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## Null and Alternative Hypotheses | Definitions & Examples

Published on 5 October 2022 by Shaun Turney . Revised on 6 December 2022.

The null and alternative hypotheses are two competing claims that researchers weigh evidence for and against using a statistical test :

- Null hypothesis (H 0 ): There’s no effect in the population .
- Alternative hypothesis (H A ): There’s an effect in the population.

The effect is usually the effect of the independent variable on the dependent variable .

## Table of contents

Answering your research question with hypotheses, what is a null hypothesis, what is an alternative hypothesis, differences between null and alternative hypotheses, how to write null and alternative hypotheses, frequently asked questions about null and alternative hypotheses.

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 hypothesis (H 0 ) answers “No, there’s no effect in the population.” On the other hand, the alternative hypothesis (H A ) answers “Yes, there is an effect in the population.”

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.

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.

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 ≤).

## Examples of null hypotheses

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.

## Examples of alternative hypotheses

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:

- They’re both answers to the research question
- They both make claims about the population
- They’re both evaluated by statistical tests.

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 ?

- Null hypothesis (H 0 ): Independent variable does not affect dependent variable .
- Alternative hypothesis (H A ): Independent variable affects dependent variable .

## Test-specific

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.

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.

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## Module 9: Hypothesis Testing With One Sample

Null and alternative hypotheses, learning outcomes.

- Describe hypothesis testing in general and in practice

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

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

H a : The alternative hypothesis : It is a claim about the population that is contradictory to H 0 and what we conclude when we reject H 0 .

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

After you have determined which hypothesis the sample supports, you make adecision. There are two options for a decision . They are “reject H 0 ” if the sample information favors the alternative hypothesis or “do not reject H 0 ” or “decline to reject H 0 ” if the sample information is insufficient to reject the null hypothesis.

Mathematical Symbols Used in H 0 and H a :

equal (=) | not equal (≠) greater than (>) less than (<) |

greater than or equal to (≥) | less than (<) |

less than or equal to (≤) | more than (>) |

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

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

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

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

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

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

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

H 0 : μ = 2.0

H a : μ ≠ 2.0

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

- H 0 : μ = 66
- H a : μ ≠ 66

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

H 0 : μ ≥ 5

H a : μ < 5

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

- H 0 : μ ≥ 45
- H a : μ < 45

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

H 0 : p ≤ 0.066

H a : p > 0.066

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

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

## Concept Review

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

## Formula Review

H 0 and H a are contradictory.

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

## 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 :

equal (=) | not equal (≠) greater than (>) less than (<) |

greater than or equal to (≥) | less than (<) |

less than or equal to (≤) | more than (>) |

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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Statistics By Jim

Making statistics intuitive

## Null Hypothesis: Definition, Rejecting & Examples

By Jim Frost 6 Comments

## What is a Null Hypothesis?

The null hypothesis in statistics states that there is no difference between groups or no relationship between variables. It is one of two mutually exclusive hypotheses about a population in a hypothesis test.

- Null Hypothesis H 0 : No effect exists in the population.
- Alternative Hypothesis H A : The effect exists in the population.

In every study or experiment, researchers assess an effect or relationship. This effect can be the effectiveness of a new drug, building material, or other intervention that has benefits. There is a benefit or connection that the researchers hope to identify. Unfortunately, no effect may exist. In statistics, we call this lack of an effect the null hypothesis. Researchers assume that this notion of no effect is correct until they have enough evidence to suggest otherwise, similar to how a trial presumes innocence.

In this context, the analysts don’t necessarily believe the null hypothesis is correct. In fact, they typically want to reject it because that leads to more exciting finds about an effect or relationship. The new vaccine works!

You can think of it as the default theory that requires sufficiently strong evidence to reject. Like a prosecutor, researchers must collect sufficient evidence to overturn the presumption of no effect. Investigators must work hard to set up a study and a data collection system to obtain evidence that can reject the null hypothesis.

Related post : What is an Effect in Statistics?

## Null Hypothesis Examples

Null hypotheses start as research questions that the investigator rephrases as a statement indicating there is no effect or relationship.

Does the vaccine prevent infections? | The vaccine does not affect the infection rate. |

Does the new additive increase product strength? | The additive does not affect mean product strength. |

Does the exercise intervention increase bone mineral density? | The intervention does not affect bone mineral density. |

As screen time increases, does test performance decrease? | There is no relationship between screen time and test performance. |

After reading these examples, you might think they’re a bit boring and pointless. However, the key is to remember that the null hypothesis defines the condition that the researchers need to discredit before suggesting an effect exists.

Let’s see how you reject the null hypothesis and get to those more exciting findings!

## When to Reject the Null Hypothesis

So, you want to reject the null hypothesis, but how and when can you do that? To start, you’ll need to perform a statistical test on your data. The following is an overview of performing a study that uses a hypothesis test.

The first step is to devise a research question and the appropriate null hypothesis. After that, the investigators need to formulate an experimental design and data collection procedures that will allow them to gather data that can answer the research question. Then they collect the data. For more information about designing a scientific study that uses statistics, read my post 5 Steps for Conducting Studies with Statistics .

After data collection is complete, statistics and hypothesis testing enter the picture. Hypothesis testing takes your sample data and evaluates how consistent they are with the null hypothesis. The p-value is a crucial part of the statistical results because it quantifies how strongly the sample data contradict the null hypothesis.

When the sample data provide sufficient evidence, you can reject the null hypothesis. In a hypothesis test, this process involves comparing the p-value to your significance level .

## Rejecting the Null Hypothesis

Reject the null hypothesis when the p-value is less than or equal to your significance level. Your sample data favor the alternative hypothesis, which suggests that the effect exists in the population. For a mnemonic device, remember—when the p-value is low, the null must go!

When you can reject the null hypothesis, your results are statistically significant. Learn more about Statistical Significance: Definition & Meaning .

## Failing to Reject the Null Hypothesis

Conversely, when the p-value is greater than your significance level, you fail to reject the null hypothesis. The sample data provides insufficient data to conclude that the effect exists in the population. When the p-value is high, the null must fly!

Note that failing to reject the null is not the same as proving it. For more information about the difference, read my post about Failing to Reject the Null .

That’s a very general look at the process. But I hope you can see how the path to more exciting findings depends on being able to rule out the less exciting null hypothesis that states there’s nothing to see here!

Let’s move on to learning how to write the null hypothesis for different types of effects, relationships, and tests.

Related posts : How Hypothesis Tests Work and Interpreting P-values

## How to Write a Null Hypothesis

The null hypothesis varies by the type of statistic and hypothesis test. Remember that inferential statistics use samples to draw conclusions about populations. Consequently, when you write a null hypothesis, it must make a claim about the relevant population parameter . Further, that claim usually indicates that the effect does not exist in the population. Below are typical examples of writing a null hypothesis for various parameters and hypothesis tests.

Related posts : Descriptive vs. Inferential Statistics and Populations, Parameters, and Samples in Inferential Statistics

## Group Means

T-tests and ANOVA assess the differences between group means. For these tests, the null hypothesis states that there is no difference between group means in the population. In other words, the experimental conditions that define the groups do not affect the mean outcome. Mu (µ) is the population parameter for the mean, and you’ll need to include it in the statement for this type of study.

For example, an experiment compares the mean bone density changes for a new osteoporosis medication. The control group does not receive the medicine, while the treatment group does. The null states that the mean bone density changes for the control and treatment groups are equal.

- Null Hypothesis H 0 : Group means are equal in the population: µ 1 = µ 2 , or µ 1 – µ 2 = 0
- Alternative Hypothesis H A : Group means are not equal in the population: µ 1 ≠ µ 2 , or µ 1 – µ 2 ≠ 0.

## Group Proportions

Proportions tests assess the differences between group proportions. For these tests, the null hypothesis states that there is no difference between group proportions. Again, the experimental conditions did not affect the proportion of events in the groups. P is the population proportion parameter that you’ll need to include.

For example, a vaccine experiment compares the infection rate in the treatment group to the control group. The treatment group receives the vaccine, while the control group does not. The null states that the infection rates for the control and treatment groups are equal.

- Null Hypothesis H 0 : Group proportions are equal in the population: p 1 = p 2 .
- Alternative Hypothesis H A : Group proportions are not equal in the population: p 1 ≠ p 2 .

## Correlation and Regression Coefficients

Some studies assess the relationship between two continuous variables rather than differences between groups.

In these studies, analysts often use either correlation or regression analysis . For these tests, the null states that there is no relationship between the variables. Specifically, it says that the correlation or regression coefficient is zero. As one variable increases, there is no tendency for the other variable to increase or decrease. Rho (ρ) is the population correlation parameter and beta (β) is the regression coefficient parameter.

For example, a study assesses the relationship between screen time and test performance. The null states that there is no correlation between this pair of variables. As screen time increases, test performance does not tend to increase or decrease.

- Null Hypothesis H 0 : The correlation in the population is zero: ρ = 0.
- Alternative Hypothesis H A : The correlation in the population is not zero: ρ ≠ 0.

