difference between null and alternative hypothesis with example

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• Null and Alternative Hypotheses | Definitions & Examples

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.

*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.

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.

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.

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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• Key Differences

Know the Differences & Comparisons

Difference Between Null and Alternative Hypothesis

Null hypothesis implies a statement that expects no difference or effect. On the contrary, an alternative hypothesis is one that expects some difference or effect. Null hypothesis This article excerpt shed light on the fundamental differences between null and alternative hypothesis.

Content: Null Hypothesis Vs Alternative Hypothesis

Comparison chart, definition of null hypothesis.

A null hypothesis is a statistical hypothesis in which there is no significant difference exist between the set of variables. It is the original or default statement, with no effect, often represented by H 0 (H-zero). It is always the hypothesis that is tested. It denotes the certain value of population parameter such as µ, s, p. A null hypothesis can be rejected, but it cannot be accepted just on the basis of a single test.

Definition of Alternative Hypothesis

A statistical hypothesis used in hypothesis testing, which states that there is a significant difference between the set of variables. It is often referred to as the hypothesis other than the null hypothesis, often denoted by H 1 (H-one). It is what the researcher seeks to prove in an indirect way, by using the test. It refers to a certain value of sample statistic, e.g., x¯, s, p

The acceptance of alternative hypothesis depends on the rejection of the null hypothesis i.e. until and unless null hypothesis is rejected, an alternative hypothesis cannot be accepted.

Key Differences Between Null and Alternative Hypothesis

The important points of differences between null and alternative hypothesis are explained as under:

• A null hypothesis is a statement, in which there is no relationship between two variables. An alternative hypothesis is a statement; that is simply the inverse of the null hypothesis, i.e. there is some statistical significance between two measured phenomenon.
• A null hypothesis is what, the researcher tries to disprove whereas an alternative hypothesis is what the researcher wants to prove.
• A null hypothesis represents, no observed effect whereas an alternative hypothesis reflects, some observed effect.
• If the null hypothesis is accepted, no changes will be made in the opinions or actions. Conversely, if the alternative hypothesis is accepted, it will result in the changes in the opinions or actions.
• As null hypothesis refers to population parameter, the testing is indirect and implicit. On the other hand, the alternative hypothesis indicates sample statistic, wherein, the testing is direct and explicit.
• A null hypothesis is labelled as H 0 (H-zero) while an alternative hypothesis is represented by H 1 (H-one).
• The mathematical formulation of a null hypothesis is an equal sign but for an alternative hypothesis is not equal to sign.
• In null hypothesis, the observations are the outcome of chance whereas, in the case of the alternative hypothesis, the observations are an outcome of real effect.

There are two outcomes of a statistical test, i.e. first, a null hypothesis is rejected and alternative hypothesis is accepted, second, null hypothesis is accepted, on the basis of the evidence. In simple terms, a null hypothesis is just opposite of alternative hypothesis.

You Might Also Like:

Zipporah Thuo says

February 22, 2018 at 6:06 pm

The comparisons between the two hypothesis i.e Null hypothesis and the Alternative hypothesis are the best.Thank you.

Getu Gamo says

March 4, 2019 at 3:42 am

Thank you so much for the detail explanation on two hypotheses. Now I understood both very well, including their differences.

Jyoti Bhardwaj says

May 28, 2019 at 6:26 am

Thanks, Surbhi! Appreciate the clarity and precision of this content.

January 9, 2020 at 6:16 am

John Jenstad says

July 20, 2020 at 2:52 am

Thanks very much, Surbhi, for your clear explanation!!

Navita says

July 2, 2021 at 11:48 am

Thanks for the Comparison chart! it clears much of my doubt.

GURU UPPALA says

July 21, 2022 at 8:36 pm

Thanks for the Comparison chart!

Enock kipkoech says

September 22, 2022 at 1:57 pm

What are the examples of null hypothesis and substantive hypothesis

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AP®︎/College Statistics

Course: ap®︎/college statistics   >   unit 10.

• Idea behind hypothesis testing

Examples of null and alternative hypotheses

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

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Video transcript

Module 9: Hypothesis Testing With One Sample

Null and alternative hypotheses, learning outcomes.

• Describe hypothesis testing in general and in practice

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

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

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

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

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

Mathematical Symbols Used in  H 0 and H a :

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

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

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

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

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

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

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

H 0 : μ = 2.0

H a : μ ≠ 2.0

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

• H 0 : μ = 66
• H a : μ ≠ 66

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

H 0 : μ ≥ 5

H a : μ < 5

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

• H 0 : μ ≥ 45
• H a : μ < 45

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

H 0 : p ≤ 0.066

H a : p > 0.066

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

• H 0 : p = 0.40
• H a : p > 0.40

Concept Review

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

Formula Review

H 0 and H a are contradictory.

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9.1 Null and Alternative Hypotheses

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

H 0 : The null hypothesis: It is a statement of no difference between the variables–they are not related. This can often be considered the status quo and as a result if you cannot accept the null it requires some action.

