what is hypothesis in math means

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.

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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Unit 12: Significance tests (hypothesis testing)

About this unit.

Significance tests give us a formal process for using sample data to evaluate the likelihood of some claim about a population value. Learn how to conduct significance tests and calculate p-values to see how likely a sample result is to occur by random chance. You'll also see how we use p-values to make conclusions about hypotheses.

The idea of significance tests

Simple hypothesis testing (Opens a modal)
Idea behind hypothesis testing (Opens a modal)
Examples of null and alternative hypotheses (Opens a modal)
P-values and significance tests (Opens a modal)
Comparing P-values to different significance levels (Opens a modal)
Estimating a P-value from a simulation (Opens a modal)
Using P-values to make conclusions (Opens a modal)
Simple hypothesis testing Get 3 of 4 questions to level up!
Writing null and alternative hypotheses Get 3 of 4 questions to level up!
Estimating P-values from simulations Get 3 of 4 questions to level up!

Error probabilities and power

Introduction to Type I and Type II errors (Opens a modal)
Type 1 errors (Opens a modal)
Examples identifying Type I and Type II errors (Opens a modal)
Introduction to power in significance tests (Opens a modal)
Examples thinking about power in significance tests (Opens a modal)
Consequences of errors and significance (Opens a modal)
Type I vs Type II error Get 3 of 4 questions to level up!
Error probabilities and power Get 3 of 4 questions to level up!

Tests about a population proportion

Constructing hypotheses for a significance test about a proportion (Opens a modal)
Conditions for a z test about a proportion (Opens a modal)
Reference: Conditions for inference on a proportion (Opens a modal)
Calculating a z statistic in a test about a proportion (Opens a modal)
Calculating a P-value given a z statistic (Opens a modal)
Making conclusions in a test about a proportion (Opens a modal)
Writing hypotheses for a test about a proportion Get 3 of 4 questions to level up!
Conditions for a z test about a proportion Get 3 of 4 questions to level up!
Calculating the test statistic in a z test for a proportion Get 3 of 4 questions to level up!
Calculating the P-value in a z test for a proportion Get 3 of 4 questions to level up!
Making conclusions in a z test for a proportion Get 3 of 4 questions to level up!

Tests about a population mean

Writing hypotheses for a significance test about a mean (Opens a modal)
Conditions for a t test about a mean (Opens a modal)
Reference: Conditions for inference on a mean (Opens a modal)
When to use z or t statistics in significance tests (Opens a modal)
Example calculating t statistic for a test about a mean (Opens a modal)
Using TI calculator for P-value from t statistic (Opens a modal)
Using a table to estimate P-value from t statistic (Opens a modal)
Comparing P-value from t statistic to significance level (Opens a modal)
Free response example: Significance test for a mean (Opens a modal)
Writing hypotheses for a test about a mean Get 3 of 4 questions to level up!
Conditions for a t test about a mean Get 3 of 4 questions to level up!
Calculating the test statistic in a t test for a mean Get 3 of 4 questions to level up!
Calculating the P-value in a t test for a mean Get 3 of 4 questions to level up!
Making conclusions in a t test for a mean Get 3 of 4 questions to level up!

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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Hypothesis Testing

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A hypothesis test is a statistical inference method used to test the significance of a proposed (hypothesized) relation between population statistics (parameters) and their corresponding sample estimators . In other words, hypothesis tests are used to determine if there is enough evidence in a sample to prove a hypothesis true for the entire population.

The test considers two hypotheses: the null hypothesis , which is a statement meant to be tested, usually something like "there is no effect" with the intention of proving this false, and the alternate hypothesis , which is the statement meant to stand after the test is performed. The two hypotheses must be mutually exclusive ; moreover, in most applications, the two are complementary (one being the negation of the other). The test works by comparing the $p$-value to the level of significance (a chosen target). If the $p$-value is less than or equal to the level of significance, then the null hypothesis is rejected.

When analyzing data, only samples of a certain size might be manageable as efficient computations. In some situations the error terms follow a continuous or infinite distribution, hence the use of samples to suggest accuracy of the chosen test statistics. The method of hypothesis testing gives an advantage over guessing what distribution or which parameters the data follows.

Definitions and Methodology

Hypothesis test and confidence intervals.

In statistical inference, properties (parameters) of a population are analyzed by sampling data sets. Given assumptions on the distribution, i.e. a statistical model of the data, certain hypotheses can be deduced from the known behavior of the model. These hypotheses must be tested against sampled data from the population.

The null hypothesis $($denoted $H_0)$ is a statement that is assumed to be true. If the null hypothesis is rejected, then there is enough evidence (statistical significance) to accept the alternate hypothesis $($denoted $H_1).$ Before doing any test for significance, both hypotheses must be clearly stated and non-conflictive, i.e. mutually exclusive, statements. Rejecting the null hypothesis, given that it is true, is called a type I error and it is denoted $\alpha$, which is also its probability of occurrence. Failing to reject the null hypothesis, given that it is false, is called a type II error and it is denoted $\beta$, which is also its probability of occurrence. Also, $\alpha$ is known as the significance level , and $1-\beta$ is known as the power of the test. $H_0$ $\textbf{is true}$$\hspace{15mm}$ $H_0$ $\textbf{is false}$ $\textbf{Reject}$ $H_0$$\hspace{10mm}$ Type I error Correct Decision $\textbf{Reject}$ $H_1$ Correct Decision Type II error The test statistic is the standardized value following the sampled data under the assumption that the null hypothesis is true, and a chosen particular test. These tests depend on the statistic to be studied and the assumed distribution it follows, e.g. the population mean following a normal distribution. The $p$-value is the probability of observing an extreme test statistic in the direction of the alternate hypothesis, given that the null hypothesis is true. The critical value is the value of the assumed distribution of the test statistic such that the probability of making a type I error is small.