For all these cases, the analysts define the hypotheses before the study. After collecting the data, they perform a hypothesis test to determine whether they can reject the null hypothesis.

The preceding examples are all for two-tailed hypothesis tests. To learn about one-tailed tests and how to write a null hypothesis for them, read my post One-Tailed vs. Two-Tailed Tests .

Related post : Understanding Correlation

Neyman, J; Pearson, E. S. (January 1, 1933). On the Problem of the most Efficient Tests of Statistical Hypotheses . Philosophical Transactions of the Royal Society A . 231 (694–706): 289–337.

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## Reader Interactions

January 11, 2024 at 2:57 pm

Thanks for the reply.

January 10, 2024 at 1:23 pm

Hi Jim, In your comment you state that equivalence test null and alternate hypotheses are reversed. For hypothesis tests of data fits to a probability distribution, the null hypothesis is that the probability distribution fits the data. Is this correct?

January 10, 2024 at 2:15 pm

Those two separate things, equivalence testing and normality tests. But, yes, you’re correct for both.

Hypotheses are switched for equivalence testing. You need to “work” (i.e., collect a large sample of good quality data) to be able to reject the null that the groups are different to be able to conclude they’re the same.

With typical hypothesis tests, if you have low quality data and a low sample size, you’ll fail to reject the null that they’re the same, concluding they’re equivalent. But that’s more a statement about the low quality and small sample size than anything to do with the groups being equal.

So, equivalence testing make you work to obtain a finding that the groups are the same (at least within some amount you define as a trivial difference).

For normality testing, and other distribution tests, the null states that the data follow the distribution (normal or whatever). If you reject the null, you have sufficient evidence to conclude that your sample data don’t follow the probability distribution. That’s a rare case where you hope to fail to reject the null. And it suffers from the problem I describe above where you might fail to reject the null simply because you have a small sample size. In that case, you’d conclude the data follow the probability distribution but it’s more that you don’t have enough data for the test to register the deviation. In this scenario, if you had a larger sample size, you’d reject the null and conclude it doesn’t follow that distribution.

I don’t know of any equivalence testing type approach for distribution fit tests where you’d need to work to show the data follow a distribution, although I haven’t looked for one either!

February 20, 2022 at 9:26 pm

Is a null hypothesis regularly (always) stated in the negative? “there is no” or “does not”

February 23, 2022 at 9:21 pm

Typically, the null hypothesis includes an equal sign. The null hypothesis states that the population parameter equals a particular value. That value is usually one that represents no effect. In the case of a one-sided hypothesis test, the null still contains an equal sign but it’s “greater than or equal to” or “less than or equal to.” If you wanted to translate the null hypothesis from its native mathematical expression, you could use the expression “there is no effect.” But the mathematical form more specifically states what it’s testing.

It’s the alternative hypothesis that typically contains does not equal.

There are some exceptions. For example, in an equivalence test where the researchers want to show that two things are equal, the null hypothesis states that they’re not equal.

In short, the null hypothesis states the condition that the researchers hope to reject. They need to work hard to set up an experiment and data collection that’ll gather enough evidence to be able to reject the null condition.

February 15, 2022 at 9:32 am

Dear sir I always read your notes on Research methods.. Kindly tell is there any available Book on all these..wonderfull Urgent

## Comments and Questions Cancel reply

## Hypothesis Testing: Null Hypothesis and Alternative Hypothesis

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Figuring out exactly what the null hypothesis and the alternative hypotheses are is not a walk in the park. Hypothesis testing is based on the knowledge that you can acquire by going over what we have previously covered about statistics in our blog.

So, if you don’t want to have a hard time keeping up, make sure you have read all the tutorials about confidence intervals , distributions , z-tables and t-tables .

We've also made a video on null hypothesis vs alternative hypothesis - you can watch it below or just scroll down if you prefer reading.

Confidence intervals provide us with an estimation of where the parameters are located. You can obtain them with our confidence interval calculator and learn more about them in the related article.

However, when we are making a decision, we need a yes or no answer. The correct approach, in this case, is to use a test .

Here we will start learning about one of the fundamental tasks in statistics - hypothesis testing !

## The Hypothesis Testing Process

First off, let’s talk about data-driven decision-making. It consists of the following steps:

- First, we must formulate a hypothesis .
- After doing that, we have to find the right test for our hypothesis .
- Then, we execute the test.
- Finally, we make a decision based on the result.

Let’s start from the beginning.

## What is a Hypothesis?

Though there are many ways to define it, the most intuitive must be:

“A hypothesis is an idea that can be tested.”

This is not the formal definition, but it explains the point very well.

So, if we say that apples in New York are expensive, this is an idea or a statement. However, it is not testable, until we have something to compare it with.

For instance, if we define expensive as: any price higher than $1.75 dollars per pound, then it immediately becomes a hypothesis .

## What Cannot Be a Hypothesis?

An example may be: would the USA do better or worse under a Clinton administration, compared to a Trump administration? Statistically speaking, this is an idea , but there is no data to test it. Therefore, it cannot be a hypothesis of a statistical test.

Actually, it is more likely to be a topic of another discipline.

Conversely, in statistics, we may compare different US presidencies that have already been completed. For example, the Obama administration and the Bush administration, as we have data on both.

## A Two-Sided Test

Alright, let’s get out of politics and get into hypotheses . Here’s a simple topic that CAN be tested.

According to Glassdoor (the popular salary information website), the mean data scientist salary in the US is 113,000 dollars.

So, we want to test if their estimate is correct.

## The Null and Alternative Hypotheses

There are two hypotheses that are made: the null hypothesis , denoted H 0 , and the alternative hypothesis , denoted H 1 or H A .

The null hypothesis is the one to be tested and the alternative is everything else. In our example:

The null hypothesis would be: The mean data scientist salary is 113,000 dollars.

While the alternative : The mean data scientist salary is not 113,000 dollars.

Author's note: If you're interested in a data scientist career, check out our articles Data Scientist Career Path , 5 Business Basics for Data Scientists , Data Science Interview Questions , and 15 Data Science Consulting Companies Hiring Now .

## An Example of a One-Sided Test

You can also form one-sided or one-tailed tests.

Say your friend, Paul, told you that he thinks data scientists earn more than 125,000 dollars per year. You doubt him, so you design a test to see who’s right.

The null hypothesis of this test would be: The mean data scientist salary is more than 125,000 dollars.

The alternative will cover everything else, thus: The mean data scientist salary is less than or equal to 125,000 dollars.

Important: The outcomes of tests refer to the population parameter rather than the sample statistic! So, the result that we get is for the population.

Important: Another crucial consideration is that, generally, the researcher is trying to reject the null hypothesis . Think about the null hypothesis as the status quo and the alternative as the change or innovation that challenges that status quo. In our example, Paul was representing the status quo, which we were challenging.

Let’s go over it once more. In statistics, the null hypothesis is the statement we are trying to reject. Therefore, the null hypothesis is the present state of affairs, while the alternative is our personal opinion.

## Why Hypothesis Testing Works

Right now, you may be feeling a little puzzled. This is normal because this whole concept is counter-intuitive at the beginning. However, there is an extremely easy way to continue your journey of exploring it. By diving into the linked tutorial, you will find out why hypothesis testing actually works.

Interested in learning more? You can take your skills from good to great with our statistics course!

Try statistics course for free

Next Tutorial: Hypothesis Testing: Significance Level and Rejection Region

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## Hypothesis Testing | A Step-by-Step Guide with Easy Examples

Published on November 8, 2019 by Rebecca Bevans . Revised on June 22, 2023.

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics . It is most often used by scientists to test specific predictions, called hypotheses, that arise from theories.

There are 5 main steps in hypothesis testing:

- State your research hypothesis as a null hypothesis and alternate hypothesis (H o ) and (H a or H 1 ).
- Collect data in a way designed to test the hypothesis.
- Perform an appropriate statistical test .
- Decide whether to reject or fail to reject your null hypothesis.
- Present the findings in your results and discussion section.

Though the specific details might vary, the procedure you will use when testing a hypothesis will always follow some version of these steps.

## Table of contents

Step 1: state your null and alternate hypothesis, step 2: collect data, step 3: perform a statistical test, step 4: decide whether to reject or fail to reject your null hypothesis, step 5: present your findings, other interesting articles, frequently asked questions about hypothesis testing.

After developing your initial research hypothesis (the prediction that you want to investigate), it is important to restate it as a null (H o ) and alternate (H a ) hypothesis so that you can test it mathematically.

The alternate hypothesis is usually your initial hypothesis that predicts a relationship between variables. The null hypothesis is a prediction of no relationship between the variables you are interested in.

- H 0 : Men are, on average, not taller than women. H a : Men are, on average, taller than women.

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For a statistical test to be valid , it is important to perform sampling and collect data in a way that is designed to test your hypothesis. If your data are not representative, then you cannot make statistical inferences about the population you are interested in.