H a : The alternative hypothesis: It is a claim about the population that is contradictory to H 0 and what we conclude when we cannot accept H 0 . This is usually what the researcher is trying to prove. The alternative hypothesis is the contender and must win with significant evidence to overthrow the status quo. This concept is sometimes referred to the tyranny of the status quo because as we will see later, to overthrow the null hypothesis takes usually 90 or greater confidence that this is the proper decision.

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 "cannot accept 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. These conclusions are all based upon a level of probability, a significance level, that is set by the analyst.

Table 9.1 presents the various hypotheses in the relevant pairs. For example, if the null hypothesis is equal to some value, the alternative has to be not equal to that value.

As a mathematical convention 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.

Example 9.1

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.

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: 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 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

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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.

Null Hypothesis and Alternative Hypothesis

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Hypothesis testing involves the careful construction of two statements: the null hypothesis and the alternative hypothesis. These hypotheses can look very similar but are actually different.

How do we know which hypothesis is the null and which one is the alternative? We will see that there are a few ways to tell the difference.

The Null Hypothesis

The null hypothesis reflects that there will be no observed effect in our experiment. In a mathematical formulation of the null hypothesis, there will typically be an equal sign. This hypothesis is denoted by H 0 .

The null hypothesis is what we attempt to find evidence against in our hypothesis test. We hope to obtain a small enough p-value that it is lower than our level of significance alpha and we are justified in rejecting the null hypothesis. If our p-value is greater than alpha, then we fail to reject the null hypothesis.

If the null hypothesis is not rejected, then we must be careful to say what this means. The thinking on this is similar to a legal verdict. Just because a person has been declared "not guilty", it does not mean that he is innocent. In the same way, just because we failed to reject a null hypothesis it does not mean that the statement is true.

For example, we may want to investigate the claim that despite what convention has told us, the mean adult body temperature is not the accepted value of 98.6 degrees Fahrenheit . The null hypothesis for an experiment to investigate this is “The mean adult body temperature for healthy individuals is 98.6 degrees Fahrenheit.” If we fail to reject the null hypothesis, then our working hypothesis remains that the average adult who is healthy has a temperature of 98.6 degrees. We do not prove that this is true.

If we are studying a new treatment, the null hypothesis is that our treatment will not change our subjects in any meaningful way. In other words, the treatment will not produce any effect in our subjects.

The Alternative Hypothesis

The alternative or experimental hypothesis reflects that there will be an observed effect for our experiment. In a mathematical formulation of the alternative hypothesis, there will typically be an inequality, or not equal to symbol. This hypothesis is denoted by either H a or by H 1 .

The alternative hypothesis is what we are attempting to demonstrate in an indirect way by the use of our hypothesis test. If the null hypothesis is rejected, then we accept the alternative hypothesis. If the null hypothesis is not rejected, then we do not accept the alternative hypothesis. Going back to the above example of mean human body temperature, the alternative hypothesis is “The average adult human body temperature is not 98.6 degrees Fahrenheit.”

If we are studying a new treatment, then the alternative hypothesis is that our treatment does, in fact, change our subjects in a meaningful and measurable way.

The following set of negations may help when you are forming your null and alternative hypotheses. Most technical papers rely on just the first formulation, even though you may see some of the others in a statistics textbook.

• Null hypothesis: “ x is equal to y .” Alternative hypothesis “ x is not equal to y .”
• Null hypothesis: “ x is at least y .” Alternative hypothesis “ x is less than y .”
• Null hypothesis: “ x is at most y .” Alternative hypothesis “ x is greater than y .”
• An Example of a Hypothesis Test
• Hypothesis Test for the Difference of Two Population Proportions
• What Is a P-Value?
• How to Conduct a Hypothesis Test
• Hypothesis Test Example
• Chi-Square Goodness of Fit Test
• What Level of Alpha Determines Statistical Significance?
• How to Do Hypothesis Tests With the Z.TEST Function in Excel
• The Difference Between Type I and Type II Errors in Hypothesis Testing
• Type I and Type II Errors in Statistics
• The Runs Test for Random Sequences
• What 'Fail to Reject' Means in a Hypothesis Test
• What Is the Difference Between Alpha and P-Values?
• An Example of Chi-Square Test for a Multinomial Experiment
• Example of a Chi-Square Goodness of Fit Test
• Null Hypothesis Definition and Examples

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Null vs. Alternative Hypothesis

04.28.2023 • 5 min read

Sarah Thomas

Subject Matter Expert

Learn about a null versus alternative hypothesis and what they show with examples for each. Also go over the main differences and similarities between them.

In This Article

What Is a Null Hypothesis?

What is an alternative hypothesis, outcomes of a hypothesis test.

Main Differences Between Null & Alternative Hypothesis

Similarities Between Null & Alternative Hypothesis

Hypothesis Testing & Errors

In statistics, you’ll draw insights or “inferences” about population parameters using data from a sample. This process is called inferential statistics.

To make statistical inferences, you need to determine if you have enough evidence to support a certain hypothesis about the population. This is where null and alternative hypotheses come into play!

In this article, we’ll explain the differences between these two types of hypotheses, and we’ll explain the role they play in hypothesis testing.