Methodologies: Given an estimator $\hat \theta$ of a population statistic $\theta$, following a probability distribution $P(T)$, computed from a sample $\mathcal{S},$ and given a significance level $\alpha$ and test statistic $t^*,$ define $H_0$ and $H_1;$ compute the test statistic $t^*.$ $p$-value Approach (most prevalent): Find the $p$-value using $t^*$ (right-tailed). If the $p$-value is at most $\alpha,$ reject $H_0$. Otherwise, reject $H_1$. Critical Value Approach: Find the critical value solving the equation $P(T\geq t_\alpha)=\alpha$ (right-tailed). If $t^*>t_\alpha$, reject $H_0$. Otherwise, reject $H_1$. Note: Failing to reject $H_0$ only means inability to accept $H_1$, and it does not mean to accept $H_0$.

Assume a normally distributed population has recorded cholesterol levels with various statistics computed. From a sample of 100 subjects in the population, the sample mean was 214.12 mg/dL (milligrams per deciliter), with a sample standard deviation of 45.71 mg/dL. Perform a hypothesis test, with significance level 0.05, to test if there is enough evidence to conclude that the population mean is larger than 200 mg/dL. Hypothesis Test We will perform a hypothesis test using the $p$-value approach with significance level $\alpha=0.05:$ Define $H_0$: $\mu=200$. Define $H_1$: $\mu>200$. Since our values are normally distributed, the test statistic is $z^*=\frac{\bar X - \mu_0}{\frac{s}{\sqrt{n}}}=\frac{214.12 - 200}{\frac{45.71}{\sqrt{100}}}\approx 3.09$. Using a standard normal distribution, we find that our $p$-value is approximately $0.001$. Since the $p$-value is at most $\alpha=0.05,$ we reject $H_0$. Therefore, we can conclude that the test shows sufficient evidence to support the claim that $\mu$ is larger than $200$ mg/dL.

If the sample size was smaller, the normal and $t$-distributions behave differently. Also, the question itself must be managed by a double-tail test instead.

Assume a population's cholesterol levels are recorded and various statistics are computed. From a sample of 25 subjects, the sample mean was 214.12 mg/dL (milligrams per deciliter), with a sample standard deviation of 45.71 mg/dL. Perform a hypothesis test, with significance level 0.05, to test if there is enough evidence to conclude that the population mean is not equal to 200 mg/dL. Hypothesis Test We will perform a hypothesis test using the $p$-value approach with significance level $\alpha=0.05$ and the $t$-distribution with 24 degrees of freedom: Define $H_0$: $\mu=200$. Define $H_1$: $\mu\neq 200$. Using the $t$-distribution, the test statistic is $t^*=\frac{\bar X - \mu_0}{\frac{s}{\sqrt{n}}}=\frac{214.12 - 200}{\frac{45.71}{\sqrt{25}}}\approx 1.54$. Using a $t$-distribution with 24 degrees of freedom, we find that our $p$-value is approximately $2(0.068)=0.136$. We have multiplied by two since this is a two-tailed argument, i.e. the mean can be smaller than or larger than. Since the $p$-value is larger than $\alpha=0.05,$ we fail to reject $H_0$. Therefore, the test does not show sufficient evidence to support the claim that $\mu$ is not equal to $200$ mg/dL.

The complement of the rejection on a two-tailed hypothesis test (with significance level $\alpha$) for a population parameter $\theta$ is equivalent to finding a confidence interval $($with confidence level $1-\alpha)$ for the population parameter $\theta$. If the assumption on the parameter $\theta$ falls inside the confidence interval, then the test has failed to reject the null hypothesis $($with $p$-value greater than $\alpha).$ Otherwise, if $\theta$ does not fall in the confidence interval, then the null hypothesis is rejected in favor of the alternate $($with $p$-value at most $\alpha).$

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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!

Still have questions? Please ask!

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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

All images/mathematical drawings were created with GeoGebra.

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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.

Definition Of Hypothesis

Hypothesis is the part of a conditional statement just after the word if.

Examples of Hypothesis

In the conditional, "If all fours sides of a quadrilateral measure the same, then the quadrilateral is a square" the hypothesis is "all fours sides of a quadrilateral measure the same".

Video Examples: Hypothesis

Solved Example on Hypothesis

Ques: in the example above, is the hypothesis "all fours sides of a quadrilateral measure the same" always, never, or sometimes true.

A. always B. never C. sometimes Correct Answer: C

Step 1: The hypothesis is sometimes true. Because, its true only for a square and a rhombus, not for the other quadrilaterals rectangle, parallelogram, or trapezoid.

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17 Introduction to Hypothesis Testing

Jenna Lehmann

What is Hypothesis Testing?

