simple definition of composite hypothesis

Simple Hypothesis and Composite Hypothesis

A simple hypothesis is one in which all parameters of the distribution are specified. For example, the heights of college students are normally distributed with $${\sigma ^2} = 4$$, and the hypothesis that its mean $$\mu $$ is, say, $$62”$$; that is, $${H_o}:\mu = 62$$. So we have stated a simple hypothesis, as the mean and variance together specify a normal distribution completely. A simple hypothesis, in general, states that $$\theta = {\theta _o}$$ where $${\theta _o}$$ is the specified value of a parameter $$\theta $$, ($$\theta $$ may represent $$\mu ,p,{\mu _1} – {\mu _2}$$ etc).

A hypothesis which is not simple (i.e. in which not all of the parameters are specified) is called a composite hypothesis. For instance, if we hypothesize that $${H_o}:\mu > 62$$ (and $${\sigma ^2} = 4$$) or$${H_o}:\mu = 62$$ and $${\sigma ^2} < 4$$, the hypothesis becomes a composite hypothesis because we cannot know the exact distribution of the population in either case. Obviously, the parameters $$\mu > 62”$$ and$${\sigma ^2} < 4$$ have more than one value and no specified values are being assigned. The general form of a composite hypothesis is $$\theta \leqslant {\theta _o}$$ or $$\theta \geqslant {\theta _o}$$; that is, the parameter $$\theta $$ does not exceed or does not fall short of a specified value $${\theta _o}$$. The concept of simple and composite hypotheses applies to both the null hypothesis and alternative hypothesis.

Hypotheses may also be classified as exact and inexact. A hypothesis is said to be an exact hypothesis if it selects a unique value for the parameter, such as $${H_o}:\mu = 62$$ or $$p > 0.5$$. A hypothesis is called an inexact hypothesis when it indicates more than one possible value for the parameter, such as $${H_o}:\mu \ne 62$$ or $${H_o}:p = 62$$. A simple hypothesis must be exact while an exact hypothesis is not necessarily a simple hypothesis. An inexact hypothesis is a composite hypothesis.

One Comment

Etini August 5 @ 7:28 pm

How can i design a sequential test for the shape parameters of the beta distribution

Leave a Reply Cancel reply

Your email address will not be published. Required fields are marked *

Definition: Simple and composite hypothesis

Definition: Let $H$ be a statistical hypothesis . Then,

$H$ is called a simple hypothesis, if it completely specifies the population distribution; in this case, the sampling distribution of the test statistic is a function of sample size alone.

$H$ is called a composite hypothesis, if it does not completely specify the population distribution; for example, the hypothesis may only specify one parameter of the distribution and leave others unspecified.

• Wikipedia (2021): "Exclusion of the null hypothesis" ; in: Wikipedia, the free encyclopedia , retrieved on 2021-03-19 ; URL: https://en.wikipedia.org/wiki/Exclusion_of_the_null_hypothesis#Terminology .

• Privacy Policy

Buy Me a Coffee

Home » What is a Hypothesis – Types, Examples and Writing Guide

What is a Hypothesis – Types, Examples and Writing Guide

Table of Contents

Definition:

Hypothesis is an educated guess or proposed explanation for a phenomenon, based on some initial observations or data. It is a tentative statement that can be tested and potentially proven or disproven through further investigation and experimentation.

Hypothesis is often used in scientific research to guide the design of experiments and the collection and analysis of data. It is an essential element of the scientific method, as it allows researchers to make predictions about the outcome of their experiments and to test those predictions to determine their accuracy.

Types of Hypothesis

Types of Hypothesis are as follows:

Research Hypothesis

A research hypothesis is a statement that predicts a relationship between variables. It is usually formulated as a specific statement that can be tested through research, and it is often used in scientific research to guide the design of experiments.

Null Hypothesis

The null hypothesis is a statement that assumes there is no significant difference or relationship between variables. It is often used as a starting point for testing the research hypothesis, and if the results of the study reject the null hypothesis, it suggests that there is a significant difference or relationship between variables.

Alternative Hypothesis

An alternative hypothesis is a statement that assumes there is a significant difference or relationship between variables. It is often used as an alternative to the null hypothesis and is tested against the null hypothesis to determine which statement is more accurate.

Directional Hypothesis

A directional hypothesis is a statement that predicts the direction of the relationship between variables. For example, a researcher might predict that increasing the amount of exercise will result in a decrease in body weight.

Non-directional Hypothesis

A non-directional hypothesis is a statement that predicts the relationship between variables but does not specify the direction. For example, a researcher might predict that there is a relationship between the amount of exercise and body weight, but they do not specify whether increasing or decreasing exercise will affect body weight.

Statistical Hypothesis

A statistical hypothesis is a statement that assumes a particular statistical model or distribution for the data. It is often used in statistical analysis to test the significance of a particular result.

Composite Hypothesis

A composite hypothesis is a statement that assumes more than one condition or outcome. It can be divided into several sub-hypotheses, each of which represents a different possible outcome.

Empirical Hypothesis

An empirical hypothesis is a statement that is based on observed phenomena or data. It is often used in scientific research to develop theories or models that explain the observed phenomena.

Simple Hypothesis

A simple hypothesis is a statement that assumes only one outcome or condition. It is often used in scientific research to test a single variable or factor.

Complex Hypothesis

A complex hypothesis is a statement that assumes multiple outcomes or conditions. It is often used in scientific research to test the effects of multiple variables or factors on a particular outcome.