Imagine you want to know what percent of Americans are vegetarians. You find a Gallup poll claiming ‌5% of the population was vegetarian in 2018, but your intuition tells you vegetarianism is on the rise and that ‌far more than 5% of Americans are vegetarian today.

To investigate further, you collect your own sample data by surveying 1,000 randomly selected Americans. You’ll use this random sample to determine whether it’s likely ‌the true population proportion of vegetarians is, in fact, 5% (as the Gallup data suggests) or whether it could be the case that the percentage of vegetarians is now higher.

Notice ‌that your investigation involves two rival hypotheses about the population. One hypothesis is that the proportion of vegetarians is 5%. The other hypothesis is that the proportion of vegetarians is greater than 5%. In statistics, we would call the first hypothesis the null hypothesis, and the second hypothesis the alternative hypothesis. The null hypothesis ( H 0 H_0 H 0 ​ ) represents the status quo or what is assumed to be true about the population at the start of your investigation.

Null Hypothesis

In hypothesis testing, the null hypothesis ( H 0 H_0 H 0 ​ ) is the default hypothesis.

It's what the status quo assumes to be true about the population.

The alternative hypothesis ( H a H_a H a ​ or H 1 H_1 H 1 ​ ) is the hypothesis that stands contrary to the null hypothesis. The alternative hypothesis ‌represents the research hypothesis—what you as the statistician are trying to prove with your data .

In medical studies, where scientists are trying to demonstrate whether a treatment has a significant effect on patient outcomes, the alternative hypothesis represents the hypothesis that the treatment does have an effect, while the null hypothesis represents the assumption that the treatment has no effect.

Alternative Hypothesis

The alternative hypothesis ( H a H_a H a ​ or H 1 H_1 H 1 ​ ) is the hypothesis being proposed in opposition to the null hypothesis.

Examples of Null and Alternative Hypotheses

In a hypothesis test, the null and alternative hypotheses must be mutually exclusive statements, meaning both hypotheses cannot be true at the same time. For example, if the null hypothesis includes an equal sign, the alternative hypothesis must state that the values being mentioned are “not equal” in some way.

Your hypotheses will also depend on the formulation of your test—are you running a one-sample T-test, a two-sample T-test, F-test for ANOVA , or a Chi-squared test? It also matters whether you are conducting a directional one-tailed test or a nondirectional two-tailed test.

Example 1: Two-Tailed T-test

Null Hypothesis: The population mean is equal to some number, x. 𝝁 = x

Alternative Hypothesis: The population mean is not equal to x. 𝝁 ≠ x

Example 2: One-tailed T-test (Right-Tailed)

Null Hypothesis: The population mean is less than or equal to some number, x. 𝝁 ≤ x Alternative Hypothesis: The population mean is greater than x. 𝝁 > x

Example 3: One-tailed T-test (Left-Tailed)

Null Hypothesis: The population mean is greater than or equal to some number, x. 𝝁 ≥ x

Alternative Hypothesis: The population mean is less than x. 𝝁 < x

By the end of a hypothesis test, you will have reached one of two conclusions.

You will run into either 2 outcomes:

Fail to reject the null hypothesis on the grounds that there's insufficient evidence to move away from the null hypothesis

Reject the null hypothesis in favor of the alternative.

If you’re ‌confused about the outcomes of a hypothesis test, a good analogy is a jury trial. In a jury trial, the defendant is innocent until proven guilty. To reach a verdict of guilt, the jury must find strong evidence (beyond a reasonable doubt) that the defendant committed the crime.

This is analogous to a statistician who must assume the null hypothesis is true unless they can uncover strong evidence ( a p-value less than or equal to the significance level) in support of the alternative hypothesis.

Notice also, that a jury never concludes a defendant is innocent—only that the defendant is guilty or not guilty. This is similar to how we never conclude that the null hypothesis is true. In a hypothesis test, we never conclude ‌that the null hypothesis is true. We can only “reject” the null hypothesis or “fail to reject” it.

In this video, let’s look at the jury example again, the reasoning behind hypothesis testing, and how to form a test. It starts by stating your null and alternative hypotheses.

Main Differences Between Null and Alternative Hypothesis

Here is a summary of the key differences between the null and the alternative hypothesis test.

The null hypothesis represents the status quo; the alternative hypothesis represents an alternative statement about the population.

The null and the alternative are mutually exclusive statements, meaning both statements cannot be true at the same time.

In a medical study, the null hypothesis represents the assumption that a treatment has no statistically significant effect on the outcome being studied. The alternative hypothesis represents the belief that the treatment does have an effect.

The null hypothesis is denoted by H_0 ; the alternative hypothesis is denoted by H_a H_1

You “fail to reject” the null hypothesis when the p-value is larger than the significance level. You “reject” the null hypothesis in favor of the alternative hypothesis when the p-value is less than or equal to your test’s significance level.

Similarities Between Null and Alternative Hypothesis

The similarities between the null and alternative hypotheses are as follows.

Both the null and the alternative are statements about the same underlying data.

Both statements provide a possible answer to a statistician’s research question.

The same hypothesis test will provide evidence for or against the null and alternative hypotheses.

