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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.
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.
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)?
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.
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.
Accept | Reject | |
---|---|---|
is true | Correct decision | Type I error |
is false | Type II error | Correct decision |
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.
Exercise. Calculate the type I error probability and type II error probability when the sample size is 12 (the rejection region remains unchanged).
0.01 | |
0.04 | |
0.06 | |
0.08 | |
0.1 |
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".
Neyman-pearson lemma.
For the case where the underlying distribution is discrete, the proof is very similar (just replace the integrals with sums), and hence omitted.
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.
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.
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.
Composite Hypothesis:
A statistical hypothesis which does not completely specify the distribution of a random variable is referred to as a composite hypothesis.
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Published on May 6, 2022 by Shaun Turney . Revised on June 22, 2023.
The null and alternative hypotheses are two competing claims that researchers weigh evidence for and against using a statistical test :
Answering your research question with hypotheses, what is a null hypothesis, what is an alternative hypothesis, similarities and differences between null and alternative hypotheses, how to write null and alternative hypotheses, other interesting articles, frequently asked questions.
The null and alternative hypotheses offer competing answers to your research question . When the research question asks “Does the independent variable affect the dependent variable?”:
The null and alternative are always claims about the population. That’s because the goal of hypothesis testing is to make inferences about a population based on a sample . Often, we infer whether there’s an effect in the population by looking at differences between groups or relationships between variables in the sample. It’s critical for your research to write strong hypotheses .
You can use a statistical test to decide whether the evidence favors the null or alternative hypothesis. Each type of statistical test comes with a specific way of phrasing the null and alternative hypothesis. However, the hypotheses can also be phrased in a general way that applies to any test.
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The null hypothesis is the claim that there’s no effect in the population.
If the sample provides enough evidence against the claim that there’s no effect in the population ( p ≤ α), then we can reject the null hypothesis . Otherwise, we fail to reject the null hypothesis.
Although “fail to reject” may sound awkward, it’s the only wording that statisticians accept . Be careful not to say you “prove” or “accept” the null hypothesis.
Null hypotheses often include phrases such as “no effect,” “no difference,” or “no relationship.” When written in mathematical terms, they always include an equality (usually =, but sometimes ≥ or ≤).
You can never know with complete certainty whether there is an effect in the population. Some percentage of the time, your inference about the population will be incorrect. When you incorrectly reject the null hypothesis, it’s called a type I error . When you incorrectly fail to reject it, it’s a type II error.
The table below gives examples of research questions and null hypotheses. There’s always more than one way to answer a research question, but these null hypotheses can help you get started.
( ) | ||
Does tooth flossing affect the number of cavities? | Tooth flossing has on the number of cavities. | test: The mean number of cavities per person does not differ between the flossing group (µ ) and the non-flossing group (µ ) in the population; µ = µ . |
Does the amount of text highlighted in the textbook affect exam scores? | The amount of text highlighted in the textbook has on exam scores. | : There is no relationship between the amount of text highlighted and exam scores in the population; β = 0. |
Does daily meditation decrease the incidence of depression? | Daily meditation the incidence of depression.* | test: The proportion of people with depression in the daily-meditation group ( ) is greater than or equal to the no-meditation group ( ) in the population; ≥ . |
*Note that some researchers prefer to always write the null hypothesis in terms of “no effect” and “=”. It would be fine to say that daily meditation has no effect on the incidence of depression and p 1 = p 2 .
The alternative hypothesis ( H a ) is the other answer to your research question . It claims that there’s an effect in the population.
Often, your alternative hypothesis is the same as your research hypothesis. In other words, it’s the claim that you expect or hope will be true.
The alternative hypothesis is the complement to the null hypothesis. Null and alternative hypotheses are exhaustive, meaning that together they cover every possible outcome. They are also mutually exclusive, meaning that only one can be true at a time.
Alternative hypotheses often include phrases such as “an effect,” “a difference,” or “a relationship.” When alternative hypotheses are written in mathematical terms, they always include an inequality (usually ≠, but sometimes < or >). As with null hypotheses, there are many acceptable ways to phrase an alternative hypothesis.
The table below gives examples of research questions and alternative hypotheses to help you get started with formulating your own.
Does tooth flossing affect the number of cavities? | Tooth flossing has an on the number of cavities. | test: The mean number of cavities per person differs between the flossing group (µ ) and the non-flossing group (µ ) in the population; µ ≠ µ . |
Does the amount of text highlighted in a textbook affect exam scores? | The amount of text highlighted in the textbook has an on exam scores. | : There is a relationship between the amount of text highlighted and exam scores in the population; β ≠ 0. |
Does daily meditation decrease the incidence of depression? | Daily meditation the incidence of depression. | test: The proportion of people with depression in the daily-meditation group ( ) is less than the no-meditation group ( ) in the population; < . |
Null and alternative hypotheses are similar in some ways:
However, there are important differences between the two types of hypotheses, summarized in the following table.