Applications of Hypothesis

Hypotheses are used in various fields to guide research and make predictions about the outcomes of experiments or observations. Here are some examples of how hypotheses are applied in different fields:

• Science : In scientific research, hypotheses are used to test the validity of theories and models that explain natural phenomena. For example, a hypothesis might be formulated to test the effects of a particular variable on a natural system, such as the effects of climate change on an ecosystem.
• Medicine : In medical research, hypotheses are used to test the effectiveness of treatments and therapies for specific conditions. For example, a hypothesis might be formulated to test the effects of a new drug on a particular disease.
• Psychology : In psychology, hypotheses are used to test theories and models of human behavior and cognition. For example, a hypothesis might be formulated to test the effects of a particular stimulus on the brain or behavior.
• Sociology : In sociology, hypotheses are used to test theories and models of social phenomena, such as the effects of social structures or institutions on human behavior. For example, a hypothesis might be formulated to test the effects of income inequality on crime rates.
• Business : In business research, hypotheses are used to test the validity of theories and models that explain business phenomena, such as consumer behavior or market trends. For example, a hypothesis might be formulated to test the effects of a new marketing campaign on consumer buying behavior.
• Engineering : In engineering, hypotheses are used to test the effectiveness of new technologies or designs. For example, a hypothesis might be formulated to test the efficiency of a new solar panel design.

How to write a Hypothesis

Here are the steps to follow when writing a hypothesis:

Identify the Research Question

The first step is to identify the research question that you want to answer through your study. This question should be clear, specific, and focused. It should be something that can be investigated empirically and that has some relevance or significance in the field.

Conduct a Literature Review

Before writing your hypothesis, it’s essential to conduct a thorough literature review to understand what is already known about the topic. This will help you to identify the research gap and formulate a hypothesis that builds on existing knowledge.

Determine the Variables

The next step is to identify the variables involved in the research question. A variable is any characteristic or factor that can vary or change. There are two types of variables: independent and dependent. The independent variable is the one that is manipulated or changed by the researcher, while the dependent variable is the one that is measured or observed as a result of the independent variable.

Formulate the Hypothesis

Based on the research question and the variables involved, you can now formulate your hypothesis. A hypothesis should be a clear and concise statement that predicts the relationship between the variables. It should be testable through empirical research and based on existing theory or evidence.

Write the Null Hypothesis

The null hypothesis is the opposite of the alternative hypothesis, which is the hypothesis that you are testing. The null hypothesis states that there is no significant difference or relationship between the variables. It is important to write the null hypothesis because it allows you to compare your results with what would be expected by chance.

Refine the Hypothesis

After formulating the hypothesis, it’s important to refine it and make it more precise. This may involve clarifying the variables, specifying the direction of the relationship, or making the hypothesis more testable.

Examples of Hypothesis

Here are a few examples of hypotheses in different fields:

• Psychology : “Increased exposure to violent video games leads to increased aggressive behavior in adolescents.”
• Biology : “Higher levels of carbon dioxide in the atmosphere will lead to increased plant growth.”
• Sociology : “Individuals who grow up in households with higher socioeconomic status will have higher levels of education and income as adults.”
• Education : “Implementing a new teaching method will result in higher student achievement scores.”
• Marketing : “Customers who receive a personalized email will be more likely to make a purchase than those who receive a generic email.”
• Physics : “An increase in temperature will cause an increase in the volume of a gas, assuming all other variables remain constant.”
• Medicine : “Consuming a diet high in saturated fats will increase the risk of developing heart disease.”

Purpose of Hypothesis

The purpose of a hypothesis is to provide a testable explanation for an observed phenomenon or a prediction of a future outcome based on existing knowledge or theories. A hypothesis is an essential part of the scientific method and helps to guide the research process by providing a clear focus for investigation. It enables scientists to design experiments or studies to gather evidence and data that can support or refute the proposed explanation or prediction.

The formulation of a hypothesis is based on existing knowledge, observations, and theories, and it should be specific, testable, and falsifiable. A specific hypothesis helps to define the research question, which is important in the research process as it guides the selection of an appropriate research design and methodology. Testability of the hypothesis means that it can be proven or disproven through empirical data collection and analysis. Falsifiability means that the hypothesis should be formulated in such a way that it can be proven wrong if it is incorrect.

In addition to guiding the research process, the testing of hypotheses can lead to new discoveries and advancements in scientific knowledge. When a hypothesis is supported by the data, it can be used to develop new theories or models to explain the observed phenomenon. When a hypothesis is not supported by the data, it can help to refine existing theories or prompt the development of new hypotheses to explain the phenomenon.

When to use Hypothesis

Here are some common situations in which hypotheses are used:

• In scientific research , hypotheses are used to guide the design of experiments and to help researchers make predictions about the outcomes of those experiments.
• In social science research , hypotheses are used to test theories about human behavior, social relationships, and other phenomena.
• I n business , hypotheses can be used to guide decisions about marketing, product development, and other areas. For example, a hypothesis might be that a new product will sell well in a particular market, and this hypothesis can be tested through market research.