Hypothesis Testing and Errors

Always remember that statistical inference provides you with inferences based on probability rather than hard truths. Anytime you conduct a hypothesis test, there is a chance that you’ll reach the wrong conclusion about your data.

In statistics, we categorize these wrong conclusions into two types of errors:

Type I Errors

Type II Errors

Type I Error (ɑ)

A Type I error occurs when you reject the null hypothesis when, in fact, the null hypothesis is true. This is sometimes called a false positive and is analogous to a jury that falsely convicts an innocent defendant. The probability of making this type of error is represented by alpha, ɑ.

Type II Error (ꞵ)

A Type II error occurs when you fail to reject the null hypothesis when, in fact, the null hypothesis is false. This is sometimes called a false negative and is analogous to a jury that reaches a verdict of “not guilty,” when, in fact, the defendant has committed the crime. The probability of making this type of error is represented by beta, ꞵ.

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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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8.1.1: Null and Alternative Hypotheses

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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 of no difference between the variables—they are not related. This can often be considered the status quo and as a result if you cannot accept the null it requires some action.

\(H_a\): The alternative hypothesis: It is a claim about the population that is contradictory to \(H_0\) and what we conclude when we reject \(H_0\). This is usually what the researcher is trying to prove.

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.

\(H_{0}\) 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.

Example \(\PageIndex{1}\)

• \(H_{0}\): No more than 30% of the registered voters in Santa Clara County voted in the primary election. \(p \leq 30\)
• \(H_{a}\): More than 30% of the registered voters in Santa Clara County voted in the primary election. \(p > 30\)

Exercise \(\PageIndex{1}\)

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%. \(p \neq 0.25\)

Example \(\PageIndex{2}\)

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:

• \(H_{0}: \mu = 2.0\)
• \(H_{a}: \mu \neq 2.0\)

Exercise \(\PageIndex{2}\)

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 \((=, \neq, \geq, <, \leq, >)\) for the null and alternative hypotheses.

• \(H_{0}: \mu \_ 66\)
• \(H_{a}: \mu \_ 66\)
• \(H_{0}: \mu = 66\)
• \(H_{a}: \mu \neq 66\)

Example \(\PageIndex{3}\)

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:

• \(H_{0}: \mu \geq 5\)
• \(H_{a}: \mu < 5\)

Exercise \(\PageIndex{3}\)

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.

• \(H_{0}: \mu \_ 45\)
• \(H_{a}: \mu \_ 45\)
• \(H_{0}: \mu \geq 45\)
• \(H_{a}: \mu < 45\)

Example \(\PageIndex{4}\)

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.

• \(H_{0}: p \leq 0.066\)
• \(H_{a}: p > 0.066\)

Exercise \(\PageIndex{4}\)

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 (\(=, \neq, \geq, <, \leq, >\)) for the null and alternative hypotheses.

• \(H_{0}: p \_ 0.40\)
• \(H_{a}: p \_ 0.40\)
• \(H_{0}: p = 0.40\)
• \(H_{a}: p > 0.40\)

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.

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 \((=, \leq \text{or} \geq)\)
• Always write the alternative hypothesis , typically denoted with \(H_{a}\) or \(H_{1}\), using less than, greater than, or not equals symbols, i.e., \((\neq, >, \text{or} <)\).
• 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.

• If \(\alpha \leq p\)-value, then do not reject \(H_{0}\).
• If\(\alpha > p\)-value, then reject \(H_{0}\).

\(\alpha\) is preconceived. Its value is set before the hypothesis test starts. The \(p\)-value is calculated from the data.References

Data from the National Institute of Mental Health. Available online at http://www.nimh.nih.gov/publicat/depression.cfm .

Research Hypothesis In Psychology: Types, & Examples

Saul Mcleod, PhD

Editor-in-Chief for Simply Psychology

BSc (Hons) Psychology, MRes, PhD, University of Manchester

Saul Mcleod, PhD., is a qualified psychology teacher with over 18 years of experience in further and higher education. He has been published in peer-reviewed journals, including the Journal of Clinical Psychology.

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Olivia Guy-Evans, MSc

Associate Editor for Simply Psychology

BSc (Hons) Psychology, MSc Psychology of Education

Olivia Guy-Evans is a writer and associate editor for Simply Psychology. She has previously worked in healthcare and educational sectors.

On This Page:

A research hypothesis, in its plural form “hypotheses,” is a specific, testable prediction about the anticipated results of a study, established at its outset. It is a key component of the scientific method .

Hypotheses connect theory to data and guide the research process towards expanding scientific understanding

Some key points about hypotheses:

• A hypothesis expresses an expected pattern or relationship. It connects the variables under investigation.
• It is stated in clear, precise terms before any data collection or analysis occurs. This makes the hypothesis testable.
• A hypothesis must be falsifiable. It should be possible, even if unlikely in practice, to collect data that disconfirms rather than supports the hypothesis.
• Hypotheses guide research. Scientists design studies to explicitly evaluate hypotheses about how nature works.
• For a hypothesis to be valid, it must be testable against empirical evidence. The evidence can then confirm or disprove the testable predictions.
• Hypotheses are informed by background knowledge and observation, but go beyond what is already known to propose an explanation of how or why something occurs.