There are a variety of statistical tests available, but they are all based on the comparison of within-group variance (how spread out the data is within a category) versus between-group variance (how different the categories are from one another).

If the between-group variance is large enough that there is little or no overlap between groups, then your statistical test will reflect that by showing a low p -value . This means it is unlikely that the differences between these groups came about by chance.

Alternatively, if there is high within-group variance and low between-group variance, then your statistical test will reflect that with a high p -value. This means it is likely that any difference you measure between groups is due to chance.

Your choice of statistical test will be based on the type of variables and the level of measurement of your collected data .

- an estimate of the difference in average height between the two groups.
- a p -value showing how likely you are to see this difference if the null hypothesis of no difference is true.

Based on the outcome of your statistical test, you will have to decide whether to reject or fail to reject your null hypothesis.

In most cases you will use the p -value generated by your statistical test to guide your decision. And in most cases, your predetermined level of significance for rejecting the null hypothesis will be 0.05 – that is, when there is a less than 5% chance that you would see these results if the null hypothesis were true.

In some cases, researchers choose a more conservative level of significance, such as 0.01 (1%). This minimizes the risk of incorrectly rejecting the null hypothesis ( Type I error ).

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The results of hypothesis testing will be presented in the results and discussion sections of your research paper , dissertation or thesis .

In the results section you should give a brief summary of the data and a summary of the results of your statistical test (for example, the estimated difference between group means and associated p -value). In the discussion , you can discuss whether your initial hypothesis was supported by your results or not.

In the formal language of hypothesis testing, we talk about rejecting or failing to reject the null hypothesis. You will probably be asked to do this in your statistics assignments.

However, when presenting research results in academic papers we rarely talk this way. Instead, we go back to our alternate hypothesis (in this case, the hypothesis that men are on average taller than women) and state whether the result of our test did or did not support the alternate hypothesis.

If your null hypothesis was rejected, this result is interpreted as “supported the alternate hypothesis.”

These are superficial differences; you can see that they mean the same thing.

You might notice that we don’t say that we reject or fail to reject the alternate hypothesis . This is because hypothesis testing is not designed to prove or disprove anything. It is only designed to test whether a pattern we measure could have arisen spuriously, or by chance.

If we reject the null hypothesis based on our research (i.e., we find that it is unlikely that the pattern arose by chance), then we can say our test lends support to our hypothesis . But if the pattern does not pass our decision rule, meaning that it could have arisen by chance, then we say the test is inconsistent with our hypothesis .

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.

- Normal distribution
- Descriptive statistics
- Measures of central tendency
- Correlation coefficient

Methodology

- Cluster sampling
- Stratified sampling
- Types of interviews
- Cohort study
- Thematic analysis

Research bias

- Implicit bias
- Cognitive bias
- Survivorship bias
- Availability heuristic
- Nonresponse bias
- Regression to the mean

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.

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

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

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.

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## AP®︎/College Statistics

Course: ap®︎/college statistics > unit 10.

- Idea behind hypothesis testing
- Examples of null and alternative hypotheses

## Writing null and alternative hypotheses

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

- (Choice A) H 0 : p = 0.1 H a : p ≠ 0.1 A H 0 : p = 0.1 H a : p ≠ 0.1
- (Choice B) H 0 : p ≠ 0.1 H a : p = 0.1 B H 0 : p ≠ 0.1 H a : p = 0.1
- (Choice C) H 0 : p = 0.1 H a : p > 0.1 C H 0 : p = 0.1 H a : p > 0.1
- (Choice D) H 0 : p = 0.1 H a : p < 0.1 D H 0 : p = 0.1 H a : p < 0.1

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## Writing Null Hypotheses in Research and Statistics

Last Updated: January 17, 2024 Fact Checked

This article was co-authored by Joseph Quinones and by wikiHow staff writer, Jennifer Mueller, JD . Joseph Quinones is a High School Physics Teacher working at South Bronx Community Charter High School. Joseph specializes in astronomy and astrophysics and is interested in science education and science outreach, currently practicing ways to make physics accessible to more students with the goal of bringing more students of color into the STEM fields. He has experience working on Astrophysics research projects at the Museum of Natural History (AMNH). Joseph recieved his Bachelor's degree in Physics from Lehman College and his Masters in Physics Education from City College of New York (CCNY). He is also a member of a network called New York City Men Teach. There are 7 references cited in this article, which can be found at the bottom of the page. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 26,811 times.

Are you working on a research project and struggling with how to write a null hypothesis? Well, you've come to the right place! Start by recognizing that the basic definition of "null" is "none" or "zero"—that's your biggest clue as to what a null hypothesis should say. Keep reading to learn everything you need to know about the null hypothesis, including how it relates to your research question and your alternative hypothesis as well as how to use it in different types of studies.

## Things You Should Know

- Write a research null hypothesis as a statement that the studied variables have no relationship to each other, or that there's no difference between 2 groups.

- Adjust the format of your null hypothesis to match the statistical method you used to test it, such as using "mean" if you're comparing the mean between 2 groups.

## What is a null hypothesis?

- Research hypothesis: States in plain language that there's no relationship between the 2 variables or there's no difference between the 2 groups being studied.
- Statistical hypothesis: States the predicted outcome of statistical analysis through a mathematical equation related to the statistical method you're using.

## Examples of Null Hypotheses

## Null Hypothesis vs. Alternative Hypothesis

- For example, your alternative hypothesis could state a positive correlation between 2 variables while your null hypothesis states there's no relationship. If there's a negative correlation, then both hypotheses are false.

- You need additional data or evidence to show that your alternative hypothesis is correct—proving the null hypothesis false is just the first step.
- In smaller studies, sometimes it's enough to show that there's some relationship and your hypothesis could be correct—you can leave the additional proof as an open question for other researchers to tackle.

## How do I test a null hypothesis?

- Group means: Compare the mean of the variable in your sample with the mean of the variable in the general population. [6] X Research source
- Group proportions: Compare the proportion of the variable in your sample with the proportion of the variable in the general population. [7] X Research source
- Correlation: Correlation analysis looks at the relationship between 2 variables—specifically, whether they tend to happen together. [8] X Research source
- Regression: Regression analysis reveals the correlation between 2 variables while also controlling for the effect of other, interrelated variables. [9] X Research source

## Templates for Null Hypotheses

- Research null hypothesis: There is no difference in the mean [dependent variable] between [group 1] and [group 2].

- Research null hypothesis: The proportion of [dependent variable] in [group 1] and [group 2] is the same.

- Research null hypothesis: There is no correlation between [independent variable] and [dependent variable] in the population.

- Research null hypothesis: There is no relationship between [independent variable] and [dependent variable] in the population.

## Expert Q&A

## You Might Also Like

## Expert Interview

Thanks for reading our article! If you’d like to learn more about physics, check out our in-depth interview with Joseph Quinones .

- ↑ https://online.stat.psu.edu/stat100/lesson/10/10.1
- ↑ https://online.stat.psu.edu/stat501/lesson/2/2.12
- ↑ https://support.minitab.com/en-us/minitab/21/help-and-how-to/statistics/basic-statistics/supporting-topics/basics/null-and-alternative-hypotheses/
- ↑ https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5635437/
- ↑ https://online.stat.psu.edu/statprogram/reviews/statistical-concepts/hypothesis-testing
- ↑ https://education.arcus.chop.edu/null-hypothesis-testing/
- ↑ https://sphweb.bumc.bu.edu/otlt/mph-modules/bs/bs704_hypothesistest-means-proportions/bs704_hypothesistest-means-proportions_print.html

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## Keyboard Shortcuts

10.1 - setting the hypotheses: examples.

A significance test examines whether the null hypothesis provides a plausible explanation of the data. The null hypothesis itself does not involve the data. It is a statement about a parameter (a numerical characteristic of the population). These population values might be proportions or means or differences between means or proportions or correlations or odds ratios or any other numerical summary of the population. The alternative hypothesis is typically the research hypothesis of interest. Here are some examples.

## Example 10.2: Hypotheses with One Sample of One Categorical Variable Section

About 10% of the human population is left-handed. Suppose a researcher at Penn State speculates that students in the College of Arts and Architecture are more likely to be left-handed than people found in the general population. We only have one sample since we will be comparing a population proportion based on a sample value to a known population value.

- Research Question : Are artists more likely to be left-handed than people found in the general population?
- Response Variable : Classification of the student as either right-handed or left-handed

## State Null and Alternative Hypotheses

- Null Hypothesis : Students in the College of Arts and Architecture are no more likely to be left-handed than people in the general population (population percent of left-handed students in the College of Art and Architecture = 10% or p = .10).
- Alternative Hypothesis : Students in the College of Arts and Architecture are more likely to be left-handed than people in the general population (population percent of left-handed students in the College of Arts and Architecture > 10% or p > .10). This is a one-sided alternative hypothesis.

## Example 10.3: Hypotheses with One Sample of One Measurement Variable Section

A generic brand of the anti-histamine Diphenhydramine markets a capsule with a 50 milligram dose. The manufacturer is worried that the machine that fills the capsules has come out of calibration and is no longer creating capsules with the appropriate dosage.