Characteristics of Hypothesis

Here are some common characteristics of a hypothesis:

• Testable : A hypothesis must be able to be tested through observation or experimentation. This means that it must be possible to collect data that will either support or refute the hypothesis.
• Falsifiable : A hypothesis must be able to be proven false if it is not supported by the data. If a hypothesis cannot be falsified, then it is not a scientific hypothesis.
• Clear and concise : A hypothesis should be stated in a clear and concise manner so that it can be easily understood and tested.
• Based on existing knowledge : A hypothesis should be based on existing knowledge and research in the field. It should not be based on personal beliefs or opinions.
• Specific : A hypothesis should be specific in terms of the variables being tested and the predicted outcome. This will help to ensure that the research is focused and well-designed.
• Tentative: A hypothesis is a tentative statement or assumption that requires further testing and evidence to be confirmed or refuted. It is not a final conclusion or assertion.
• Relevant : A hypothesis should be relevant to the research question or problem being studied. It should address a gap in knowledge or provide a new perspective on the issue.

Advantages of Hypothesis

Hypotheses have several advantages in scientific research and experimentation:

• Guides research: A hypothesis provides a clear and specific direction for research. It helps to focus the research question, select appropriate methods and variables, and interpret the results.
• Predictive powe r: A hypothesis makes predictions about the outcome of research, which can be tested through experimentation. This allows researchers to evaluate the validity of the hypothesis and make new discoveries.
• Facilitates communication: A hypothesis provides a common language and framework for scientists to communicate with one another about their research. This helps to facilitate the exchange of ideas and promotes collaboration.
• Efficient use of resources: A hypothesis helps researchers to use their time, resources, and funding efficiently by directing them towards specific research questions and methods that are most likely to yield results.
• Provides a basis for further research: A hypothesis that is supported by data provides a basis for further research and exploration. It can lead to new hypotheses, theories, and discoveries.
• Increases objectivity: A hypothesis can help to increase objectivity in research by providing a clear and specific framework for testing and interpreting results. This can reduce bias and increase the reliability of research findings.

Limitations of Hypothesis

Some Limitations of the Hypothesis are as follows:

• Limited to observable phenomena: Hypotheses are limited to observable phenomena and cannot account for unobservable or intangible factors. This means that some research questions may not be amenable to hypothesis testing.
• May be inaccurate or incomplete: Hypotheses are based on existing knowledge and research, which may be incomplete or inaccurate. This can lead to flawed hypotheses and erroneous conclusions.
• May be biased: Hypotheses may be biased by the researcher’s own beliefs, values, or assumptions. This can lead to selective interpretation of data and a lack of objectivity in research.
• Cannot prove causation: A hypothesis can only show a correlation between variables, but it cannot prove causation. This requires further experimentation and analysis.
• Limited to specific contexts: Hypotheses are limited to specific contexts and may not be generalizable to other situations or populations. This means that results may not be applicable in other contexts or may require further testing.
• May be affected by chance : Hypotheses may be affected by chance or random variation, which can obscure or distort the true relationship between variables.

About the author

Muhammad Hassan

Researcher, Academic Writer, Web developer

You may also like

Data Collection – Methods Types and Examples

Delimitations in Research – Types, Examples and...

Research Process – Steps, Examples and Tips

Research Design – Types, Methods and Examples

Institutional Review Board – Application Sample...

Evaluating Research – Process, Examples and...

What is the difference between Simple and Composite Hypothesis.

What is a hypothesis?

A h ypothesis is an educated guess about how something works. In the scientific method, a hypothesis is an idea that can be tested. If the hypothesis is correct, then the experiment will support the hypothesis. If the hypothesis is incorrect, the experiment will not support the hypothesis.

A hypothesis is simple if it specifies the population completely, i.e., it specifies the population distribution uniquely, while a composite hypothesis leads to two or more possibilities.

Before diving further into their differences, let’s first define a few terms that are handy in understanding the concept of a hypothesis.

Let’s dive in;

Difference between hypothesis and theory.

A hypothesis is a proposed explanation for a phenomenon. A scientific theory is a well-substantiated explanation for an aspect of the natural world supported by a vast body of evidence. Theories are generally much broader in scope than hypotheses and are often not as specific.

The objective of statistics is to make inferences about a population based on information contained in the sample.

There are two major areas of statistical inference, namely;

• Estimation of parameter
• Hypothesis testing

We will develop general methods for testing hypotheses and then apply them to common problems.

Statistical hypothesis

A statistical hypothesis is a testable statement about a population parameter. The statement is based on an assumption about the population parameter. This assumption is usually made about the population parameters based on past research or experience. The statistical hypothesis is used to make predictions about future events. These predictions are based on the assumption that the population parameters will remain the same.

A statistical hypothesis is about a population parameter, usually denoted by some symbol, such as μ or θ.

Statistical hypothesis testing is a method of statistical inference. There are two types of statistical hypothesis tests:

• A point null hypothesis specifies that a population parameter (such as the mean) equals a specific value. For example, the null hypothesis could be that μ=0.
• A composite null hypothesis specifies that a population parameter is less than, greater than, or not equal to a specific value. For example, the null hypothesis could be that μ≠0.

The alternative hypothesis is the hypothesis that is being tested against the null hypothesis. The alternative hypothesis could be that μ>0 or μ<0.

A statistical hypothesis test determines whether or not to reject the null hypothesis. The null hypothesis is rejected if the test statistic is greater than or less than the critical value.

Hypothesis Testing

A hypothesis is a statement or claims about how two variables are related. Hypothesis testing is a statistical procedure used to assess whether the null hypothesis—a statement that there is no difference between two groups or no association between two variables—can be rejected based on sample data. There are four steps in hypothesis testing:

• State the null and alternative hypotheses.
• Select a significance level.
• Calculate the test statistic.
• Interpret the results.

The first step is to state the null and alternative hypotheses. The null hypothesis is that the two variables have no difference or association. The alternative hypothesis is the statement that there is a difference or an association between two variables.