Predictions typically arise from a thorough knowledge of the research literature, curiosity about real-world problems or implications, and integrating this to advance theory. They build on existing literature while providing new insight.

Types of Research Hypotheses

Alternative hypothesis.

The research hypothesis is often called the alternative or experimental hypothesis in experimental research.

It typically suggests a potential relationship between two key variables: the independent variable, which the researcher manipulates, and the dependent variable, which is measured based on those changes.

The alternative hypothesis states a relationship exists between the two variables being studied (one variable affects the other).

A hypothesis is a testable statement or prediction about the relationship between two or more variables. It is a key component of the scientific method. Some key points about hypotheses:

• Important hypotheses lead to predictions that can be tested empirically. The evidence can then confirm or disprove the testable predictions.

In summary, a hypothesis is a precise, testable statement of what researchers expect to happen in a study and why. Hypotheses connect theory to data and guide the research process towards expanding scientific understanding.

An experimental hypothesis predicts what change(s) will occur in the dependent variable when the independent variable is manipulated.

It states that the results are not due to chance and are significant in supporting the theory being investigated.

The alternative hypothesis can be directional, indicating a specific direction of the effect, or non-directional, suggesting a difference without specifying its nature. It’s what researchers aim to support or demonstrate through their study.

Null Hypothesis

The null hypothesis states no relationship exists between the two variables being studied (one variable does not affect the other). There will be no changes in the dependent variable due to manipulating the independent variable.

It states results are due to chance and are not significant in supporting the idea being investigated.

The null hypothesis, positing no effect or relationship, is a foundational contrast to the research hypothesis in scientific inquiry. It establishes a baseline for statistical testing, promoting objectivity by initiating research from a neutral stance.

Many statistical methods are tailored to test the null hypothesis, determining the likelihood of observed results if no true effect exists.

This dual-hypothesis approach provides clarity, ensuring that research intentions are explicit, and fosters consistency across scientific studies, enhancing the standardization and interpretability of research outcomes.

Nondirectional Hypothesis

A non-directional hypothesis, also known as a two-tailed hypothesis, predicts that there is a difference or relationship between two variables but does not specify the direction of this relationship.

It merely indicates that a change or effect will occur without predicting which group will have higher or lower values.

For example, “There is a difference in performance between Group A and Group B” is a non-directional hypothesis.

Directional Hypothesis

A directional (one-tailed) hypothesis predicts the nature of the effect of the independent variable on the dependent variable. It predicts in which direction the change will take place. (i.e., greater, smaller, less, more)

It specifies whether one variable is greater, lesser, or different from another, rather than just indicating that there’s a difference without specifying its nature.

For example, “Exercise increases weight loss” is a directional hypothesis.

Falsifiability

The Falsification Principle, proposed by Karl Popper , is a way of demarcating science from non-science. It suggests that for a theory or hypothesis to be considered scientific, it must be testable and irrefutable.

Falsifiability emphasizes that scientific claims shouldn’t just be confirmable but should also have the potential to be proven wrong.

It means that there should exist some potential evidence or experiment that could prove the proposition false.

However many confirming instances exist for a theory, it only takes one counter observation to falsify it. For example, the hypothesis that “all swans are white,” can be falsified by observing a black swan.

For Popper, science should attempt to disprove a theory rather than attempt to continually provide evidence to support a research hypothesis.

Can a Hypothesis be Proven?

Hypotheses make probabilistic predictions. They state the expected outcome if a particular relationship exists. However, a study result supporting a hypothesis does not definitively prove it is true.

All studies have limitations. There may be unknown confounding factors or issues that limit the certainty of conclusions. Additional studies may yield different results.

In science, hypotheses can realistically only be supported with some degree of confidence, not proven. The process of science is to incrementally accumulate evidence for and against hypothesized relationships in an ongoing pursuit of better models and explanations that best fit the empirical data. But hypotheses remain open to revision and rejection if that is where the evidence leads.

• Disproving a hypothesis is definitive. Solid disconfirmatory evidence will falsify a hypothesis and require altering or discarding it based on the evidence.
• However, confirming evidence is always open to revision. Other explanations may account for the same results, and additional or contradictory evidence may emerge over time.

We can never 100% prove the alternative hypothesis. Instead, we see if we can disprove, or reject the null hypothesis.

If we reject the null hypothesis, this doesn’t mean that our alternative hypothesis is correct but does support the alternative/experimental hypothesis.

Upon analysis of the results, an alternative hypothesis can be rejected or supported, but it can never be proven to be correct. We must avoid any reference to results proving a theory as this implies 100% certainty, and there is always a chance that evidence may exist which could refute a theory.