Hypothesis testing is a big part of what we would actually consider testing for inferential statistics. It’s a procedure and set of rules that allow us to move from descriptive statistics to make inferences about a population based on sample data. It is a statistical method that uses sample data to evaluate a hypothesis about a population.

This type of test is usually used within the context of research. If we expect to see a difference between a treated and untreated group (in some cases the untreated group is the parameters we know about the population), we expect there to be a difference in the means between the two groups, but that the standard deviation remains the same, as if each individual score has had a value added or subtracted from it.

Steps of Hypothesis Testing

The following steps will be tailored to fit the first kind of hypothesis testing we will learn first: single-sample z-tests. There are many other kinds of tests, so keep this in mind.

Null Hypothesis (H0): states that in the general population there is no change, no difference, or no relationship, or in the context of an experiment, it predicts that the independent variable has no effect on the dependent variable.
Alternative Hypothesis (H1): states that there is a change, a difference, or a relationship for the general population, or in the context of an experiment, it predicts that the independent variable has an effect on the dependent variable.

$\alpha = 0.05,$

Critical Region: Composed of the extreme sample values that are very unlikely to be obtained if the null hypothesis is true. Determined by alpha level. If sample data fall in the critical region, the null hypothesis is rejected, because it’s very unlikely they’ve fallen there by chance.
After collecting the data, we find the sample mean. Now we can compare the sample mean with the null hypothesis by computing a z-score that describes where the sample mean is located relative to the hypothesized population mean. We use the z-score formula.
We decided previously what the two z-score boundaries are for a critical score. If the z-score we get after plugging the numbers in the aforementioned equation is outside of that critical region, we reject the null hypothesis. Otherwise, we would say that we failed to reject the null hypothesis.

Regions of the Distribution

Because we’re making judgments based on probability and proportion, our normal distributions and certain regions within them come into play.

The Critical Region is composed of the extreme sample values that are very unlikely to be obtained if the null hypothesis is true. Determined by alpha level. If sample data fall in the critical region, the null hypothesis is rejected, because it’s very unlikely they’ve fallen there by chance.

These regions come into play when talking about different errors.

A Type I Error occurs when a researcher rejects a null hypothesis that is actually true; the researcher concludes that a treatment has an effect when it actually doesn’t. This happens when a researcher unknowingly obtains an extreme, non-representative sample. This goes back to alpha level: it’s the probability that the test will lead to a Type I error if the null hypothesis is true.

$(\beta)$

A result is said to be significant or statistically significant if it is very unlikely to occur when the null hypothesis is true. That is, the result is sufficient to reject the null hypothesis. For instance, two means can be significantly different from one another.

Factors that Influence and Assumptions of Hypothesis Testing

Assumptions of Hypothesis Testing:

Random sampling: it is assumed that the participants used in the study were selected randomly so that we can confidently generalize our findings from the sample to the population.
Independent observation: two observations are independent if there is no consistent, predictable relationship between the first observation and the second. The value of σ is unchanged by the treatment; if the population standard deviation is unknown, we assume that the standard deviation for the unknown population (after treatment) is the same as it was for the population before treatment. There are ways of checking to see if this is true in SPSS or Excel.
Normal sampling distribution: in order to use the unit normal table to identify the critical region, we need the distribution of sample means to be normal (which means we need the population to be distributed normally and/or each sample size needs to be 30 or greater based on what we know about the central limit theorem).

Factors that influence hypothesis testing:

The variability of the scores, which is measured by either the standard deviation or the variance. The variability influences the size of the standard error in the denominator of the z-score.
The number of scores in the sample. This value also influences the size of the standard error in the denominator.

Test statistic: indicates that the sample data are converted into a single, specific statistic that is used to test the hypothesis (in this case, the z-score statistic).

Directional Hypotheses and Tailed Tests

In a directional hypothesis test , also known as a one-tailed test, the statistical hypotheses specify with an increase or decrease in the population mean. That is, they make a statement about the direction of the effect.

The Hypotheses for a Directional Test:

H0: The test scores are not increased/decreased (the treatment doesn’t work)
H1: The test scores are increased/decreased (the treatment works as predicted)

Because we’re only worried about scores that are either greater or less than the scores predicted by the null hypothesis, we only worry about what’s going on in one tail meaning that the critical region only exists within one tail. This means that all of the alpha is contained in one tail rather than split up into both (so the whole 5% is located in the tail we care about, rather than 2.5% in each tail). So before, we cared about what’s going on at the 0.025 mark of the unit normal table to look at both tails, but now we care about 0.05 because we’re only looking at one tail.

A one-tailed test allows you to reject the null hypothesis when the difference between the sample and the population is relatively small, as long as that difference is in the direction that you predicted. A two-tailed test, on the other hand, requires a relatively large difference independent of direction. In practice, researchers hypothesize using a one-tailed method but base their findings off of whether the results fall into the critical region of a two-tailed method. For the purposes of this class, make sure to calculate your results using the test that is specified in the problem.

Effect Size

A measure of effect size is intended to provide a measurement of the absolute magnitude of a treatment effect, independent of the size of the sample(s) being used. Usually done with Cohen’s d. If you imagine the two distributions, they’re layered over one another. The more they overlap, the smaller the effect size (the means of the two distributions are close). The more they are spread apart, the greater the effect size (the means of the two distributions are farther apart).