The second step is to select a significance level. The significance level is the probability of rejecting the null hypothesis when it is true. The most common significance levels are 0.05 and 0.01.

The third step is to calculate the test statistic. The test statistic measures the difference between the null and alternative hypotheses. There are many different test statistics, and the choice of test statistic depends on the data type and hypothesis test.

The fourth and final step is to interpret the results. The results of a hypothesis test are either significant or not significant. A significant result means that the null hypothesis can be rejected. A non-significant result means that the null hypothesis cannot be rejected.

Null Hypothesis vs Alternative Hypothesis

In statistics, a null hypothesis is a statement one seeks to disprove, reject or nullify. Most commonly, it is a statement that the phenomenon being studied produces no effect or makes no difference. For example, if one were testing the efficacy of a new drug, the null hypothesis would be that the drug does not affect the treated condition.

The null hypothesis is usually denoted H0, and the alternate hypothesis is denoted H1. If the null hypothesis is rejected in favor of the alternative hypothesis, it is said to be “statistically significant.” The null hypothesis is often assumed to be true until it can be proved otherwise.

Many different types of tests can be used to test a null hypothesis. The most common is the Student’s t-test, which compares the means of two groups. If the t-test is significant, there is a statistically significant difference between the two groups.

Other tests that can be used to test the null hypothesis include the chi-square, Fisher’s exact, and Wilcoxon rank-sum tests.

The alternative hypothesis is the hypothesis that is being tested in a statistical test. This is the hypothesis that is the opposite of the null hypothesis. We are trying to find evidence for the alternative hypothesis in a test.

Simple and Composite Hypothesis

Simple hypothesis.

Hypotheses can be composite or simple, and both are useful depending on the research question and the available evidence.

A simple hypothesis is a straightforward statement that proposes a relationship between two variables. It is a clear, concise statement that is easy to test and evaluate. A simple hypothesis is often used in experimental research where the researcher wants to test the effect of one variable on another.

Examples of hypothesis :

An example of a simple hypothesis is “students who study more will get better grades.” This hypothesis proposes a direct relationship between the amount of time a student spends studying and their academic performance. This hypothesis is testable by comparing the grades of students who study more with those who study less.

Another example of a simple hypothesis is “increased exposure to sunlight will result in higher vitamin D levels.” This hypothesis proposes a direct relationship between sunlight exposure and vitamin D levels. This hypothesis is testable by measuring the vitamin D levels of individuals with varying levels of sunlight exposure.

Simple hypotheses are advantageous because they are easy to test and evaluate. They also allow researchers to focus on a specific research question and avoid unnecessary complexity. Simple hypotheses are particularly useful in experimental research where researchers manipulate one variable to observe its effect on another.

However, simple hypotheses also have limitations. They may oversimplify complex phenomena, and their results may not generalize to a larger population. The available evidence may also limit simple hypotheses, and additional research may be necessary to understand the relationship between variables fully.

In essence, a simple hypothesis is a straightforward statement that proposes a relationship between two variables. Simple hypotheses are useful in experimental research and allow researchers to focus on a specific research question. However, simple hypotheses also have limitations and should be evaluated in the context of the available evidence and research question.

Composite Hypothesis

A composite hypothesis, on the other hand, proposes multiple relationships between two or more variables. For example, a composite hypothesis might state that “there is a significant difference between the average heights of men and women, and there is also a significant difference between the average heights of people from different continents.”

Composite hypothesis testing is a statistical technique used to determine the probability of an event or phenomenon based on observed data. This technique is often used in scientific research, quality control, and decision-making processes where the outcome of a particular experiment or test is uncertain.

A composite hypothesis is an alternative hypothesis encompassing a range of possible outcomes. It is defined as a hypothesis with more than one parameter value. For example, if we are testing the hypothesis that the mean of a population is greater than a certain value, we could define the composite hypothesis as follows:

H1: μ > μ0, where μ is the population means, and μ0 is the hypothesized value of the mean.

The composite hypothesis, in this case, includes all values of μ greater than μ0. This means we are not specifying a specific value of μ, but rather a range of possible values.

Composite hypothesis testing involves evaluating the probability of observing a particular result under the null hypothesis and then comparing it to the probability of observing the same result under the composite hypothesis. The result is considered significant if the probability of observing it under the composite hypothesis is sufficiently low.

We use statistical tests such as the t-test, F-test, or chi-square test to test a composite hypothesis. Given the null hypothesis and the observed data, these tests allow us to calculate the probability of observing a particular result.

In conclusion, composite hypothesis testing is a valuable statistical technique used to determine the probability of an event or phenomenon based on observed data. It allows us to test hypotheses that encompass a range of possible outcomes and is an essential tool for scientific research, quality control , and decision-making processes.

Understanding composite hypothesis testing is essential for anyone working in these fields and can help ensure that decisions are made based on solid statistical evidence.

Post navigation

Previous post.

Beginner’s Guide to Tidying Up Your Datasets: Data Cleaning 101 with Proven Strategies

How to use Which Function in R (With Examples)

How to perform Kolmogorov-Smirnov test in R

How to Plot ROC Curve in R

Statistics/Hypothesis Testing

• 1 Introduction
• 2 Basic concepts and terminologies
• 3 Evaluating a hypothesis test
• 4.1 Neyman-Pearson lemma
• 4.2 Likelihood-ratio test
• 5 Relationship between hypothesis testing and confidence intervals

Introduction [ edit | edit source ]

In previous chapters, we have discussed two methods for estimating unknown parameters , namely point estimation and interval estimation . Estimating unknown parameters is an important area in statistical inference, and in this chapter we will discuss another important area, namely hypothesis testing , which is related to decision making . Indeed, the concepts of confidence intervals and hypothesis testing are closely related, as we will demonstrate.