- Research Question : Does the data suggest that the population mean dosage of this brand is different than 50 mg?
- Response Variable : dosage of the active ingredient found by a chemical assay.
- Null Hypothesis : On the average, the dosage sold under this brand is 50 mg (population mean dosage = 50 mg).
- Alternative Hypothesis : On the average, the dosage sold under this brand is not 50 mg (population mean dosage ≠ 50 mg). This is a two-sided alternative hypothesis.

## Example 10.4: Hypotheses with Two Samples of One Categorical Variable Section

Many people are starting to prefer vegetarian meals on a regular basis. Specifically, a researcher believes that females are more likely than males to eat vegetarian meals on a regular basis.

- Research Question : Does the data suggest that females are more likely than males to eat vegetarian meals on a regular basis?
- Response Variable : Classification of whether or not a person eats vegetarian meals on a regular basis
- Explanatory (Grouping) Variable: Sex
- Null Hypothesis : There is no sex effect regarding those who eat vegetarian meals on a regular basis (population percent of females who eat vegetarian meals on a regular basis = population percent of males who eat vegetarian meals on a regular basis or p females = p males ).
- Alternative Hypothesis : Females are more likely than males to eat vegetarian meals on a regular basis (population percent of females who eat vegetarian meals on a regular basis > population percent of males who eat vegetarian meals on a regular basis or p females > p males ). This is a one-sided alternative hypothesis.

## Example 10.5: Hypotheses with Two Samples of One Measurement Variable Section

Obesity is a major health problem today. Research is starting to show that people may be able to lose more weight on a low carbohydrate diet than on a low fat diet.

- Research Question : Does the data suggest that, on the average, people are able to lose more weight on a low carbohydrate diet than on a low fat diet?
- Response Variable : Weight loss (pounds)
- Explanatory (Grouping) Variable : Type of diet
- Null Hypothesis : There is no difference in the mean amount of weight loss when comparing a low carbohydrate diet with a low fat diet (population mean weight loss on a low carbohydrate diet = population mean weight loss on a low fat diet).
- Alternative Hypothesis : The mean weight loss should be greater for those on a low carbohydrate diet when compared with those on a low fat diet (population mean weight loss on a low carbohydrate diet > population mean weight loss on a low fat diet). This is a one-sided alternative hypothesis.

## Example 10.6: Hypotheses about the relationship between Two Categorical Variables Section

- Research Question : Do the odds of having a stroke increase if you inhale second hand smoke ? A case-control study of non-smoking stroke patients and controls of the same age and occupation are asked if someone in their household smokes.
- Variables : There are two different categorical variables (Stroke patient vs control and whether the subject lives in the same household as a smoker). Living with a smoker (or not) is the natural explanatory variable and having a stroke (or not) is the natural response variable in this situation.
- Null Hypothesis : There is no relationship between whether or not a person has a stroke and whether or not a person lives with a smoker (odds ratio between stroke and second-hand smoke situation is = 1).
- Alternative Hypothesis : There is a relationship between whether or not a person has a stroke and whether or not a person lives with a smoker (odds ratio between stroke and second-hand smoke situation is > 1). This is a one-tailed alternative.

This research question might also be addressed like example 11.4 by making the hypotheses about comparing the proportion of stroke patients that live with smokers to the proportion of controls that live with smokers.

## Example 10.7: Hypotheses about the relationship between Two Measurement Variables Section

- Research Question : A financial analyst believes there might be a positive association between the change in a stock's price and the amount of the stock purchased by non-management employees the previous day (stock trading by management being under "insider-trading" regulatory restrictions).
- Variables : Daily price change information (the response variable) and previous day stock purchases by non-management employees (explanatory variable). These are two different measurement variables.
- Null Hypothesis : The correlation between the daily stock price change (\$) and the daily stock purchases by non-management employees (\$) = 0.
- Alternative Hypothesis : The correlation between the daily stock price change (\$) and the daily stock purchases by non-management employees (\$) > 0. This is a one-sided alternative hypothesis.

## Example 10.8: Hypotheses about comparing the relationship between Two Measurement Variables in Two Samples Section

- Research Question : Is there a linear relationship between the amount of the bill (\$) at a restaurant and the tip (\$) that was left. Is the strength of this association different for family restaurants than for fine dining restaurants?
- Variables : There are two different measurement variables. The size of the tip would depend on the size of the bill so the amount of the bill would be the explanatory variable and the size of the tip would be the response variable.
- Null Hypothesis : The correlation between the amount of the bill (\$) at a restaurant and the tip (\$) that was left is the same at family restaurants as it is at fine dining restaurants.
- Alternative Hypothesis : The correlation between the amount of the bill (\$) at a restaurant and the tip (\$) that was left is the difference at family restaurants then it is at fine dining restaurants. This is a two-sided alternative hypothesis.

Want to create or adapt books like this? Learn more about how Pressbooks supports open publishing practices.

## 8.2 Null and Alternative Hypotheses

Learning objectives.

- Describe hypothesis testing in general and in practice.

A hypothesis test begins by considering two hypotheses . They are called the null hypothesis and the alternative hypothesis . These hypotheses contain opposing viewpoints and only one of these hypotheses is true. The hypothesis test determines which hypothesis is most likely true.

- The null hypothesis is a claim that a population parameter equals some value. For example, [latex]H_0: \mu=5[/latex].
- The alternative hypothesis is a claim that a population parameter is greater than, less than, or not equal to some value. For example, [latex]H_a: \mu>5[/latex], [latex]H_a: \mu<5[/latex], or [latex]H_a: \mu \neq 5[/latex]. The form of the alternative hypothesis depends on the wording of the hypothesis test.
- An alternative notation for [latex]H_a[/latex] is [latex]H_1[/latex].

Because the null and alternative hypotheses are contradictory, we must examine evidence to decide if we have enough evidence to reject the null hypothesis or not reject the null hypothesis. The evidence is in the form of sample data. After we have determined which hypothesis the sample data supports, we make a decision. There are two options for a decision . They are “ reject [latex]H_0[/latex] ” if the sample information favors the alternative hypothesis or “ do not reject [latex]H_0[/latex] ” if the sample information is insufficient to reject the null hypothesis.

Watch this video: Simple hypothesis testing | Probability and Statistics | Khan Academy by Khan Academy [6:24]

A candidate in a local election claims that 30% of registered voters voted in a recent election. Information provided by the returning office suggests that the percentage is higher than the 30% claimed.

The parameter under study is the proportion of registered voters, so we use [latex]p[/latex] in the statements of the hypotheses. The hypotheses are

[latex]\begin{eqnarray*} \\ H_0: & & p=30\% \\ \\ H_a: & & p \gt 30\% \\ \\ \end{eqnarray*}[/latex]

- The null hypothesis [latex]H_0[/latex] is the claim that the proportion of registered voters that voted equals 30%.
- The alternative hypothesis [latex]H_a[/latex] is the claim that the proportion of registered voters that voted is greater than (i.e. higher) than 30%.

A medical researcher believes that a new medicine reduces cholesterol by 25%. A medical trial suggests that the percent reduction is different than claimed. State the null and alternative hypotheses.

[latex]\begin{eqnarray*} H_0: & & p=25\% \\ \\ H_a: & & p \neq 25\% \end{eqnarray*}[/latex]

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

[latex]\begin{eqnarray*} H_0: & & \mu=2 \mbox{ points} \\ \\ H_a: & & \mu \neq 2 \mbox{ points} \end{eqnarray*}[/latex]

We want to test whether or not the mean height of eighth graders is 66 inches. State the null and alternative hypotheses.

[latex]\begin{eqnarray*} H_0: & & \mu=66 \mbox{ inches} \\ \\ H_a: & & \mu \neq 66 \mbox{ inches} \end{eqnarray*}[/latex]

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

[latex]\begin{eqnarray*} H_0: & & \mu=5 \mbox{ years} \\ \\ H_a: & & \mu \lt 5 \mbox{ years} \end{eqnarray*}[/latex]

We want to test if it takes fewer than 45 minutes to teach a lesson plan. State the null and alternative hypotheses.

[latex]\begin{eqnarray*} H_0: & & \mu=45 \mbox{ minutes} \\ \\ H_a: & & \mu \lt 45 \mbox{ minutes} \end{eqnarray*}[/latex]

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

[latex]\begin{eqnarray*} H_0: & & p=6.6\% \\ \\ H_a: & & p \gt 6.6\% \end{eqnarray*}[/latex]

On a state driver’s test, about 40% pass the test on the first try. We want to test if more than 40% pass on the first try. State the null and alternative hypotheses.