A claim that there is in the population. | A claim that there is in the population. | |
| ||
Equality symbol (=, ≥, or ≤) | Inequality symbol (≠, <, or >) | |
Rejected | Supported | |
Failed to reject | Not supported |
To help you write your hypotheses, you can use the template sentences below. If you know which statistical test you’re going to use, you can use the test-specific template sentences. Otherwise, you can use the general template sentences.
The only thing you need to know to use these general template sentences are your dependent and independent variables. To write your research question, null hypothesis, and alternative hypothesis, fill in the following sentences with your variables:
Does independent variable affect dependent variable ?
Once you know the statistical test you’ll be using, you can write your hypotheses in a more precise and mathematical way specific to the test you chose. The table below provides template sentences for common statistical tests.
( ) | ||
test
with two groups | The mean dependent variable does not differ between group 1 (µ ) and group 2 (µ ) in the population; µ = µ . | The mean dependent variable differs between group 1 (µ ) and group 2 (µ ) in the population; µ ≠ µ . |
with three groups | The mean dependent variable does not differ between group 1 (µ ), group 2 (µ ), and group 3 (µ ) in the population; µ = µ = µ . | The mean dependent variable of group 1 (µ ), group 2 (µ ), and group 3 (µ ) are not all equal in the population. |
There is no correlation between independent variable and dependent variable in the population; ρ = 0. | There is a correlation between independent variable and dependent variable in the population; ρ ≠ 0. | |
There is no relationship between independent variable and dependent variable in the population; β = 0. | There is a relationship between independent variable and dependent variable in the population; β ≠ 0. | |
Two-proportions test | The dependent variable expressed as a proportion does not differ between group 1 ( ) and group 2 ( ) in the population; = . | The dependent variable expressed as a proportion differs between group 1 ( ) and group 2 ( ) in the population; ≠ . |
Note: The template sentences above assume that you’re performing one-tailed tests . One-tailed tests are appropriate for most studies.
If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.
Methodology
Research bias
Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.
Null and alternative hypotheses are used in statistical hypothesis testing . The null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship.
The null hypothesis is often abbreviated as H 0 . When the null hypothesis is written using mathematical symbols, it always includes an equality symbol (usually =, but sometimes ≥ or ≤).
The alternative hypothesis is often abbreviated as H a or H 1 . When the alternative hypothesis is written using mathematical symbols, it always includes an inequality symbol (usually ≠, but sometimes < or >).
A research hypothesis is your proposed answer to your research question. The research hypothesis usually includes an explanation (“ x affects y because …”).
A statistical hypothesis, on the other hand, is a mathematical statement about a population parameter. Statistical hypotheses always come in pairs: the null and alternative hypotheses . In a well-designed study , the statistical hypotheses correspond logically to the research hypothesis.
If you want to cite this source, you can copy and paste the citation or click the “Cite this Scribbr article” button to automatically add the citation to our free Citation Generator.
Turney, S. (2023, June 22). Null & Alternative Hypotheses | Definitions, Templates & Examples. Scribbr. Retrieved July 30, 2024, from https://www.scribbr.com/statistics/null-and-alternative-hypotheses/
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So if we have
$H_0 :\theta=\theta_0$ vs $H_1 :\theta=\theta_1$
It is easy to see that this is a case of simple vs simple hypothesis (assuming that $\theta$ is the only unknown parameter of our distribution)
$H_0 :\theta\leq\theta_0$ vs $H_1 :\theta>\theta_0$
Is this composite vs composite or simple vs composite?
Since it is somewhat equivalent to
$H_0 :\theta=\theta_0$ vs $H_1 :\theta>\theta_0$
Which I guess it's a simple vs composite hypothesis
And last, if we have two unkown parameters, is $H_0 :\alpha=\alpha_0 , \beta\geq\beta_0$
Simple or composite?
$H_0 :\theta=\theta_0$ vs $H_1 :\theta>\theta_0$ is a composite hypothesis since for $H_1$ you can have many different $\theta$s.
You can check these links the explanations are pretty clear.
http://www.emathzone.com/tutorials/basic-statistics/simple-hypothesis-and-composite-hypothesis.html
http://isites.harvard.edu/fs/docs/icb.topic1383356.files/Lecture%2014%20-%20Intro%20to%20Hypothesis%20Testing%20-%204%20per%20page.pdf
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0versus H. 1: µ>µ. 0, 278. Introduction to the Science of Statistics Composite Hypotheses. we use the test statistic from the likelihood ratio test and reject H. 0if the statistic x¯ is too large. Thus, the critical region C = {x;¯x k(µ. 0)}. If µ is the true mean, then the power function ⇡(µ)=P.
A composite hypothesis test contains more than one parameter and more than one model. In a simple hypothesis test, the probability density functions for both the null hypothesis (H 0) and alternate hypothesis (H 1) are known. In academic and hypothetical situations, the simple hypothesis test works for most cases.