Statistical Power

The power of a statistical test is the probability that the test will correctly reject a false null hypothesis. It’s usually what we’re hoping to get when we run an experiment. It’s displayed in the table posted above. Power and effect size are connected. So, we know that the greater the distance between the means, the greater the effect size. If the two distributions overlapped very little, there would be a greater chance of selecting a sample that leads to rejecting the null hypothesis.

This chapter was originally posted to the Math Support Center blog at the University of Baltimore on June 11, 2019.

Math and Statistics Guides from UB's Math & Statistics Center Copyright © by Jenna Lehmann is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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9.1 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: a statement of no difference between sample means or proportions or no difference between a sample mean or proportion and a population mean or proportion. In other words, the difference equals 0.

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

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 use = in the null hypothesis, even with > or < as the symbol in the alternative hypothesis. This practice is acceptable because we only make the decision to reject or not reject the null hypothesis.

Example 9.1

H 0 : No more than 30 percent of the registered voters in Santa Clara County voted in the primary election. p ≤ 30 H a : More than 30 percent 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 percent. State the null and alternative hypotheses.

Example 9.2

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

Example 9.3

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

Example 9.4

An article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third of the students pass. The same article stated that 6.6 percent of U.S. students take advanced placement exams and 4.4 percent pass. Test if the percentage of U.S. students who take advanced placement exams is more than 6.6 percent. State the null and alternative hypotheses. H 0 : p ≤ 0.066 H a : p > 0.066

On a state driver’s test, about 40 percent pass the test on the first try. We want to test if more than 40 percent 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

Collaborative Exercise

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

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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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What Is a Hypothesis Test?

The null hypothesis significance testing (nhst) framework, our simple scenario.

Our null hypothesis is $H_0\colon \pi=\frac{1}{2}$. This says that the proportion is what we believe it should be.
Our alternative hypothesis is $H_1\colon \pi\ne\frac{1}{2}$. This says that the proportion has changed.

Testing our hypotheses

We can work out the critical region for $X$, that is, those extreme values of $X$ which would lead us to reject the null hypothesis at 5% significance. (This can be done even before performing the experiment.) The probability of $X$ taking a value in this critical region, assuming that the null hypothesis is true, should be 5%, or as close at we can get to 5% without going over it. In symbols, we can say: $$\mathrm{P}(\text{$X$ in critical region} | \text{$H_0$ is true}) \le 0.05.$$ Then we reject the null hypothesis if $X$ lies in that region.
We can work out the probability of $X$ taking the value it did or a more extreme value, assuming that the null hypothesis is true. This is known as the p-value . If the p-value is less than 0.05, then we will reject the null hypothesis at 5% significance. [ note 1 ] In symbols, we can write $$\text{p-value} = \mathrm{P}(\text{$X$ taking this or a more extreme value} | \text{$H_0$ is true}).$$

Other types of scenario

Does this drug/treatment/intervention/... have any effect?
Which of these drugs/... is more effective, or are they equally effective?
Is the mean height/mass/intelligence/test score/... of this population equal to some predicted value?
Is the standard deviation of the height/mass/... equal to some predicted value?
For two distinct groups of people, is their mean height/mass/... of each group the same?
Does this group of people's heights/masses/... appear to be following the probability distribution we expect?
Do these two populations' heights/masses/... appear to have the same distribution as each other?
Do this population's heights and weights appear to be correlated?

Interpreting the results

The key question that hypothesis testing (nhst) answers, what a hypothesis test does not tell us, a non-significant result.

It could be that the null hypothesis is true. In this case, we would have to be unlucky to get a significant p-value, so most of the time, we will end up accepting the null hypothesis. (If the null hypothesis is true, we would reject it with a probability of only 0.05.)
On the other hand, it could be that the alternative hypothesis is true, but we did not use a large enough sample to obtain a significant result (or we were just unlucky). In such a case, we could say that our test was insensitive . In this situation (the alternative hypothesis is true but we do not reject the null hypothesis), we say that we have made a Type II error . The probability of this happening depends on the sample size and on how different the true $\pi$ is from $\frac{1}{2}$ (or whatever our null hypothesis says), as is explored in Powerful Hypothesis Testing .

A significant result

It could be that the null hypothesis is true. In this case, we reject the null hypothesis with a probability of $0.05=\frac{1}{20}$, that is, one time in 20 (at a significance level of 5%), so we were just unlucky.
On the other hand, the alternative hypothesis could indeed be true. Either the sample was large enough to obtain a significant result, or the sample size wasn't that large, but we were just lucky.