Basic concepts and terminologies [ edit | edit source ]

Before discussing how to conduct hypothesis testing, and evaluate the "goodness" of a hypothesis test, let us introduce some basic concepts and terminologies related to hypothesis testing first.

Definition. (Hypothesis) A (statistical) hypothesis is a statement about population parameter(s).

There are two terms that classify hypotheses:

Definition. (Simple and composite hypothesis) A hypothesis is a simple hypothesis if it completely specifies the distribution of the population (that is, the distribution is completely known, without any unknown parameters involved), and is a composite hypothesis otherwise.

Sometimes, it is not immediately clear that whether a hypothesis is simple or composite. To understand the classification of hypotheses more clearly, let us consider the following example.

• (a) and (b) are simple hypotheses, since they all completely specifies the distribution.

In hypothesis tests, we consider two hypotheses:

Example. Suppose your friend gives you a coin for tossing, and we do not know whether it is fair or not. However, since the coin is given by your friend, you believe that the coin is fair unless there are sufficient evidences suggesting otherwise. What is the null hypothesis and alternative hypothesis in this context (suppose the coin never land on edge)?

• Of course, in some other places, the saying of "accepting null hypothesis" is avoided because of these philosophical issues.

Now, we are facing with two questions. First, what evidences should we consider? Second, what is meant by "sufficient"? For the first question, a natural answer is that we should consider the observed samples , right? This is because we are making hypothesis about the population, and the samples are taken from, and thus closely related to the population, which should help us make the decision.

Let us formally define the terms related to hypothesis testing in the following.

• Graphically, it looks like

• We use the terminology "tail" since the rejection region includes the values that are located at the "extreme portions" (i.e., very left (with small values) or very right (with large values) portions) (called tails) of distributions.

• We sometimes also call upper-tailed and lower-tailed tests as one-sided tests , and two-tailed tests as two-sided tests .

Exercise. What is the type of this hypothesis test?

Right-tailed test.

As we have mentioned, the decisions made by hypothesis test should not be perfect, and errors occur. Indeed, when we think carefully, there are actually two types of errors, as follows:

We can illustrate these two types of errors more clearly using the following table.

• The power function will be our basis in evaluating the goodness of a test or comparing two different tests.

You notice that the type II error of this hypothesis test can be quite large, so you want to revise the test to lower the type II error.

To describe "control the type I error probability at this level" in a more precise way, let us define the following term.

• Intuitively, we choose the maximum probability of type I error to be the size so that the size can tell us how probable type I error occurs in the worst situation , to show that how "well" can the test control the type I error [4] .

Exercise. Calculate the type I error probability and type II error probability when the sample size is 12 (the rejection region remains unchanged).

• Case 3 : The test is two-tailed.

• For case 3 subcase 1 , consider the following diagram:
• For case 3 subcase 2 , consider the following diagram:

Evaluating a hypothesis test [ edit | edit source ]

After discussing some basic concepts and terminologies, let us now study some ways to evaluate goodness of a hypothesis test. As we have previously mentioned, we want the probability of making type I errors and type II errors to be small, but we have mentioned that it is generally impossible to make both probabilities to be arbitrarily small. Hence, we have suggested to control the type I error, using the size of a test, and the "best" test should the one with the smallest probability of making type II error, after controlling the type I error.

These ideas lead us to the following definitions.

Using this definition, instead of saying "best" test (test with the smallest type II error probability), we can say "a test with the most power", or in other words, the "most powerful test".

Constructing a hypothesis test [ edit | edit source ]

Neyman-pearson lemma [ edit | edit source ].

For the case where the underlying distribution is discrete, the proof is very similar (just replace the integrals with sums), and hence omitted.

• In fact, the MP test constructed by Neyman-Pearson lemma is a variant from the likelihood-ratio test , which is more general in the sense that the likelihood-ratio test can also be constructed for composite null and alternative hypotheses, apart from simple null and alternative hypotheses directly. But, the likelihood-ratio test may not be (U)MP. We will discuss likelihood-ratio test later.

• This rejection region has appeared in a previous example.

Now, let us consider another example where the underlying distribution is discrete.

Exercise. Calculate the probability of making type II error for the above test.

Likelihood-ratio test [ edit | edit source ]

Previously, we have suggested using the Neyman-Pearson lemma to construct MPT for testing simple null hypothesis against simple alternative hypothesis. However, when the hypotheses are composite, we may not be able to use the Neyman-Pearson lemma. So, in the following, we will give a general method for constructing tests for any hypotheses, not limited to simple hypotheses. But we should notice that the tests constructed are not necessarily UMPT.

• When the simple and alternative hypotheses are simple, the likelihood ratio test will be the same as the test suggested in the Neyman-Pearson lemma.

Relationship between hypothesis testing and confidence intervals [ edit | edit source ]

We have mentioned that there are similarities between hypothesis testing and confidence intervals. In this section, we will introduce a theorem suggesting how to construct a hypothesis test from a confidence interval (or in general, confidence set ), and vice versa.

• ↑ Thus, a natural measure of "goodness" of a hypothesis test is its "size of errors". We will discuss these later in this chapter.