[latex]\begin{eqnarray*} H_0: & & p=40\% \\ \\ H_a: & & p \gt 40\% \end{eqnarray*}[/latex]

## Concept Review

In a hypothesis test , sample data is evaluated in order to arrive at a decision about some type of claim. If certain conditions about the sample are satisfied, then the claim can be evaluated for a population. In a hypothesis test, we evaluate the null hypothesis , typically denoted with [latex]H_0[/latex]. The null hypothesis is not rejected unless the hypothesis test shows otherwise. The null hypothesis always contain an equal sign ([latex]=[/latex]). Always write the alternative hypothesis , typically denoted with [latex]H_a[/latex] or [latex]H_1[/latex], using less than, greater than, or not equals symbols ([latex]\lt[/latex], [latex]\gt[/latex], [latex]\neq[/latex]). If we reject the null hypothesis, then we can assume there is enough evidence to support the alternative hypothesis. But we can never state that a claim is proven true or false. All we can conclude from the hypothesis test is which of the hypothesis is most likely true. Because the underlying facts about hypothesis testing is based on probability laws, we can talk only in terms of non-absolute certainties.

## Attribution

“ 9.1 Null and Alternative Hypotheses “ in Introductory Statistics by OpenStax is licensed under a Creative Commons Attribution 4.0 International License.

Introduction to Statistics Copyright © 2022 by Valerie Watts is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

## Null Hypothesis Definition and Examples, How to State

What is the null hypothesis, how to state the null hypothesis, null hypothesis overview.

## Why is it Called the “Null”?

The word “null” in this context means that it’s a commonly accepted fact that researchers work to nullify . It doesn’t mean that the statement is null (i.e. amounts to nothing) itself! (Perhaps the term should be called the “nullifiable hypothesis” as that might cause less confusion).

## Why Do I need to Test it? Why not just prove an alternate one?

The short answer is, as a scientist, you are required to ; It’s part of the scientific process. Science uses a battery of processes to prove or disprove theories, making sure than any new hypothesis has no flaws. Including both a null and an alternate hypothesis is one safeguard to ensure your research isn’t flawed. Not including the null hypothesis in your research is considered very bad practice by the scientific community. If you set out to prove an alternate hypothesis without considering it, you are likely setting yourself up for failure. At a minimum, your experiment will likely not be taken seriously.

- Null hypothesis : H 0 : The world is flat.
- Alternate hypothesis: The world is round.

Several scientists, including Copernicus , set out to disprove the null hypothesis. This eventually led to the rejection of the null and the acceptance of the alternate. Most people accepted it — the ones that didn’t created the Flat Earth Society !. What would have happened if Copernicus had not disproved the it and merely proved the alternate? No one would have listened to him. In order to change people’s thinking, he first had to prove that their thinking was wrong .

## How to State the Null Hypothesis from a Word Problem

You’ll be asked to convert a word problem into a hypothesis statement in statistics that will include a null hypothesis and an alternate hypothesis . Breaking your problem into a few small steps makes these problems much easier to handle.

Step 2: Convert the hypothesis to math . Remember that the average is sometimes written as μ.

H 1 : μ > 8.2

Broken down into (somewhat) English, that’s H 1 (The hypothesis): μ (the average) > (is greater than) 8.2

Step 3: State what will happen if the hypothesis doesn’t come true. If the recovery time isn’t greater than 8.2 weeks, there are only two possibilities, that the recovery time is equal to 8.2 weeks or less than 8.2 weeks.

H 0 : μ ≤ 8.2

Broken down again into English, that’s H 0 (The null hypothesis): μ (the average) ≤ (is less than or equal to) 8.2

## How to State the Null Hypothesis: Part Two

But what if the researcher doesn’t have any idea what will happen.

Example Problem: A researcher is studying the effects of radical exercise program on knee surgery patients. There is a good chance the therapy will improve recovery time, but there’s also the possibility it will make it worse. Average recovery times for knee surgery patients is 8.2 weeks.

Step 1: State what will happen if the experiment doesn’t make any difference. That’s the null hypothesis–that nothing will happen. In this experiment, if nothing happens, then the recovery time will stay at 8.2 weeks.

H 0 : μ = 8.2

Broken down into English, that’s H 0 (The null hypothesis): μ (the average) = (is equal to) 8.2

Step 2: Figure out the alternate hypothesis . The alternate hypothesis is the opposite of the null hypothesis. In other words, what happens if our experiment makes a difference?

H 1 : μ ≠ 8.2

In English again, that’s H 1 (The alternate hypothesis): μ (the average) ≠ (is not equal to) 8.2

That’s How to State the Null Hypothesis!

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## Educational Research Basics by Del Siegle

Null and alternative hypotheses.

Converting research questions to hypothesis is a simple task. Take the questions and make it a positive statement that says a relationship exists (correlation studies) or a difference exists between the groups (experiment study) and you have the alternative hypothesis. Write the statement such that a relationship does not exist or a difference does not exist and you have the null hypothesis. You can reverse the process if you have a hypothesis and wish to write a research question.

When you are comparing two groups, the groups are the independent variable. When you are testing whether something affects something else, the cause is the independent variable. The independent variable is the one you manipulate.

Teachers given higher pay will have more positive attitudes toward children than teachers given lower pay. The first step is to ask yourself “Are there two or more groups being compared?” The answer is “Yes.” What are the groups? Teachers who are given higher pay and teachers who are given lower pay. The independent variable is teacher pay. The dependent variable (the outcome) is attitude towards school.

You could also approach is another way. “Is something causing something else?” The answer is “Yes.” What is causing what? Teacher pay is causing attitude towards school. Therefore, teacher pay is the independent variable (cause) and attitude towards school is the dependent variable (outcome).

By tradition, we try to disprove (reject) the null hypothesis. We can never prove a null hypothesis, because it is impossible to prove something does not exist. We can disprove something does not exist by finding an example of it. Therefore, in research we try to disprove the null hypothesis. When we do find that a relationship (or difference) exists then we reject the null and accept the alternative. If we do not find that a relationship (or difference) exists, we fail to reject the null hypothesis (and go with it). We never say we accept the null hypothesis because it is never possible to prove something does not exist. That is why we say that we failed to reject the null hypothesis, rather than we accepted it.

Del Siegle, Ph.D. Neag School of Education – University of Connecticut [email protected] www.delsiegle.com

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Once you have developed a clear and focused research question or set of research questions, you’ll be ready to conduct further research, a literature review, on the topic to help you make an educated guess about the answer to your question(s). This educated guess is called a hypothesis.

In research, there are two types of hypotheses: null and alternative. They work as a complementary pair, each stating that the other is wrong.

- Null Hypothesis (H 0 ) – This can be thought of as the implied hypothesis. “Null” meaning “nothing.” This hypothesis states that there is no difference between groups or no relationship between variables. The null hypothesis is a presumption of status quo or no change.
- Alternative Hypothesis (H a ) – This is also known as the claim. This hypothesis should state what you expect the data to show, based on your research on the topic. This is your answer to your research question.

Null Hypothesis: H 0 : There is no difference in the salary of factory workers based on gender. Alternative Hypothesis : H a : Male factory workers have a higher salary than female factory workers.

Null Hypothesis : H 0 : There is no relationship between height and shoe size. Alternative Hypothesis : H a : There is a positive relationship between height and shoe size.

Null Hypothesis : H 0 : Experience on the job has no impact on the quality of a brick mason’s work. Alternative Hypothesis : H a : The quality of a brick mason’s work is influenced by on-the-job experience.

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## Null Hypothesis Examples

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In statistical analysis, the null hypothesis assumes there is no meaningful relationship between two variables. Testing the null hypothesis can tell you whether your results are due to the effect of manipulating a dependent variable or due to chance. It's often used in conjunction with an alternative hypothesis, which assumes there is, in fact, a relationship between two variables.

The null hypothesis is among the easiest hypothesis to test using statistical analysis, making it perhaps the most valuable hypothesis for the scientific method. By evaluating a null hypothesis in addition to another hypothesis, researchers can support their conclusions with a higher level of confidence. Below are examples of how you might formulate a null hypothesis to fit certain questions.

## What Is the Null Hypothesis?

The null hypothesis states there is no relationship between the measured phenomenon (the dependent variable ) and the independent variable , which is the variable an experimenter typically controls or changes. You do not need to believe that the null hypothesis is true to test it. On the contrary, you will likely suspect there is a relationship between a set of variables. One way to prove that this is the case is to reject the null hypothesis. Rejecting a hypothesis does not mean an experiment was "bad" or that it didn't produce results. In fact, it is often one of the first steps toward further inquiry.

To distinguish it from other hypotheses , the null hypothesis is written as H 0 (which is read as “H-nought,” "H-null," or "H-zero"). A significance test is used to determine the likelihood that the results supporting the null hypothesis are not due to chance. A confidence level of 95% or 99% is common. Keep in mind, even if the confidence level is high, there is still a small chance the null hypothesis is not true, perhaps because the experimenter did not account for a critical factor or because of chance. This is one reason why it's important to repeat experiments.

## Examples of the Null Hypothesis

To write a null hypothesis, first start by asking a question. Rephrase that question in a form that assumes no relationship between the variables. In other words, assume a treatment has no effect. Write your hypothesis in a way that reflects this.