Using this tree diagram, we can work out the probabilities of $H_0$ being true or $H_1$ being true given our experimental results. To avoid the expressions becoming unwieldy, we will write $H_0$ for "$\text{$H_0$ true}$", $H_1$ for "$\text{$H_1$ true}$" and "$\text{p}^+$" for "observed p-value or more extreme". Then we can write (conditional) probabilities on the branches of the tree diagram leading to our observed p-value: [ note 2 ]

The two routes which give our observed p-value (or more extreme) have the following probabilities: $$\begin{align*} \mathrm{P}(H_0\cap \text{p}^+) &= \mathrm{P}(H_0) \times \mathrm{P}(\text{p}^+ | H_0) \\ \mathrm{P}(H_1\cap \text{p}^+) &= \mathrm{P}(H_1) \times \mathrm{P}(\text{p}^+ | H_1) \end{align*}$$ (Recall that $\mathrm{P}(H_0\cap \text{p}^+)$ means "the probability of $H_0$ being true and the p-value being that observed or more extreme".) We can therefore work out the probability of the alternative hypothesis being true given the observed p-value, using conditional probability: $$\begin{align*} \mathrm{P}(H_1|\text{p}^+) &= \frac{\mathrm{P}(H_1\cap \text{p}^+)}{\mathrm{P}(\text{p}^+)} \\ &= \frac{\mathrm{P}(H_1\cap \text{p}^+)}{\mathrm{P}(H_0\cap\text{p}^+)+\mathrm{P}(H_1\cap\text{p}^+)} \\ &= \frac{\mathrm{P}(H_1) \times \mathrm{P}(\text{p}^+ | H_1)}{\mathrm{P}(H_0) \times \mathrm{P}(\text{p}^+ | H_0) + \mathrm{P}(H_1) \times \mathrm{P}(\text{p}^+ | H_1)} \end{align*}$$ Though this is a mouthful, it is a calculation which only involves the four probabilities on the above tree diagram. (This is an example of Bayes' Theorem , discussed further in this resource .) However, we immediately hit a big difficulty if we try to calculate this for a given experiment. We know $\mathrm{P}(\text{p}^+ | H_0)$: this is just the p-value itself. (The p-value tells us the probability of obtaining a result at least this extreme given that the null hypothesis is true.) But we don't know the probability of the null hypothesis being true or false (that is, $\mathrm{P}(H_0)$ and $\mathrm{P}(H_1)=1-\mathrm{P}(H_0)$), nor do we know the probability of the observed result if the alternative hypothesis is true ($P(\text{p}^+|H_1)$), as knowing that the proportion of greens is not $\frac{1}{2}$ does not tell us what it actually is. (Similar issues apply to all the other contexts of hypothesis testing listed above.) So we are quite stuck: in the null hypothesis significance testing model, it is impossible to give a numerical answer to our key question: "Given our results, what is the probability that the alternative hypothesis is true?" This is because we don't know two of the three probabilities that we need in order to answer the question. An example might highlight the issue a little better. Let us suppose that we are trying to work out whether a coin is biased (alternative hypothesis), or whether the probability of heads is exactly $\frac{1}{2}$ (null hypothesis). We toss the coin 50 times and obtain a p-value of 0.02. Do we now believe that the coin is biased? Most people believe that coins are not biased, and so are much more likely to attribute this result to chance or poor coin-tossing technique than to the coin being biased. On the other hand, consider a case of a road planner who introduces a traffic-calming feature to reduce the number of fatalities along a certain stretch of road. The null hypothesis is that there is no change in fatality rate, while the alternative hypothesis is that the fatality rate has decreased. A hypothesis test is performed on data collected for 24 months before and 24 months after the feature is built. Again, the p-value was 0.02. Do we believe that the alternative hypothesis is true? In this case, we are more likely to believe that the alternative hypothesis is true, because it makes a lot of sense that this feature will reduce the number of fatalities. Our "instinctive" responses to these results are tied up with assigning values to the unknown probabilities in the formula above. For the coin, we would probably take $\mathrm{P}(H_0)$ to be close to 1, say $0.99$, as we think it is very unlikely that the coin is biased, and $\mathrm{P}(\text{p}^+|H_1)$ will be, say, $0.1$: if the coin is biased, the bias is not likely to be very large, and so it is only a bit more likely that the result will be significant in this case. Putting these figures into the formula above gives: $$\mathrm{P}(H_1|\text{p}^+) = \frac{0.01 \times 0.1}{0.99 \times 0.02 + 0.01 \times 0.1} \approx 0.05,$$ that is, we are still very doubtful that this coin is biased, even after performing the experiment. Note that in this case, the probability of these results given that the null hypothesis is true is 0.02, whereas the probability that the null hypothesis is true given these results is $1-0.05=0.95$, which is very different. This shows how dramatically different the answers to the two questions can be. On the other hand, for the fatalities situation, we might assume quite the opposite: we are pretty confident that the traffic-calming feature will help, so we might take $\mathrm{P}(H_0)$ to be $0.4$, and $\mathrm{P}(\text{p}^+|H_1)$ will be, say, $0.25$ (though the traffic-calming may help, the impact may be relatively small). Putting these figures into the formula gives: $$\mathrm{P}(H_1|\text{p}^+) = \frac{0.6 \times 0.25}{0.4 \times 0.02 + 0.6 \times 0.25} \approx 0.95,$$ so we are now much more convinced that the traffic-calming feature is helping than we were before we had the data. This time, the probability of these results given that the null hypothesis is true is still 0.02, whereas the probability that the null hypothesis is true given these results is $1-0.95=0.05$, which is not that different. This approach may seem very disturbing, as we have to make assumptions about what we believe before we do the hypothesis test. But as we have seen, we cannot answer our key question without making such assumptions.