• Book:Statistics

Navigation menu

26.1 - Neyman-Pearson Lemma

As we learned from our work in the previous lesson, whenever we perform a hypothesis test, we should make sure that the test we are conducting has sufficient power to detect a meaningful difference from the null hypothesis. That said, how can we be sure that the T -test for a mean \(\mu\) is the "most powerful" test we could use? Is there instead a K -test or a V -test or you-name-the-letter-of-the-alphabet-test that would provide us with more power? A very important result, known as the Neyman Pearson Lemma, will reassure us that each of the tests we learned in Section 7 is the most powerful test for testing statistical hypotheses about the parameter under the assumed probability distribution. Before we can present the lemma, however, we need to:

• Define some notation
• Learn the distinction between simple and composite hypotheses
• Define what it means to have a best critical region of size \(\alpha\). First, the notation.

If \(X_1 , X_2 , \dots , X_n\) is a random sample of size n from a distribution with probability density (or mass) function \f(x; \theta)\), then the joint probability density (or mass) function of \(X_1 , X_2 , \dots , X_n\) is denoted by the likelihood function \(L (\theta)\). That is, the joint p.d.f. or p.m.f. is:

\(L(\theta) =L(\theta; x_1, x_2, ... , x_n) = f(x_1;\theta) \times f(x_2;\theta) \times ... \times f(x_n;\theta)\)

Note that for the sake of ease, we drop the reference to the sample \(X_1 , X_2 , \dots , X_n\) in using \(L (\theta)\) as the notation for the likelihood function. We'll want to keep in mind though that the likelihood \(L (\theta)\) still depends on the sample data.

Now, the definition of simple and composite hypotheses.

If a random sample is taken from a distribution with parameter \(\theta\), a hypothesis is said to be a simple hypothesis if the hypothesis uniquely specifies the distribution of the population from which the sample is taken. Any hypothesis that is not a simple hypothesis is called a composite hypothesis .

Example 26-1

Suppose \(X_1 , X_2 , \dots , X_n\) is a random sample from an exponential distribution with parameter \(\theta\). Is the hypothesis \(H \colon \theta = 3\) a simple or a composite hypothesis?

The p.d.f. of an exponential random variable is:

\(f(x) = \dfrac{1}{\theta}e^{-x/\theta} \)

for \(x ≥ 0\). Under the hypothesis \(H \colon \theta = 3\), the p.d.f. of an exponential random variable is:

\(f(x) = \dfrac{1}{3}e^{-x/3} \)

for \(x ≥ 0\). Because we can uniquely specify the p.d.f. under the hypothesis \(H \colon \theta = 3\), the hypothesis is a simple hypothesis.

Example 26-2

Suppose \(X_1 , X_2 , \dots , X_n\) is a random sample from an exponential distribution with parameter \(\theta\). Is the hypothesis \(H \colon \theta > 2\) a simple or a composite hypothesis?

Again, the p.d.f. of an exponential random variable is:

for \(x ≥ 0\). Under the hypothesis \(H \colon \theta > 2\), the p.d.f. of an exponential random variable could be:

for \(x ≥ 0\). Or, the p.d.f. could be:

\(f(x) = \dfrac{1}{22}e^{-x/22} \)

for \(x ≥ 0\). The p.d.f. could, in fact, be any of an infinite number of possible exponential probability density functions. Because the p.d.f. is not uniquely specified under the hypothesis \(H \colon \theta > 2\), the hypothesis is a composite hypothesis.

Example 26-3

Suppose \(X_1 , X_2 , \dots , X_n\) is a random sample from a normal distribution with mean \(\mu\) and unknown variance \(\sigma^2\). Is the hypothesis \(H \colon \mu = 12\) a simple or a composite hypothesis?

The p.d.f. of a normal random variable is:

\(f(x)= \dfrac{1}{\sigma\sqrt{2\pi}} exp \left[-\dfrac{(x-\mu)^2}{2\sigma^2} \right] \)

for \(−∞ < x <  ∞, −∞ < \mu < ∞\), and \(\sigma > 0\). Under the hypothesis \(H \colon \mu = 12\), the p.d.f. of a normal random variable is:

\(f(x)= \dfrac{1}{\sigma\sqrt{2\pi}} exp \left[-\dfrac{(x-12)^2}{2\sigma^2} \right] \)

for \(−∞ < x < ∞\) and \(\sigma > 0\). In this case, the mean parameter \( \mu = 12\) is uniquely specified in the p.d.f., but the variance \(\sigma^2\) is not. Therefore, the hypothesis \(H \colon \mu = 12\) is a composite hypothesis.

And, finally, the definition of a best critical region of size \(\alpha\).

Consider the test of the simple null hypothesis \(H_0 \colon \theta = \theta_0\) against the simple alternative hypothesis \(H_A \colon \theta = \theta_a\). Let C and D be critical regions of size \(\alpha\), that is, let:

\(\alpha = P(C;\theta_0) \) and \(\alpha = P(D;\theta_0) \)

Then, C is a best critical region of size \(\alpha\) if the power of the test at \(\theta = \theta_a\) is the largest among all possible hypothesis tests. More formally, C is the best critical region of size \(\alpha\) if, for every other critical region D of size \(\alpha\), we have:

\(P(C;\theta_\alpha) \ge P(D;\theta_\alpha)\)

that is, C is the best critical region of size \(\alpha\) if the power of C is at least as great as the power of every other critical region D of size \(\alpha\). We say that C is the most powerful size \(\alpha\) test .