How to Write a Hypothesis

• Identify variables . The researcher manipulates the independent variable and the dependent variable is the measured outcome.
• Operationalized the variables being investigated . Operationalization of a hypothesis refers to the process of making the variables physically measurable or testable, e.g. if you are about to study aggression, you might count the number of punches given by participants.
• Decide on a direction for your prediction . If there is evidence in the literature to support a specific effect of the independent variable on the dependent variable, write a directional (one-tailed) hypothesis. If there are limited or ambiguous findings in the literature regarding the effect of the independent variable on the dependent variable, write a non-directional (two-tailed) hypothesis.
• Make it Testable : Ensure your hypothesis can be tested through experimentation or observation. It should be possible to prove it false (principle of falsifiability).
• Clear & concise language . A strong hypothesis is concise (typically one to two sentences long), and formulated using clear and straightforward language, ensuring it’s easily understood and testable.

Consider a hypothesis many teachers might subscribe to: students work better on Monday morning than on Friday afternoon (IV=Day, DV= Standard of work).

Now, if we decide to study this by giving the same group of students a lesson on a Monday morning and a Friday afternoon and then measuring their immediate recall of the material covered in each session, we would end up with the following:

• The alternative hypothesis states that students will recall significantly more information on a Monday morning than on a Friday afternoon.
• The null hypothesis states that there will be no significant difference in the amount recalled on a Monday morning compared to a Friday afternoon. Any difference will be due to chance or confounding factors.

More Examples

• Memory : Participants exposed to classical music during study sessions will recall more items from a list than those who studied in silence.
• Social Psychology : Individuals who frequently engage in social media use will report higher levels of perceived social isolation compared to those who use it infrequently.
• Developmental Psychology : Children who engage in regular imaginative play have better problem-solving skills than those who don’t.
• Clinical Psychology : Cognitive-behavioral therapy will be more effective in reducing symptoms of anxiety over a 6-month period compared to traditional talk therapy.
• Cognitive Psychology : Individuals who multitask between various electronic devices will have shorter attention spans on focused tasks than those who single-task.
• Health Psychology : Patients who practice mindfulness meditation will experience lower levels of chronic pain compared to those who don’t meditate.
• Organizational Psychology : Employees in open-plan offices will report higher levels of stress than those in private offices.
• Behavioral Psychology : Rats rewarded with food after pressing a lever will press it more frequently than rats who receive no reward.

• Alternative vs Null Hypothesis: Pros, Cons, Uses & Examples

All research starts with a problem that needs to be solved. From this problem, hypotheses are developed to provide the researcher with a clear statement of the problem.

To understand alternative hypotheses also known as alternate hypotheses, you must first understand what the hypothesis is .

When you hear the word hypothesis it means the accurate explanations in relation to a set of facts that can be analyzed when studied, using some specific method of research.

There are primarily two types of hypothesis which are null hypothesis and alternative hypothesis.

When you think about the word “null” what should come to mind is something that can not change, what you expect is what you get, unlike alternate hypotheses which can change.

Now, the research problems or questions which could be in the form of null hypothesis or alternative hypothesis are expressed as the relationship that exists between two or more variables. The process for this states that the questions should be what expresses the relationship between two variables that can be measured.

Both null hypotheses and alternative hypotheses are used by statisticians and researchers to conduct research in various industries or fields such as mathematics, psychology, science, medicine, and technology.

We are going to discuss alternative hypotheses and null hypotheses in this post and how they work in research.

What is an Alternative Hypothesis?

Alternative hypothesis simply put is another viable option to the null hypothesis. It means looking for a substantial change or option that can allow you to reject the null hypothesis.

It is an opposing theory to a null hypothesis.

If you develop a null hypothesis, you make an informed guess on whether a thing is true or whether there is a relationship between that thing and another variable. An alternate hypothesis will always take an opposite stand against a null hypothesis. So if according to a null hypothesis something is correct to an alternate hypothesis that same thing will be incorrect.

For example, let’s assume that you develop a null hypothesis that states “I”m going to be $500 richer” the alternate hypothesis will be “I’m going to get $500 or be richer”

When you are trying to disprove a null hypothesis, that is when you test an alternate hypothesis. If there is enough data to back up the alternative hypothesis then you can dispose of the null hypothesis. 

Get Answers: What is Empirical Research Study? [Examples & Method]

What is a Null Hypothesis?

The null hypothesis is best explained as the statement showing that no relationship exists between two variables that are being considered or that two groups are not related. As we have earlier established, a hypothesis is an assumed statement that has not been proven with sufficient data that could serve as a piece of evidence. 

The null hypothesis is now the statement that a researcher or an investigator wants to disprove. The null hypothesis is capable of being tested, being verifiable, and also capable of being rejected.

For example, if you want to conduct a study that will compare the relationship between project A and project B if the study is based on the assumption that both projects are of equal standard, the assumption is referred to as the null hypothesis.

This is because the null hypothesis should be specific at all times.

Learn: Hypothesis Testing in Research: Definition, Procedure, Uses, Limitations + Examples

Advantages of the Alternative Hypothesis 

• Alternative hypothesis gives a researcher specific clarifications on the research questions or problems .
• It provides a study with the direction that can be used to collect data and obtain results of interest by the researcher.
• An alternative hypothesis is always selected before commencing the studies which gives the researcher the opportunity to prove that the restatement is backed up by evidence and not just from the researcher’s ideas or values.
• Another good thing about alternative hypotheses is that it provides the opportunity to discover new theories that a researcher can use to disprove an existing theory that may not have been backed up by evidence .
• An alternate hypothesis is also useful to prove that there is a relationship between two selected variables and the outcomes of the conducted study are relevant.