Other approaches and some warnings

Because our test is two-tailed (in the alternative hypothesis, the true proportion could be less than $\frac{1}{2}$ or more than $\frac{1}{2}$), we must be careful when calculating the p-value: we calculate the probability of the observed outcome or more extreme occurring, and then double the answer to account for the other tail. We could also compare the probability of the value or more extreme to 0.025 instead of 0.05, but that would not be called a p-value. Likewise, when we determine the critical region, we will have two parts: a tail with large values of $X$ and a tail with small values of $X$; we require that the probability of $X$ lying in the large-value tail is as close as possible to 0.025 without going over it, and the same for the probability of $X$ lying in the small-value tail.
There are complications here when working with two-tail tests as opposed to one-tail tests. We will ignore this problem, as it does not significantly affect the overall discussion.
"Likelihood" is a technical term. For a discrete test statistic $X$, the likelihood of $H_0$ given the data $X=x$ means $P(X=x|H_0)$, in other words, how likely would this data be if $H_0$ were true. It is not the probability of $H_0$ being true given the data.

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.

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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.

Definition of a Hypothesis

What it is and how it's used in sociology

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A hypothesis is a prediction of what will be found at the outcome of a research project and is typically focused on the relationship between two different variables studied in the research. It is usually based on both theoretical expectations about how things work and already existing scientific evidence.

Within social science, a hypothesis can take two forms. It can predict that there is no relationship between two variables, in which case it is a null hypothesis . Or, it can predict the existence of a relationship between variables, which is known as an alternative hypothesis.

In either case, the variable that is thought to either affect or not affect the outcome is known as the independent variable, and the variable that is thought to either be affected or not is the dependent variable.

Researchers seek to determine whether or not their hypothesis, or hypotheses if they have more than one, will prove true. Sometimes they do, and sometimes they do not. Either way, the research is considered successful if one can conclude whether or not a hypothesis is true.

Null Hypothesis

A researcher has a null hypothesis when she or he believes, based on theory and existing scientific evidence, that there will not be a relationship between two variables. For example, when examining what factors influence a person's highest level of education within the U.S., a researcher might expect that place of birth, number of siblings, and religion would not have an impact on the level of education. This would mean the researcher has stated three null hypotheses.

Alternative Hypothesis

Taking the same example, a researcher might expect that the economic class and educational attainment of one's parents, and the race of the person in question are likely to have an effect on one's educational attainment. Existing evidence and social theories that recognize the connections between wealth and cultural resources , and how race affects access to rights and resources in the U.S. , would suggest that both economic class and educational attainment of the one's parents would have a positive effect on educational attainment. In this case, economic class and educational attainment of one's parents are independent variables, and one's educational attainment is the dependent variable—it is hypothesized to be dependent on the other two.

Conversely, an informed researcher would expect that being a race other than white in the U.S. is likely to have a negative impact on a person's educational attainment. This would be characterized as a negative relationship, wherein being a person of color has a negative effect on one's educational attainment. In reality, this hypothesis proves true, with the exception of Asian Americans , who go to college at a higher rate than whites do. However, Blacks and Hispanics and Latinos are far less likely than whites and Asian Americans to go to college.

Formulating a Hypothesis

Formulating a hypothesis can take place at the very beginning of a research project , or after a bit of research has already been done. Sometimes a researcher knows right from the start which variables she is interested in studying, and she may already have a hunch about their relationships. Other times, a researcher may have an interest in a particular topic, trend, or phenomenon, but he may not know enough about it to identify variables or formulate a hypothesis.

Whenever a hypothesis is formulated, the most important thing is to be precise about what one's variables are, what the nature of the relationship between them might be, and how one can go about conducting a study of them.

Updated by Nicki Lisa Cole, Ph.D

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How to Write a Great Hypothesis

Hypothesis Definition, Format, Examples, and Tips

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

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

Verywell / Alex Dos Diaz

The Scientific Method

Hypothesis Format

Falsifiability of a hypothesis.

Operationalization

Hypothesis Types

Hypotheses examples.

Collecting Data

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

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

At a Glance

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

The Hypothesis in the Scientific Method

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

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

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

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

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

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

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

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

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

Elements of a Good Hypothesis

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

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

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

How to Formulate a Good Hypothesis

To form a hypothesis, you should take these steps:

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

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

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

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

The Importance of Operational Definitions

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

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

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

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

Replicability

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

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

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

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

Hypothesis Checklist

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

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

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

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

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

A few examples of simple hypotheses:

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

Examples of a complex hypothesis include:

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

Examples of a null hypothesis include:

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

Examples of an alternative hypothesis:

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

Collecting Data on Your Hypothesis

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

Descriptive Research Methods

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

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

Experimental Research Methods

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

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

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

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

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

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

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

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

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

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

Choose Your Test

Sat / act prep online guides and tips, what is a hypothesis and how do i write one.

General Education

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

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

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

So let’s get started!

What Is a Hypothesis?

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

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

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

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

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

Independent and Dependent Variables

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

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

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

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

Elements of a Good Hypothesis

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

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

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

Writing Your Hypothesis

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

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

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

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

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

The Two Types of Hypotheses

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

#1: If-Then Hypotheses

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

If ____ happens, then ____ will happen.

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

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

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

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

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

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

#2: Null Hypotheses

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

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

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

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

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

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

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

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

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

4 Tips to Write the Best Hypothesis

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

#1: Plausibility

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

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

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

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

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

#2: Defined Concepts

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

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

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

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

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

Good hypotheses ensure that you can observe the results.