Now that we have clearly defined what we mean for a critical region C to be "best," we're ready to turn to the Neyman Pearson Lemma to learn what form a hypothesis test must take in order for it to be the best, that is, to be the most powerful test.

Suppose we have a random sample \(X_1 , X_2 , \dots , X_n\) from a probability distribution with parameter \(\theta\). Then, if C is a critical region of size \(\alpha\) and k is a constant such that:

\( \dfrac{L(\theta_0)}{L(\theta_\alpha)} \le k \) inside the critical region C

\( \dfrac{L(\theta_0)}{L(\theta_\alpha)} \ge k \) outside the critical region C

then C is the best, that is, most powerful, critical region for testing the simple null hypothesis \(H_0 \colon \theta = \theta_0\) against the simple alternative hypothesis \(H_A \colon \theta = \theta_a\).

See Hogg and Tanis, pages 400-401 (8th edition pages 513-14).

Well, okay, so perhaps the proof isn't all that particularly enlightening, but perhaps if we take a look at a simple example, we'll become more enlightened. Suppose X is a single observation (that's one data point!) from a normal population with unknown mean \(\mu\) and known standard deviation \(\sigma = 1/3\). Then, we can apply the Nehman Pearson Lemma when testing the simple null hypothesis \(H_0 \colon \mu = 3\) against the simple alternative hypothesis \(H_A \colon \mu = 4\). The lemma tells us that, in order to be the most powerful test, the ratio of the likelihoods:

\(\dfrac{L(\mu_0)}{L(\mu_\alpha)} = \dfrac{L(3)}{L(4)} \)

should be small for sample points X inside the critical region C ("less than or equal to some constant k ") and large for sample points X outside of the critical region ("greater than or equal to some constant k "). In this case, because we are dealing with just one observation X , the ratio of the likelihoods equals the ratio of the normal probability curves:

\( \dfrac{L(3)}{L(4)}= \dfrac{f(x; 3, 1/9)}{f(x; 4, 1/9)} \)

Then, the following drawing summarizes the situation:

In short, it makes intuitive sense that we would want to reject \(H_0 \colon \mu = 3\) in favor of \(H_A \colon \mu = 4\) if our observed x is large, that is, if our observed x falls in the critical region C . Well, as the drawing illustrates, it is those large X values in C for which the ratio of the likelihoods is small; and, it is for the small X values not in C for which the ratio of the likelihoods is large. Just as the Neyman Pearson Lemma suggests!

Well, okay, that's the intuition behind the Neyman Pearson Lemma. Now, let's take a look at a few examples of the lemma in action.

Example 26-4

Suppose X is a single observation (again, one data point!) from a population with probabilitiy density function given by:

\(f(x) = \theta x^{\theta -1}\)

for 0 < x < 1. Find the test with the best critical region, that is, find the most powerful test, with significance level \(\alpha = 0.05\), for testing the simple null hypothesis \(H_{0} \colon \theta = 3 \) against the simple alternative hypothesis \(H_{A} \colon \theta = 2 \).

Because both the null and alternative hypotheses are simple hypotheses, we can apply the Neyman Pearson Lemma in an attempt to find the most powerful test. The lemma tells us that the ratio of the likelihoods under the null and alternative must be less than some constant k . Again, because we are dealing with just one observation X , the ratio of the likelihoods equals the ratio of the probability density functions, giving us:

\( \dfrac{L(\theta_0)}{L(\theta_\alpha)}= \dfrac{3x^{3-1}}{2x^{2-1}}= \dfrac{3}{2}x \le k \)

That is, the lemma tells us that the form of the rejection region for the most powerful test is:

\( \dfrac{3}{2}x \le k \)

or alternatively, since (2/3) k is just a new constant \(k^*\), the rejection region for the most powerful test is of the form:

\(x < \dfrac{2}{3}k = k^* \)

Now, it's just a matter of finding \(k^*\), and our work is done. We want \(\alpha\) = P(Type I Error) = P (rejecting the null hypothesis when the null hypothesis is true) to equal 0.05. In order for that to happen, the following must hold:

\(\alpha = P( X < k^* \text{ when } \theta = 3) = \int_{0}^{k^*} 3x^2dx = 0.05 \)

Doing the integration, we get:

\( \left[ x^3\right]^{x=k^*}_{x=0} = (k^*)^3 =0.05 \)

And, solving for \(k^*\), we get:

\(k^* =(0.05)^{1/3} = 0.368 \)

That is, the Neyman Pearson Lemma tells us that the rejection region of the most powerful test for testing \(H_{0} \colon \theta = 3 \) against \(H_{A} \colon \theta = 2 \), under the assumed probability distribution, is:

\(x < 0.368 \)

That is, among all of the possible tests for testing \(H_{0} \colon \theta = 3 \) against \(H_{A} \colon \theta = 2 \), based on a single observation X and with a significance level of 0.05, this test has the largest possible value for the power under the alternative hypohthesis, that is, when \(\theta = 2\).

Example 26-5

Suppose \(X_1 , X_2 , \dots , X_n\) is a random sample from a normal population with mean \(\mu\) and variance 16. Find the test with the best critical region, that is, find the most powerful test, with a sample size of \(n = 16\) and a significance level \(\alpha = 0.05\) to test the simple null hypothesis \(H_{0} \colon \mu = 10 \) against the simple alternative hypothesis \(H_{A} \colon \mu = 15 \).