Principles of the Alternative Hypothesis

• Alternative hypotheses will be accepted if the amount of data that is gone is insignificant within the significance level. This means that the null hypothesis will be rejected.
• Another principle of the alternative hypothesis is that the data gathered from random samples go through a statistical tool that analyzes the effect of the amount of data leaving the null hypothesis.

For the curious: Sampling Bias: Definition, Types + [Examples]

Purpose of the Null Hypothesis 

Here are the purposes of the null hypothesis in an experiment or study:

• The primary purpose of a null hypothesis is to disprove an assumption.
• Null hypotheses can help to further progress a theory in some scientific cases.
• You can also use a null hypothesis to ascertain how consistent the outcomes of multiple studies are.

Principle of the Null Hypothesis 

Now, these are the principles of the null hypothesis:

1. The primary principle of the null hypothesis is to prove that the assumed statement is true. This is done by collecting data and analyzing in the study , what chance the collected data has in the random sample.

2. If the collected data does not meet the expectation of the null hypothesis, it is determined that the data lacks sufficient evidence to back up the null hypothesis therefore the null hypothesis statement is rejected.

Just as in the case of the alternative hypothesis the collected data in a null hypothesis is analyzed using some statistical tools that are made to measure the extent to which data left the null hypothesis.

The process will determine whether the data that left the null hypothesis is larger than a set value. If the data collected from the random sample is enough to serve as evidence to prove the null hypothesis then the null hypothesis will be accepted as true. And also defined that it has no relationship with other variables .

Learn About: Research Reports: Definition, Types + [Writing Guide]

Types of Alternative Hypothesis (Advantages of Each and When to Use)

There are four types of alternative hypotheses, and we will briefly discuss them below.

• One-tailed directional: For one of the sampling distributions one tail, this type of alternative hypothesis focuses on the rejected part only.
• Two-tailed directional: In an alternative hypothesis, a two-tailed directional focus on the two parts or directions that were rejected in the sampling distribution.
• Point: Point is another alternate hypothesis. It occurs in hypothesis testing when the sample population in the alternate hypothesis has been completely defined in a distribution. If there are no known parameters, the hypothesis will serve no interest. They are, however, important to the foundation of the statistical inferences.
• Non-directional: In an alternate hypothesis, a non-directional does not focus on the two directions of rejection. The only focus of the nondirectional alternative hypothesis is to prove that the null hypothesis is incorrect.

Read: Type I vs Type II Errors: Definition, Examples & Prevention

Difference between Null Hypothesis and Alternative Hypothesis 

We are going to look at the differences between the alternate hypothesis and the null hypothesis based on these six factors which are:

• Mathematical expression
• Observation
• Acceptance criteria
• The difference in Mathematical expression

Null hypothesis is followed by an ‘equals to’ (=) sign. While the Alternative hypothesis is followed by these three signs; 

• The difference in how they are observed

In the null hypothesis, it is believed that the results that are observed are as a result of chance. While In the alternative hypothesis, it is believed that the observed results are the outcome of some real causes.

• Differences in results

The result of the null hypothesis always shows that there have been no changes in statements or opinions. While the result of the alternative hypothesis shows that there have been significant changes in statements and opinions.

• Differences in Acceptance criteria

If the p-value in a null hypothesis is greater than the significance level, then the null hypothesis is accepted.

If the p-value in an alternate hypothesis is smaller than the significance level, then the alternative hypothesis is accepted.

• Differences in importance

The null hypothesis accepts true existing theories and also if there has been consistency in multiple experiments of similar hypotheses.

The alternative hypothesis establishes whether a relationship exists between two variables, and the result will then lead to new improved theories.

Read: T-testing: Definition, Formula & Interpretations

Examples of an Alternative Hypothesis and Null Hypothesis

Here are some examples of the alternative hypothesis:.

A researcher assumes that a bridge’s bearing capacity is over 10 tons, the researcher will then develop an hypothesis to support this study. The hypothesis will be:

For the null hypothesis H0: µ= 10 tons

For the alternate hypothesis Ha: µ>10 tons

In another study being conducted, the researcher wants to find out whether there is a noticeable difference or change in a patient’s heart arrest medicine and the patient’s heart condition.

For the alternate hypothesis: The hypothesis is that there might indeed be a relationship between the new medicine and the frequency or chances of heart arrest in a patient.

Here are the examples of the null hypothesis

The hypothesis from example 2 in the alternate hypothesis implies that the use of one specific medicine can reduce the frequency and chances of heart arrest.

For the null hypothesis: The hypothesis will be that the use of that particular medicine cannot reduce the chance and frequency of heart arrest in a patient.

An alternate hypothesis states that the random exam scores are collected from both men and women. But are the scores of the two groups (men and women) the same or are they different?

For the null hypothesis: The hypothesis will state that the calculated mean of the men’s exam score is equal to the exam score of the women.

This is represented as

H0= The null hypothesis

µ1= The calculated mean score of men

µ2= The calculated mean score of women

Read: What is Empirical Research Study? [Examples & Method]

• Can you reject an alternative hypothesis?