#3: Observability

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

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

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

#4: Generalizability

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

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

Hypothesis Testing Examples

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

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

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

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

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

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

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

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

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

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

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

Here’s what your hypotheses might look like:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Key Takeaways: Hypothesis Writing

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

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

What’s Next?

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

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

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

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

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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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What Is A Research (Scientific) Hypothesis? A plain-language explainer + examples

By: Derek Jansen (MBA) | Reviewed By: Dr Eunice Rautenbach | June 2020

If you’re new to the world of research, or it’s your first time writing a dissertation or thesis, you’re probably noticing that the words “research hypothesis” and “scientific hypothesis” are used quite a bit, and you’re wondering what they mean in a research context .

“Hypothesis” is one of those words that people use loosely, thinking they understand what it means. However, it has a very specific meaning within academic research. So, it’s important to understand the exact meaning before you start hypothesizing.

Research Hypothesis 101

What is a hypothesis ?
What is a research hypothesis (scientific hypothesis)?
Requirements for a research hypothesis
Definition of a research hypothesis
The null hypothesis

What is a hypothesis?

Let’s start with the general definition of a hypothesis (not a research hypothesis or scientific hypothesis), according to the Cambridge Dictionary:

Hypothesis: an idea or explanation for something that is based on known facts but has not yet been proved.

In other words, it’s a statement that provides an explanation for why or how something works, based on facts (or some reasonable assumptions), but that has not yet been specifically tested . For example, a hypothesis might look something like this:

Hypothesis: sleep impacts academic performance.

This statement predicts that academic performance will be influenced by the amount and/or quality of sleep a student engages in – sounds reasonable, right? It’s based on reasonable assumptions , underpinned by what we currently know about sleep and health (from the existing literature). So, loosely speaking, we could call it a hypothesis, at least by the dictionary definition.

But that’s not good enough…

Unfortunately, that’s not quite sophisticated enough to describe a research hypothesis (also sometimes called a scientific hypothesis), and it wouldn’t be acceptable in a dissertation, thesis or research paper . In the world of academic research, a statement needs a few more criteria to constitute a true research hypothesis .

What is a research hypothesis?

A research hypothesis (also called a scientific hypothesis) is a statement about the expected outcome of a study (for example, a dissertation or thesis). To constitute a quality hypothesis, the statement needs to have three attributes – specificity , clarity and testability .

Let’s take a look at these more closely.

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Hypothesis Essential #1: Specificity & Clarity

A good research hypothesis needs to be extremely clear and articulate about both what’ s being assessed (who or what variables are involved ) and the expected outcome (for example, a difference between groups, a relationship between variables, etc.).

Let’s stick with our sleepy students example and look at how this statement could be more specific and clear.

Hypothesis: Students who sleep at least 8 hours per night will, on average, achieve higher grades in standardised tests than students who sleep less than 8 hours a night.

As you can see, the statement is very specific as it identifies the variables involved (sleep hours and test grades), the parties involved (two groups of students), as well as the predicted relationship type (a positive relationship). There’s no ambiguity or uncertainty about who or what is involved in the statement, and the expected outcome is clear.

Contrast that to the original hypothesis we looked at – “Sleep impacts academic performance” – and you can see the difference. “Sleep” and “academic performance” are both comparatively vague , and there’s no indication of what the expected relationship direction is (more sleep or less sleep). As you can see, specificity and clarity are key.

A good research hypothesis needs to be very clear about what’s being assessed and very specific about the expected outcome.

Hypothesis Essential #2: Testability (Provability)

A statement must be testable to qualify as a research hypothesis. In other words, there needs to be a way to prove (or disprove) the statement. If it’s not testable, it’s not a hypothesis – simple as that.

For example, consider the hypothesis we mentioned earlier:

Hypothesis: Students who sleep at least 8 hours per night will, on average, achieve higher grades in standardised tests than students who sleep less than 8 hours a night.

We could test this statement by undertaking a quantitative study involving two groups of students, one that gets 8 or more hours of sleep per night for a fixed period, and one that gets less. We could then compare the standardised test results for both groups to see if there’s a statistically significant difference.

Again, if you compare this to the original hypothesis we looked at – “Sleep impacts academic performance” – you can see that it would be quite difficult to test that statement, primarily because it isn’t specific enough. How much sleep? By who? What type of academic performance?

So, remember the mantra – if you can’t test it, it’s not a hypothesis 🙂

A good research hypothesis must be testable. In other words, you must able to collect observable data in a scientifically rigorous fashion to test it.

Defining A Research Hypothesis

You’re still with us? Great! Let’s recap and pin down a clear definition of a hypothesis.

A research hypothesis (or scientific hypothesis) is a statement about an expected relationship between variables, or explanation of an occurrence, that is clear, specific and testable.

So, when you write up hypotheses for your dissertation or thesis, make sure that they meet all these criteria. If you do, you’ll not only have rock-solid hypotheses but you’ll also ensure a clear focus for your entire research project.

What about the null hypothesis?

You may have also heard the terms null hypothesis , alternative hypothesis, or H-zero thrown around. At a simple level, the null hypothesis is the counter-proposal to the original hypothesis.

