics ... 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.
Null and Alternative Hypotheses
The actual test begins by considering two hypotheses.They are called the null hypothesis and the alternative hypothesis.These hypotheses contain opposing viewpoints. H 0: The null hypothesis: It is a statement about the population that either is believed to be true or is used to put forth an argument unless it can be shown to be incorrect beyond a reasonable doubt.
3.6: Mathematical Induction
Outline for Mathematical Induction. To show that a propositional function P(n) is true for all integers n ≥ a, follow these steps: Base Step: Verify that P(a) is true. Inductive Step: Show that if P(k) is true for some integer k ≥ a, then P(k + 1) is also true. Assume P(n) is true for an arbitrary integer, k with k ≥ a .
What Is a Hypothesis? The Scientific Method
A hypothesis (plural hypotheses) is a proposed explanation for an observation. The definition depends on the subject. In science, a hypothesis is part of the scientific method. It is a prediction or explanation that is tested by an experiment. Observations and experiments may disprove a scientific hypothesis, but can never entirely prove one.
8.1: The Elements of Hypothesis Testing
Hypothesis testing is a statistical procedure in which a choice is made between a null hypothesis and an alternative hypothesis based on information in a sample. The end result of a hypotheses testing procedure is a choice of one of the following two possible conclusions: Reject H0 (and therefore accept Ha ), or.
Riemann hypothesis
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1 / 2. Many consider it to be the most important unsolved problem in pure mathematics.
Hypothesis Testing
Hypothesis testing is a tool for making statistical inferences about the population data. It is an analysis tool that tests assumptions and determines how likely something is within a given standard of accuracy. Hypothesis testing provides a way to verify whether the results of an experiment are valid. A null hypothesis and an alternative ...
What is a Hypothesis?
Thus the hypothesis is what we must assume in order to be positive that the conclusion will hold. Whenever you are asked to state a theorem, be sure to include the hypothesis. In order to know when you may apply the theorem, you need to know what constraints you have. So in the example above, if we know that a function is differentiable, we may ...
Hypothesis Definition
Types of Hypothesis. The hypothesis can be broadly classified into different types. They are: Simple Hypothesis. A simple hypothesis is a hypothesis that there exists a relationship between two variables. One is called a dependent variable, and the other is called an independent variable. Complex Hypothesis.
What is Hypothesis
Branches of Mathematics; Summary - Hypothesis. A hypothesis is a testable statement serving as an initial explanation for phenomena, based on observations, theories, or existing knowledge. It acts as a guiding light for scientific research, proposing potential relationships between variables that can be empirically tested through experiments ...
How to Write a Strong Hypothesis
The specific group being studied. The predicted outcome of the experiment or analysis. 5. Phrase your hypothesis in three ways. To identify the variables, you can write a simple prediction in if…then form. The first part of the sentence states the independent variable and the second part states the dependent variable.
Understanding the Role of Hypotheses and Conclusions in Mathematical
Hypothesis and conclusion. In the context of mathematics and logic, a hypothesis is a statement or proposition that is assumed to be true for the purpose of a logical argument or investigation. It is usually denoted by "H" or "P" and is the starting point for many mathematical proofs. For example, let's consider the hypothesis: "If ...
Hypothesis test
Hypothesis test. A significance test, also referred to as a statistical hypothesis test, is a method of statistical inference in which observed data is compared to a claim (referred to as a hypothesis) in order to assess the truth of the claim. For example, one might wonder whether age affects the number of apples a person can eat, and may use a significance test to determine whether there is ...
[2404.06306] Disproof of the Riemann Hypothesis
The Riemann Hypothesis is a conjecture that all non-trivial zeros of Riemann Zeta function are located on the critical line in the complex plane. Hundreds of propositions in function theory and analytic number theory rely on this hypothesis. However, the problem has been unresolved for over a century. Here we show that at least one set of quadruplet-zeros exists outside the critical line ...
Quantum impossibility is made possible using 'exotic' mathematics
The eigenstate-thermalization hypothesis, which concerns the ability of mathematics to describe the motions of quantum systems as wholes, had been an assumption -- an educated guess -- in physics ...
Galileo's Hypothesis_4.docx
Galileo's Hypothesis: All objects experience the same acceleration downward when falling without air resistance. Measuring the acceleration of free-fall, g, is not a simple thing to do. Galileo had a great deal of trouble in his attempts to measure g because he lacked good timing devices and the motion changed much too quickly.

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Examples of null and alternative hypotheses

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Video transcript

Module 9: Hypothesis Testing With One Sample

Concept Review

Formula Review

What Is a Hypothesis? (Science)

Writing a Hypothesis

Example of a Hypothesis

8.1: The Elements of Hypothesis Testing

Learning Objectives

Types of Hypotheses

Definition: null hypothesis and alternative hypothesis

Definition: statistical procedure

Example \(\PageIndex{1}\)

Example \(\PageIndex{2}\)

The Logic of Hypothesis Testing

The Rejection Region

Definition: critical values

Example \(\PageIndex{3}\)

Definitions: Test classifications

Two Types of Errors

Definition: Type I and Type II errors

Definition: level of significance

Standardizing the Test Statistic

Definition: hypothesis test

Systematic Hypothesis Testing Procedure: Critical Value Approach

Key Takeaway

Hypothesis Testing

What is Hypothesis Testing in Statistics?

Hypothesis Testing Definition

Null Hypothesis

Alternative Hypothesis

Hypothesis Testing P Value

Hypothesis Testing Critical region

Hypothesis Testing Formula

Types of Hypothesis Testing

Hypothesis Testing Z Test

Hypothesis Testing t Test

Hypothesis Testing Chi Square

One Tailed Hypothesis Testing

Two Tailed Hypothesis Testing