Because the variance is specified, both the null and alternative hypotheses are simple hypotheses. Therefore, we can apply the Neyman Pearson Lemma in an attempt to find the most powerful test. The lemma tells us that the ratio of the likelihoods under the null and alternative must be less than some constant k :

\( \dfrac{L(10)}{L(15)}= \dfrac{(32\pi)^{-16/2} exp \left[ -(1/32)\sum_{i=1}^{16}(x_i -10)^2 \right]}{(32\pi)^{-16/2} exp \left[ -(1/32)\sum_{i=1}^{16}(x_i -15)^2 \right]} \le k \)

Simplifying, we get:

\(exp \left[ - \left( \dfrac{1}{32} \right) \left( \sum_{i=1}^{16}(x_i -10)^2 - \sum_{i=1}^{16}(x_i -15)^2 \right) \right] \le k \)

And, simplifying yet more, we get:

Now, taking the natural logarithm of both sides of the inequality, collecting like terms, and multiplying through by 32, we get:

\(-10\Sigma x_i +2000 \le 32ln(k)\)

And, moving the constant term on the left-side of the inequality to the right-side, and dividing through by −160, we get:

\(\dfrac{1}{16}\Sigma x_i \ge -\frac{1}{160}(32ln(k)-2000) \)

That is, the Neyman Pearson Lemma tells us that the rejection region for the most powerful test for testing \(H_{0} \colon \mu = 10 \) against \(H_{A} \colon \mu = 15 \), under the normal probability model, is of the form:

\(\bar{x} \ge k^* \)

where \(k^*\) is selected so that the size of the critical region is \(\alpha = 0.05\). That's simple enough, as it just involves a normal probabilty calculation! Under the null hypothesis, the sample mean is normally distributed with mean 10 and standard deviation 4/4 = 1. Therefore, the critical value \(k^*\) is deemed to be 11.645:

That is, the Neyman Pearson Lemma tells us that the rejection region for the most powerful test for testing \(H_{0} \colon \mu = 10 \) against \(H_{A} \colon \mu = 15 \), under the normal probability model, is:

\(\bar{x} \ge 11.645 \)

The power of such a test when \(\mu = 15\) is:

\( P(\bar{X} > 11.645 \text{ when } \mu = 15) = P \left( Z > \dfrac{11.645-15}{\sqrt{16} / \sqrt{16} } \right) = P(Z > -3.36) = 0.9996 \)

The power can't get much better than that, and the Neyman Pearson Lemma tells us that we shouldn't expect it to get better! That is, the Lemma tells us that there is no other test out there that will give us greater power for testing \(H_{0} \colon \mu = 10 \) against \(H_{A} \colon \mu = 15 \).

What does "Composite Hypothesis" mean?

Definition of Composite Hypothesis in the context of A/B testing (online controlled experiments).

What is a Composite Hypothesis?

In hypothesis testing a composite hypothesis is a hypothesis which covers a set of values from the parameter space. For example, if the entire parameter space covers -∞ to +∞ a composite hypothesis could be μ ≤ 0. It could be any other number as well, such 1, 2 or 3,1245. The alternative hypothesis is always a composite hypothesis : either one-sided hypothesis if the null is composite or a two-sided one if the null is a point null. The "composite" part means that such a hypothesis is the union of many simple point hypotheses.

In a Null Hypothesis Statistical Test only the null hypothesis can be a point hypothesis. Also, a composite hypothesis usually spans from -∞ to zero or some value of practical significance or from such a value to +∞.

Testing a composite null is what is most often of interest in an A/B testing scenario as we are usually interested in detecting and estimating effects in only one direction: either an increase in conversion rate or average revenue per user, or a decrease in unsubscribe events would be of interest and not its opposite. In fact, running a test so long as to detect a statistically significant negative outcome can result in significant business harm.

Like this glossary entry? For an in-depth and comprehensive reading on A/B testing stats, check out the book "Statistical Methods in Online A/B Testing" by the author of this glossary, Georgi Georgiev.

Articles on Composite Hypothesis

One-tailed vs Two-tailed Tests of Significance in A/B Testing blog.analytics-toolkit.com

Related A/B Testing terms

Statistical Methods in Online A/B Testing

Take your A/B testing program to the next level with the most comprehensive book on user testing statistics in e-commerce.

Glossary index by letter

Select a letter to see all A/B testing terms starting with that letter or visit the Glossary homepage to see all.

Biostatistics

• Data Science
• Programming
• Social Science
• Certificates
• Undergraduate
• For Businesses
• FAQs and Knowledge Base

Test Yourself

• Instructors

Composite Hypothesis

Composite Hypothesis:

A statistical hypothesis which does not completely specify the distribution of a random variable is referred to as a composite hypothesis.

Browse Other Glossary Entries

Planning on taking an introductory statistics course, but not sure if you need to start at the beginning? Review the course description for each of our introductory statistics courses and estimate which best matches your level, then take the self test for that course. If you get all or almost all the questions correct, move on and take the next test.

Data Analytics

Considering becoming adata scientist, customer analyst or our data science certificate program?

Advanced Statistics Quiz

Looking at statistics for graduate programs or to enhance your foundational knowledge?

Regression Quiz

Entering the biostatistics field? Test your skill here.

Stay Informed

Read up on our latest blogs

Learn about our certificate programs

Find the right course for you

We'd love to answer your questions

Our mentors and academic advisors are standing by to help guide you towards the courses or program that makes the most sense for you and your goals.

300 W Main St STE 301, Charlottesville, VA 22903

(434) 973-7673

[email protected]

By submitting your information, you agree to receive email communications from Statistics.com. All information submitted is subject to our privacy policy . You may opt out of receiving communications at any time.