For example, if the hypothesis predicts that there is a relationship between two variables (for example, sleep and academic performance), the null hypothesis would predict that there is no relationship between those variables.

At a more technical level, the null hypothesis proposes that no statistical significance exists in a set of given observations and that any differences are due to chance alone.

And there you have it – hypotheses in a nutshell.

If you have any questions, be sure to leave a comment below and we’ll do our best to help you. If you need hands-on help developing and testing your hypotheses, consider our private coaching service , where we hold your hand through the research journey.

Psst... there’s more!

This post was based on one of our popular Research Bootcamps . If you're working on a research project, you'll definitely want to check this out ...

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16 Comments

Very useful information. I benefit more from getting more information in this regard.

Very great insight,educative and informative. Please give meet deep critics on many research data of public international Law like human rights, environment, natural resources, law of the sea etc

In a book I read a distinction is made between null, research, and alternative hypothesis. As far as I understand, alternative and research hypotheses are the same. Can you please elaborate? Best Afshin

This is a self explanatory, easy going site. I will recommend this to my friends and colleagues.

Very good definition. How can I cite your definition in my thesis? Thank you. Is nul hypothesis compulsory in a research?

It’s a counter-proposal to be proven as a rejection

Please what is the difference between alternate hypothesis and research hypothesis?

It is a very good explanation. However, it limits hypotheses to statistically tasteable ideas. What about for qualitative researches or other researches that involve quantitative data that don’t need statistical tests?

In qualitative research, one typically uses propositions, not hypotheses.

could you please elaborate it more

I’ve benefited greatly from these notes, thank you.

This is very helpful

well articulated ideas are presented here, thank you for being reliable sources of information

Excellent. Thanks for being clear and sound about the research methodology and hypothesis (quantitative research)

I have only a simple question regarding the null hypothesis. – Is the null hypothesis (Ho) known as the reversible hypothesis of the alternative hypothesis (H1? – How to test it in academic research?

this is very important note help me much more

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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 ...
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.
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 ...
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Then, if the null hypothesis is wrong, then the data will tend to group at a point that is not the value in the null hypothesis (1.2), and then our p-value will wind up being very small. If the null hypothesis is correct, or close to being correct, then the p-value will be larger, because the data values will group around the value we hypothesized.
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Significance tests give us a formal process for using sample data to evaluate the likelihood of some claim about a population value. Learn how to conduct significance tests and calculate p-values to see how likely a sample result is to occur by random chance. You'll also see how we use p-values to make conclusions about hypotheses.
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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 ...
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A hypothesis test is a statistical inference method used to test the significance of a proposed (hypothesized) relation between population statistics (parameters) and their corresponding sample estimators. In other words, hypothesis tests are used to determine if there is enough evidence in a sample to prove a hypothesis true for the entire population. The test considers two hypotheses: the ...
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 ...
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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 H0.
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Solution: Step 1: The hypothesis is sometimes true. Because, its true only for a square and a rhombus, not for the other quadrilaterals rectangle, parallelogram, or trapezoid. Hypothesis is the part of a conditional statement...Complete information about the hypothesis, definition of an hypothesis, examples of an hypothesis, step by step ...
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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.
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Step 3: Collect Data and Compute Sample Statistics. After collecting the data, we find the sample mean. Now we can compare the sample mean with the null hypothesis by computing a z-score that describes where the sample mean is located relative to the hypothesized population mean. We use the z-score formula. Step 4: Make a Decision.
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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.
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The null hypothesis significance testing (NHST) framework. The general situation is this: we want to find out about some aspect of the real world, and we do this by performing an experiment. From the data collected in the experiment, we want to make a deduction about reality, a process known as statistical inference .
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A hypothesis is a prediction of what will be found at the outcome of a research project and is typically focused on the relationship between two different variables studied in the research. It is usually based on both theoretical expectations about how things work and already existing scientific evidence. Within social science, a hypothesis can ...
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What Is A Research Hypothesis? A Simple Definition
A research hypothesis (also called a scientific hypothesis) is a statement about the expected outcome of a study (for example, a dissertation or thesis). To constitute a quality hypothesis, the statement needs to have three attributes - specificity, clarity and testability. Let's take a look at these more closely.

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Hypothesis Testing

Definitions and Methodology

Explanation of Hypothesis

Important Terms Related to Hypothesis

Numerical Examples

Hour Hand Definition < Glossary Index > Identity Definition

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

Definition Of Hypothesis

Examples of Hypothesis

Video Examples: Hypothesis

Solved Example on Hypothesis

Related Worksheet

HighSchool Math

17 Introduction to Hypothesis Testing

What is Hypothesis Testing?

Steps of Hypothesis Testing

Regions of the Distribution

Factors that Influence and Assumptions of Hypothesis Testing

Directional Hypotheses and Tailed Tests

Effect Size

Statistical Power

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9.1 Null and Alternative Hypotheses

Example 9.1

Example 9.2

Example 9.3

Example 9.4

Collaborative Exercise

Hypothesis Definition

Hypothesis Definition in Statistics

Types of Hypothesis

Characteristics of Hypothesis

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Number and algebra

Geometry and measure

Probability and statistics

Working mathematically

For younger learners

Advanced mathematics

What Is a Hypothesis Test?

Testing our hypotheses

Other types of scenario

Interpreting the results

A significant result

Other approaches and some warnings