Are teens better at math than adults? | Age has no effect on mathematical ability. |

Does taking aspirin every day reduce the chance of having a heart attack? | Taking aspirin daily does not affect heart attack risk. |

Do teens use cell phones to access the internet more than adults? | Age has no effect on how cell phones are used for internet access. |

Do cats care about the color of their food? | Cats express no food preference based on color. |

Does chewing willow bark relieve pain? | There is no difference in pain relief after chewing willow bark versus taking a placebo. |

## Other Types of Hypotheses

In addition to the null hypothesis, the alternative hypothesis is also a staple in traditional significance tests . It's essentially the opposite of the null hypothesis because it assumes the claim in question is true. For the first item in the table above, for example, an alternative hypothesis might be "Age does have an effect on mathematical ability."

## Key Takeaways

- In hypothesis testing, the null hypothesis assumes no relationship between two variables, providing a baseline for statistical analysis.
- Rejecting the null hypothesis suggests there is evidence of a relationship between variables.
- By formulating a null hypothesis, researchers can systematically test assumptions and draw more reliable conclusions from their experiments.
- Difference Between Independent and Dependent Variables
- Examples of Independent and Dependent Variables
- What Is a Hypothesis? (Science)
- What 'Fail to Reject' Means in a Hypothesis Test
- Definition of a Hypothesis
- Null Hypothesis Definition and Examples
- Scientific Method Vocabulary Terms
- Null Hypothesis and Alternative Hypothesis
- Hypothesis Test for the Difference of Two Population Proportions
- How to Conduct a Hypothesis Test
- What Is a P-Value?
- What Are the Elements of a Good Hypothesis?
- Hypothesis Test Example
- What Is the Difference Between Alpha and P-Values?
- Understanding Path Analysis
- An Example of a Hypothesis Test

Statistics Made Easy

## What is an Alternative Hypothesis in Statistics?

Often in statistics we want to test whether or not some assumption is true about a population parameter .

For example, we might assume that the mean weight of a certain population of turtle is 300 pounds.

To determine if this assumption is true, we’ll go out and collect a sample of turtles and weigh each of them. Using this sample data, we’ll conduct a hypothesis test .

The first step in a hypothesis test is to define the null and alternative hypotheses .

These two hypotheses need to be mutually exclusive, so if one is true then the other must be false.

These two hypotheses are defined as follows:

Null hypothesis (H 0 ): The sample data is consistent with the prevailing belief about the population parameter.

Alternative hypothesis (H A ): The sample data suggests that the assumption made in the null hypothesis is not true. In other words, there is some non-random cause influencing the data.

## Types of Alternative Hypotheses

There are two types of alternative hypotheses:

A one-tailed hypothesis involves making a “greater than” or “less than ” statement. For example, suppose we assume the mean height of a male in the U.S. is greater than or equal to 70 inches.

The null and alternative hypotheses in this case would be:

- Null hypothesis: µ ≥ 70 inches
- Alternative hypothesis: µ < 70 inches

A two-tailed hypothesis involves making an “equal to” or “not equal to” statement. For example, suppose we assume the mean height of a male in the U.S. is equal to 70 inches.

- Null hypothesis: µ = 70 inches
- Alternative hypothesis: µ ≠ 70 inches

Note: The “equal” sign is always included in the null hypothesis, whether it is =, ≥, or ≤.

## Examples of Alternative Hypotheses

The following examples illustrate how to define the null and alternative hypotheses for different research problems.

Example 1: A biologist wants to test if the mean weight of a certain population of turtle is different from the widely-accepted mean weight of 300 pounds.

The null and alternative hypothesis for this research study would be:

- Null hypothesis: µ = 300 pounds
- Alternative hypothesis: µ ≠ 300 pounds

If we reject the null hypothesis, this means we have sufficient evidence from the sample data to say that the true mean weight of this population of turtles is different from 300 pounds.

Example 2: An engineer wants to test whether a new battery can produce higher mean watts than the current industry standard of 50 watts.

- Null hypothesis: µ ≤ 50 watts
- Alternative hypothesis: µ > 50 watts

If we reject the null hypothesis, this means we have sufficient evidence from the sample data to say that the true mean watts produced by the new battery is greater than the current industry standard of 50 watts.

Example 3: A botanist wants to know if a new gardening method produces less waste than the standard gardening method that produces 20 pounds of waste.

- Null hypothesis: µ ≥ 20 pounds
- Alternative hypothesis: µ < 20 pounds

If we reject the null hypothesis, this means we have sufficient evidence from the sample data to say that the true mean weight produced by this new gardening method is less than 20 pounds.

## When to Reject the Null Hypothesis

Whenever we conduct a hypothesis test, we use sample data to calculate a test-statistic and a corresponding p-value.

If the p-value is less than some significance level (common choices are 0.10, 0.05, and 0.01), then we reject the null hypothesis.

This means we have sufficient evidence from the sample data to say that the assumption made by the null hypothesis is not true.

If the p-value is not less than some significance level, then we fail to reject the null hypothesis.

This means our sample data did not provide us with evidence that the assumption made by the null hypothesis was not true.

Additional Resource: An Explanation of P-Values and Statistical Significance

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## How to choose the null and alternative hypothesis?

I'm practicing with the hypothesis test and I find myself in trouble with the decision about how to set a null and an alternative hypothesis. My main issue is to determine, in every situation, a "general rule" on how I can decide correctly which is the null and which is the alternative hypothesis.. can someone help me?

Here is an example: As an established scholar, you are requested to evaluate if Customer Relationship Management affects the financial performance of firms. The main issue will be solved by means of a test of hypothesis. Two hypothesis will be tested one against the other: CRM is related to performance, CRM is not related.

- hypothesis-testing

- 1 $\begingroup$ This tutorial video is a very good explanation if you want to learn about hypothesis testing in an intuitive approach - youtube.com/watch?v=UApFKiK4Hi8 $\endgroup$ – London guy Commented Nov 9, 2014 at 18:58

## 3 Answers 3

The rule for the proper formulation of a hypothesis test is that the alternative or research hypothesis is the statement that, if true, is strongly supported by the evidence furnished by the data.

The null hypothesis is generally the complement of the alternative hypothesis. Frequently, it is (or contains) the assumption that you are making about how the data are distributed in order to calculate the test statistic.

Here are a few examples to help you understand how these are properly chosen.

Suppose I am an epidemiologist in public health, and I'm investigating whether the incidence of smoking among a certain ethnic group is greater than the population as a whole, and therefore there is a need to target anti-smoking campaigns for this sub-population through greater community outreach and education. From previous studies that have been published in the literature, I find that the incidence among the general population is $p_0$. I can then go about collecting sample data (that's actually the hard part!) to test $$H_0 : p = p_0 \quad \mathrm{vs.} \quad H_a : p > p_0.$$ This is a one-sided binomial proportion test. $H_a$ is the statement that, if it were true, would need to be strongly supported by the data we collected. It is the statement that carries the burden of proof . This is because any conclusion we draw from the test is conditional upon assuming that the null is true: either $H_a$ is accepted, or the test is inconclusive and there is insufficient evidence from the data to suggest $H_a$ is true. The choice of $H_0$ reflects the underlying assumption that there is no difference in the smoking rates of the sub-population compared to the whole.

Now suppose I am a researcher investigating a new drug that I believe to be equally effective to an existing standard of treatment, but with fewer side effects and therefore a more desirable safety profile. I would like to demonstrate the equal efficacy by conducting a bioequivalence test. If $\mu_0$ is the mean existing standard treatment effect, then my hypothesis might look like this: $$H_0 : |\mu - \mu_0| \ge \Delta \quad \mathrm{vs.} \quad H_a : |\mu - \mu_0| < \Delta,$$ for some choice of margin $\Delta$ that I consider to be clinically significant. For example, a clinician might say that two treatments are sufficiently bioequivalent if there is less than a $\Delta = 10\%$ difference in treatment effect. Note again that $H_a$ is the statement that carries the burden of proof: the data we collect must strongly support it, in order for us to accept it; otherwise, it could still be true but we don't have the evidence to support the claim .

Now suppose I am doing an analysis for a small business owner who sells three products $A$, $B$, $C$. They suspect that there is a statistically significant preference for these three products. Then my hypothesis is $$H_0 : \mu_A = \mu_B = \mu_C \quad \mathrm{vs.} \quad H_a : \exists i \ne j \text{ such that } \mu_i \ne \mu_j.$$ Really, all that $H_a$ is saying is that there are two means that are not equal to each other, which would then suggest that some difference in preference exists.