Composite Hypothesis. ... A statistical hypothesis is a statement concerning the probability distribution of a random variable or population parameters that are inherent in a probability distribution. ... Definition 6.1.1. A hypothesis is said to be a simple hypothesis if that hypothesis uniquely specifies the distribution from which the sample ...
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 Ho: μ = 62 H o: μ = 62 or p > 0.5 p > 0.5. A hypothesis is called an ...
Definition: Simple and composite hypothesis. Definition: Let H H be a statistical hypothesis. Then, H 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 H is called a composite hypothesis, if it does not ...
STATS 200: Introduction to Statistical Inference Autumn 2016 Lecture 7 | Composite hypotheses and the t-test 7.1 Composite null and alternative hypotheses This week we will discuss various hypothesis testing problems involving a composite null hypothesis and a compositive alternative hypothesis. To motivate the discussion, consider
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 ...
Introduction to Statistical Methodology Composite Hypotheses In reality, incorrect decisions are made. Thus, for 2 0, ˇ( ) is the probability of making a type I error; i. e., rejecting the null hypothesis when it is indeed true, For 2 1, 1 ˇ( ) is the probability of making a type II error; i.e., failing to reject the null hypothesis when it ...
What Composite hypothesis is. Composite hypothesis is a type of statistical hypothesis test that combines two or more simple hypotheses into a single test. It is used to test the overall effect of a set of variables on a response variable. The steps for performing a composite hypothesis test are as follows: State the null and alternative ...
The composite hypothesis is a statistical term that refers to the combination of two or more individual hypotheses. This approach is often used in hypothesis testing, where a researcher is trying to determine whether a certain result is statistically significant.In other words, the composite hypothesis is a way of testing multiple hypotheses at once, allowing for a more comprehensive ...
For a composite hypothesis, the parameter space is divided into two disjoint regions, 0 and 1. The test is written H 0: 2 0 versus H 1: 2 1 with H 0 is called the null hypothesis and H 1 the alternative hypothesis. Rejection and failure to reject the null hypothesis, critical regions, C, and type I and type II errors have the same
then we have a simple hypothesis, as discussed in past lectures. When a set contains more than one parameter value, then the hypothesis is called a composite hypothesis, because it involves more than one model. The name is even clearer if we consider the following equivalent expression for the hypotheses above. H 0: X ˘p 0; p 0 2fp 0(xj 0)g 02 ...
Updated on 04/19/2018. a statistical hypothesis that is not specific about all relevant features of a population or that does not give a single value for a characteristic of a population but allows for a range of acceptable values. For example, a statement that the average age of employees in academia exceeds 50 is a composite hypothesis, as ...
Definition : A statistical hypothesis is an assertion or conjecture about the distribution of one or more random variables. If a statistical hypothesis completely specifies the distribution, it is referred to as a simple hypothesis; if not, it is referred to as a composite hypothesis. §2. General Steps in Hypothesis Testing
Hypothesis testing is a statistical method used to determine if there is enough evidence in a sample data to draw conclusions about a population. It involves formulating two competing hypotheses, the null hypothesis (H0) and the alternative hypothesis (Ha), and then collecting data to assess the evidence.
Statistical hypothesis: A statement about the nature of a population. It is often stated in terms of a population parameter. ... A composite score from 0 to 10 points, representing both recall and persuasion, has been produced for each consumer in the sample. ... Definition 3.1. The null hypothesis is a statement about the values of one or more ...
A statistical hypothesis test is a method of statistical inference used to decide whether the data sufficiently support a particular hypothesis. A statistical hypothesis test typically involves a calculation of a test statistic. Then a decision is made, either by comparing the test statistic to a critical value or equivalently by evaluating a p ...
(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 ...
Reminder - Frequentist test statistics and p-values • Definition of 'p-value': Probability to observe this outcome or more extreme in future repeated measurements is x%, if ... Composite hypothesis testing in the asymptotic regime • For 'histogram example': what is p-value of null-hypothesis t 0 =34.77 t 0 =0.02 t 0 = - 2 ln
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. Test Yourself. Planning on taking an introductory statistics course, but not sure if you need to start at the beginning? Review the course description for ...
A statistical hypothesis, on the other hand, is a mathematical statement about a population parameter. Statistical hypotheses always come in pairs: the null and alternative hypotheses . In a well-designed study , the statistical hypotheses correspond logically to the research hypothesis.
In practice, one may evaluate different choices of t ( s) to determine the empirically optimal one. For a composite hypothesis testing problem in (1), we define Θ 1 ⊆ R as a neighborhood of the true value of θ1. The corresponding notations for θ2, η 1 and η 2 are Θ 2 ⊆ R, H 1 ⊆ R w and H 2 ⊆ R w, respectively.
3. So if we have. H0: θ = θ0 H 0: θ = θ 0 vs H1: θ = θ1 H 1: θ = θ 1. It is easy to see that this is a case of simple vs simple hypothesis (assuming that θ θ is the only unknown parameter of our distribution) what about. H0: θ ≤ θ0 H 0: θ ≤ θ 0 vs H1: θ > θ0 H 1: θ > θ 0.