- $\begingroup$ Thank you so much.. Also your response has been of great help :D $\endgroup$ – RedViper16 Commented Nov 10, 2014 at 21:25
- $\begingroup$ The third example seems worse than the first two in that, in reality, with a finite size sample, no two distinct products will almost never sell at exactly same amount and $H_0$ as formulated above will almost always be rejected? $\endgroup$ – qazwsx Commented Nov 21, 2015 at 20:23
- 1 $\begingroup$ @qazwsx What one believes about the likelihood of rejecting the null hypothesis has no bearing on the appropriate construction of the hypothesis test that correctly tests the inference of interest. That is to say, if one is interested in demonstrating with a high degree of confidence that there is a difference between two (or more) means, then the way to test this hypothesis is to use the structure specified in (3), no matter what the data says , because the hypothesis is necessarily pre-specified . $\endgroup$ – heropup Commented Nov 22, 2015 at 5:22
- 1 $\begingroup$ @MSIS You've omitted a critical detail that is present in my original characterization: "the alternative hypothesis is the statement that, if true , is strongly supported by the evidence furnished by the data." You want to choose the alternative in such a way that if you conduct the test and reject the null in favor of the alternative, a high standard of evidence has been met to support this decision. But this does not mean the null is "usually" true or that the alternative "should be" true. It is like the legal principle of "innocent until proven guilty beyond a reasonable doubt." $\endgroup$ – heropup Commented Dec 4, 2019 at 8:44
- 2 $\begingroup$ @MSIS (cont.) To illustrate the analogy, in a criminal court of law, a defendant is assumed innocent of a crime, and the prosecution must establish guilt through presenting evidence. The strength of this evidence must be such that no reasonable logic could exculpate the defendant. Otherwise, the burden of proof is not met. That is not to say the defendant is truly innocent: it simply means the evidence is lacking to establish guilt beyond a reasonable doubt. Similarly, failure to reject the null does not imply the null is true; rather, the data is simply lacking. $\endgroup$ – heropup Commented Dec 4, 2019 at 8:48

The null hypothesis is nearly always "something didn't happen" or "there is no effect" or "there is no relationship" or something similar. But it need not be this.

In your case, the null would be "there is no relationship between CRM and performance"

The usual method is to test the null at some significance level (most often, 0.05). Whether this is a good method is another matter, but it is what is commonly done.

In science proofs, you can never prove anything, you can only demonstrate that your model describes the data better than another model. You want your alternate hypothesis to come from the new model under test, and the null hypothesis to be from a different model.

The null hypothesis should come from a model which others would choose to use when challenging your scientific claims! The most common pattern for a scientific claim is "I think that X is a factor in process Y. If everyone already believes X is a factor in the process, then there is nothing to prove, and everyone can just go out and talk about it over drinks. Scientific arguments with null hypothesis are interesting because, if someone takes the opposing view, "X is not a factor in process Y, then there is a disagreement. This is where science does its thing.

If you believe "X is a factor in process Y" enough to run an experiment, you should generally know what you're looking to see in the results. So now your phrase becomes "X is a factor in process Y, producing visible outcome Z."

This is where you pick your null hypothesis. If someone believes X is not a factor, and your experiment does indeed show Z, then they need an explanation for Z. With your choice of null hypothesis, you are effectively challenging their explanation . The dead simplest explanation is always "Z was caused by random chance because science is based on statistics." Accordingly, most null hypothesis are in the form of "The outcome should be predicted using the previously accepted model plus some random chance to account for statistics.

Both hypothesis should be phrased in terms of the visible outcome, NOT the model you intend to prove. [note] You never start with an alternate hypothesis of "I believe X is a factor." You phrase it "I expect to see this result when I observe Z." The null hypothesis will be phrased similarly, "The status quo predicts that we will see this different result when I observe Z." There is always a statistical phrasing in there such as "I expect to observe a normal distribution on Z when I do this experiment over and over." Once you observe results that defend your alternate hypothesis and reject the null hypothsis, you are THEN in a position to make claims about the validity of your model.

[note] This bolded statement is my opinion, but I feel confident enough in its wording choice to post it. The hypotheses draw a strong line between the intuitive portion of the science, and the data and analysis of the science. If your phrasing is too close to the model, it becomes hard to separate the model from the data, and makes it harder for the next scientist to use your data

In the case of our simple model with process Y and visible outcome Z, the existing belief is that Z will fit a distribution that everyone is already comfortable with, such as "the randomness expected by your particular laboratory equipment setup" or "the purity of the reagents used in the experiment." When you "reject the null hypothesis" what you are saying is most literally, "I have run this experiment, and it is so tremendously unlikely that random chance generated the observed behavior, that everybody should start considering that maybe there's more to this than meets the eye."

The alternative hypothesis is what you offer to the world to replace the null hypothesis . It is one thing to go do experiments to poke at holes in other's models, but that doesn't promote science nearly as well as poking holes in other's models and then replacing them with new models that do a better job.

With the null and alternate hypothesis, you are trying to challenge the current conventional thinking of the day. Choose the hypotheses so that they effectively declare "Here is a result everybody would expect (null hypothesis). However, I actually went out and did the experiment and gathered data, and it is VERY unlikely that the null hypothesis is true. Here is the result I expected (the alternate hypothesis). Nobody expected this hypothesis to be true but me, but when I gathered the data and did the statistics, it is very likely that my model does a better job of describing reality than the existing model . Accordingly, I reject the null hypothesis, accept my hypothesis, and challenge my fellow scientists to work from this new data."

And the fellow scientists are free to:

- Rejoice and accept your data and model with open arms.
- Ignore your data or model (sorry, it happens... welcome to real life)
- Reject your data, and spend their effort running experiments to show different data (It is very common for the science community to say "We do not trust your sample size of 10. We are going to redo your experiment with a sample size of 1000.)
- Accept your data, but not your model. They then must spend their effort generating a new model which explains the data in a different manner.

The last outcome causes strife and bickering, but is ABSOLUTELY part of the scientific process. By using the scientific method to publish your results, you accept that others are free to use the scientific method to contradict your results. They will do so, and publish their results.

At this point, the scientific community will make a political decision: who has to go out and spend the money to test their model, and whose model do we accept. TYPICALLY, because you published the model and the data first, and they are refuting your data, the onus is on them to run the experiments which proves why their model is better than yours. But this is now WELL beyond the hypothesis that caused the strife in the first place, so I leave you to experience them in your lifetime!

- $\begingroup$ Thank you so much for your response. You have been very precise and of help! I've now a stronger idea of what this hypothesis means :D $\endgroup$ – RedViper16 Commented Nov 10, 2014 at 21:24

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## spearmanr #

Calculate a Spearman correlation coefficient with associated p-value.

The Spearman rank-order correlation coefficient is a nonparametric measure of the monotonicity of the relationship between two datasets. Like other correlation coefficients, this one varies between -1 and +1 with 0 implying no correlation. Correlations of -1 or +1 imply an exact monotonic relationship. Positive correlations imply that as x increases, so does y. Negative correlations imply that as x increases, y decreases.

The p-value roughly indicates the probability of an uncorrelated system producing datasets that have a Spearman correlation at least as extreme as the one computed from these datasets. Although calculation of the p-value does not make strong assumptions about the distributions underlying the samples, it is only accurate for very large samples (>500 observations). For smaller sample sizes, consider a permutation test (see Examples section below).

One or two 1-D or 2-D arrays containing multiple variables and observations. When these are 1-D, each represents a vector of observations of a single variable. For the behavior in the 2-D case, see under axis , below. Both arrays need to have the same length in the axis dimension.

If axis=0 (default), then each column represents a variable, with observations in the rows. If axis=1, the relationship is transposed: each row represents a variable, while the columns contain observations. If axis=None, then both arrays will be raveled.

Defines how to handle when input contains nan. The following options are available (default is ‘propagate’):

‘propagate’: returns nan

‘raise’: throws an error

‘omit’: performs the calculations ignoring nan values

Defines the alternative hypothesis. Default is ‘two-sided’. The following options are available:

‘two-sided’: the correlation is nonzero

‘less’: the correlation is negative (less than zero)

‘greater’: the correlation is positive (greater than zero)

Added in version 1.7.0.

An object containing attributes:

Spearman correlation matrix or correlation coefficient (if only 2 variables are given as parameters). Correlation matrix is square with length equal to total number of variables (columns or rows) in a and b combined.

The p-value for a hypothesis test whose null hypothesis is that two samples have no ordinal correlation. See alternative above for alternative hypotheses. pvalue has the same shape as statistic .

Raised if an input is a constant array. The correlation coefficient is not defined in this case, so np.nan is returned.

Zwillinger, D. and Kokoska, S. (2000). CRC Standard Probability and Statistics Tables and Formulae. Chapman & Hall: New York. 2000. Section 14.7

Kendall, M. G. and Stuart, A. (1973). The Advanced Theory of Statistics, Volume 2: Inference and Relationship. Griffin. 1973. Section 31.18

Kershenobich, D., Fierro, F. J., & Rojkind, M. (1970). The relationship between the free pool of proline and collagen content in human liver cirrhosis. The Journal of Clinical Investigation, 49(12), 2246-2249.

Hollander, M., Wolfe, D. A., & Chicken, E. (2013). Nonparametric statistical methods. John Wiley & Sons.

B. Phipson and G. K. Smyth. “Permutation P-values Should Never Be Zero: Calculating Exact P-values When Permutations Are Randomly Drawn.” Statistical Applications in Genetics and Molecular Biology 9.1 (2010).

Ludbrook, J., & Dudley, H. (1998). Why permutation tests are superior to t and F tests in biomedical research. The American Statistician, 52(2), 127-132.

Consider the following data from [3] , which studied the relationship between free proline (an amino acid) and total collagen (a protein often found in connective tissue) in unhealthy human livers.

The x and y arrays below record measurements of the two compounds. The observations are paired: each free proline measurement was taken from the same liver as the total collagen measurement at the same index.

These data were analyzed in [4] using Spearman’s correlation coefficient, a statistic sensitive to monotonic correlation between the samples.

The value of this statistic tends to be high (close to 1) for samples with a strongly positive ordinal correlation, low (close to -1) for samples with a strongly negative ordinal correlation, and small in magnitude (close to zero) for samples with weak ordinal correlation.

The test is performed by comparing the observed value of the statistic against the null distribution: the distribution of statistic values derived under the null hypothesis that total collagen and free proline measurements are independent.

For this test, the statistic can be transformed such that the null distribution for large samples is Student’s t distribution with len(x) - 2 degrees of freedom.

The comparison is quantified by the p-value: the proportion of values in the null distribution as extreme or more extreme than the observed value of the statistic. In a two-sided test in which the statistic is positive, elements of the null distribution greater than the transformed statistic and elements of the null distribution less than the negative of the observed statistic are both considered “more extreme”.

If the p-value is “small” - that is, if there is a low probability of sampling data from independent distributions that produces such an extreme value of the statistic - this may be taken as evidence against the null hypothesis in favor of the alternative: the distribution of total collagen and free proline are not independent. Note that:

The inverse is not true; that is, the test is not used to provide evidence for the null hypothesis.

The threshold for values that will be considered “small” is a choice that should be made before the data is analyzed [5] with consideration of the risks of both false positives (incorrectly rejecting the null hypothesis) and false negatives (failure to reject a false null hypothesis).

Small p-values are not evidence for a large effect; rather, they can only provide evidence for a “significant” effect, meaning that they are unlikely to have occurred under the null hypothesis.

Suppose that before performing the experiment, the authors had reason to predict a positive correlation between the total collagen and free proline measurements, and that they had chosen to assess the plausibility of the null hypothesis against a one-sided alternative: free proline has a positive ordinal correlation with total collagen. In this case, only those values in the null distribution that are as great or greater than the observed statistic are considered to be more extreme.

Note that the t-distribution provides an asymptotic approximation of the null distribution; it is only accurate for samples with many observations. For small samples, it may be more appropriate to perform a permutation test: Under the null hypothesis that total collagen and free proline are independent, each of the free proline measurements were equally likely to have been observed with any of the total collagen measurements. Therefore, we can form an exact null distribution by calculating the statistic under each possible pairing of elements between x and y .

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Learn how to formulate null and alternative hypotheses for different statistical tests. The null hypothesis is the claim that there's no effect in the population, while the alternative hypothesis is the claim that there's an effect.

Learn how to write a null hypothesis and an alternative hypothesis for different scenarios using population parameters. See examples of null hypotheses for mean, proportion, and weight tests.

Learn how to write null and alternative hypotheses for different statistical tests. The null hypothesis is the claim that there's no effect in the population, while the alternative hypothesis is the claim that there's an effect.

Review. In a hypothesis test, sample data is evaluated in order to arrive at a decision about some type of claim.If certain conditions about the sample are satisfied, then the claim can be evaluated for a population. In a hypothesis test, we: Evaluate the null hypothesis, typically denoted with \(H_{0}\).The null is not rejected unless the hypothesis test shows otherwise.

The actual test begins by considering two hypotheses.They are called the null hypothesis and the alternative hypothesis.These hypotheses contain opposing viewpoints. H 0: The null hypothesis: It is a statement about the population that either is believed to be true or is used to put forth an argument unless it can be shown to be incorrect beyond a reasonable doubt.

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.

Consequently, when you write a null hypothesis, it must make a claim about the relevant population parameter. Further, that claim usually indicates that the effect does not exist in the population. Below are typical examples of writing a null hypothesis for various parameters and hypothesis tests. ... Alternative Hypothesis H A: Group ...

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. If you suspect that girls take longer to get ready for school than boys, then: Alternative: girls time > boys time. Null: girls time <= boys time.

The Null and Alternative Hypotheses. There are two hypotheses that are made: the null hypothesis, denoted H 0, and the alternative hypothesis, denoted H 1 or H A. The null hypothesis is the one to be tested and the alternative is everything else. In our example: The null hypothesis would be: The mean data scientist salary is 113,000 dollars.

Present the findings in your results and discussion section. Though the specific details might vary, the procedure you will use when testing a hypothesis will always follow some version of these steps. Table of contents. Step 1: State your null and alternate hypothesis. Step 2: Collect data. Step 3: Perform a statistical test.

The actual test begins by considering two hypotheses. They are called the null hypothesis and the alternative hypothesis. These hypotheses contain opposing viewpoints. H0 H 0: The null hypothesis: It is a statement of no difference between a sample mean or proportion and a population mean or proportion. In other words, the difference equals 0.

Writing null and alternative hypotheses. A ketchup company regularly receives large shipments of tomatoes. For each shipment that is received, a supervisor takes a random sample of 500 tomatoes to see what percent of the sample is bruised and performs a significance test. If the sample shows convincing evidence that more than 10 % of the entire ...

Write a statistical null hypothesis as a mathematical equation, such as. μ 1 = μ 2 {\displaystyle \mu _ {1}=\mu _ {2}} if you're comparing group means. Adjust the format of your null hypothesis to match the statistical method you used to test it, such as using "mean" if you're comparing the mean between 2 groups.

10.1 - Setting the Hypotheses: Examples. A significance test examines whether the null hypothesis provides a plausible explanation of the data. The null hypothesis itself does not involve the data. It is a statement about a parameter (a numerical characteristic of the population). These population values might be proportions or means or ...

The null hypothesis is a claim that a population parameter equals some value. For example, H 0: μ = 5 H 0: μ = 5. The alternative hypothesis is denoted H a H a. It is a claim about the population that is contradictory to the null hypothesis and is what we conclude is true when we reject H 0 H 0. The alternative hypothesis is a claim that a ...

Step 1: Figure out the hypothesis from the problem. The hypothesis is usually hidden in a word problem, and is sometimes a statement of what you expect to happen in the experiment. The hypothesis in the above question is "I expect the average recovery period to be greater than 8.2 weeks.". Step 2: Convert the hypothesis to math.

The general procedure for testing the null hypothesis is as follows: State the null and alternative hypotheses. Specify α and the sample size. Select an appropriate statistical test. Collect data (note that the previous steps should be done before collecting data) Compute the test statistic based on the sample data.

If we do not find that a relationship (or difference) exists, we fail to reject the null hypothesis (and go with it). We never say we accept the null hypothesis because it is never possible to prove something does not exist. That is why we say that we failed to reject the null hypothesis, rather than we accepted it. Del Siegle, Ph.D.

The alternative hypothesis ( Ha H a) is a claim about the population that is contradictory to H0 H 0 and what we conclude when we reject H0 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 ...

In research, there are two types of hypotheses: null and alternative. They work as a complementary pair, each stating that the other is wrong. Null Hypothesis (H0) - This can be thought of as the implied hypothesis. "Null" meaning "nothing.". This hypothesis states that there is no difference between groups or no relationship between ...

Examples of the Null Hypothesis. To write a null hypothesis, first start by asking a question. Rephrase that question in a form that assumes no relationship between the variables. In other words, assume a treatment has no effect. Write your hypothesis in a way that reflects this. Question.

Null hypothesis: µ ≥ 70 inches. Alternative hypothesis: µ < 70 inches. A two-tailed hypothesis involves making an "equal to" or "not equal to" statement. For example, suppose we assume the mean height of a male in the U.S. is equal to 70 inches. The null and alternative hypotheses in this case would be: Null hypothesis: µ = 70 inches.

Review. In a hypothesis test, sample data is evaluated in order to arrive at a decision about some type of claim.If certain conditions about the sample are satisfied, then the claim can be evaluated for a population. In a hypothesis test, we: Evaluate the null hypothesis, typically denoted with \(H_{0}\).The null is not rejected unless the hypothesis test shows otherwise.

14. The rule for the proper formulation of a hypothesis test is that the alternative or research hypothesis is the statement that, if true, is strongly supported by the evidence furnished by the data. The null hypothesis is generally the complement of the alternative hypothesis. Frequently, it is (or contains) the assumption that you are making ...

The p-value for a hypothesis test whose null hypothesis is that two samples have no ordinal correlation. See alternative above for alternative hypotheses. pvalue has the same shape as statistic. Warns: ConstantInputWarning. Raised if an input is a constant array. The correlation coefficient is not defined in this case, so np.nan is returned ...