It is quite inappropriate to say or report that an alternate hypothesis was rejected. It is much better to use the phrase “the alternate hypothesis was rather not supported”.

The reason behind this use of words is that only the null hypothesis is designed to be rejected in a study. The alternative hypothesis is designed to prove the null hypothesis incorrect, to introduce new facts that can disprove the null hypothesis but it is not designed to be rejected.

It can either be accepted or not supported.

• How do you identify alternative hypotheses?

A researcher can use this formula to identify the alternate hypothesis in a study or experiment.

H0 and Ha are in contrast.

Therefore, if Ho has:

Equal to (=)

Greater than or equal to (≥)

Less than or equal to (≤)

And then Ha has:

Not equal (≠) 

Greater than (>) or less than (

Less than ( )

If in a study, α ≤ p-value, then the researcher should not reject H0.

If in a study, α > p-value, then the researcher should reject H0.

α is preconceived. The value of α is determined even before the hypothesis test is conducted. While the p-value is derived from the calculation in the data.

• Which is better in formulating hypotheses of your study alternative or null?

The study a researcher wants to conduct will determine what hypothesis should be developed. However, the researcher should keep in mind what the purpose of the null and alternative two hypotheses are while developing the study hypothesis. So while the null hypothesis will accept existing theories that it found to be true or correct, and measure the consistency of multiple experiments, alternative hypotheses will find the relationship that exists (if any) between two phenomena and may lead to the development of a new and improved theory.

In this article, it has been clearly defined the relationship that exists between the null hypothesis and the alternative hypothesis. While the null hypothesis is always an assumption that needs to be proven with evidence for it to be accepted, the alternative hypothesis puts in all the effort to make sure the null hypothesis is disproved. 

Researchers should note that for every null hypothesis, one or more alternate hypotheses can be developed.

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Hypothesis is a statement or an assumption that may be true or false. There are six types of hypotheses mainly the Simple hypothesis, Complex hypothesis, Directional hypothesis, Associative hypothesis, and Null hypothesis. Usually, the hypothesis is the start point of any scientific investigation, It gives the right direction to the process of investigation. It avoids the blind search and gives direction to the search. It acts as a compass in the process.

Null hypothesis suggests that there is no relationship between the two variables. Null hypothesis is also exactly the opposite of the alternative hypothesis. Null hypothesis is generally what researchers or scientists try to disprove and if the null hypothesis gets accepted then we have to make changes in our opinion i.e. we have to make changes in our original opinion or statement in order to match null hypothesis. Null hypothesis is represented as H0. If my alternative hypothesis is that 55% of boys in my town are taller than girls then my alternative hypothesis will be that 55% of boys in my town are not taller than girls. 

Alternative hypothesis is a method for reaching a conclusion and making inferences and judgements about certain facts or a statement. This is done on the basis of the data which is available. Usually, the statement which we check regarding the null hypothesis is commonly known as the alternative hypothesis. Most of the times alternative hypothesis is exactly the opposite of the null hypothesis. This is what generally researchers or scientists try to approve. Alternative hypothesis is represented as Ha or H1. If my null hypothesis is that 55% of boys in my town are not taller than girls then my alternative hypothesis will be that 55% of boys in my town are taller than girls.

Following is the difference between the null hypothesis and alternate hypothesis:

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9.1: Null and Alternative Hypotheses

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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 of no difference between the variables—they are not related. This can often be considered the status quo and as a result if you cannot accept the null it requires some action.

\(H_a\): The alternative hypothesis: It is a claim about the population that is contradictory to \(H_0\) and what we conclude when we reject \(H_0\). This is usually what the researcher is trying to prove.

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.

\(H_{0}\) 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.

Example \(\PageIndex{1}\)

• \(H_{0}\): No more than 30% of the registered voters in Santa Clara County voted in the primary election. \(p \leq 30\)
• \(H_{a}\): More than 30% of the registered voters in Santa Clara County voted in the primary election. \(p > 30\)

Exercise \(\PageIndex{1}\)

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%. \(p \neq 0.25\)

Example \(\PageIndex{2}\)

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:

• \(H_{0}: \mu = 2.0\)
• \(H_{a}: \mu \neq 2.0\)

Exercise \(\PageIndex{2}\)

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 \((=, \neq, \geq, <, \leq, >)\) for the null and alternative hypotheses.

• \(H_{0}: \mu \_ 66\)
• \(H_{a}: \mu \_ 66\)
• \(H_{0}: \mu = 66\)
• \(H_{a}: \mu \neq 66\)

Example \(\PageIndex{3}\)

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:

• \(H_{0}: \mu \geq 5\)
• \(H_{a}: \mu < 5\)

Exercise \(\PageIndex{3}\)

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.

• \(H_{0}: \mu \_ 45\)
• \(H_{a}: \mu \_ 45\)
• \(H_{0}: \mu \geq 45\)
• \(H_{a}: \mu < 45\)

Example \(\PageIndex{4}\)

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.

• \(H_{0}: p \leq 0.066\)
• \(H_{a}: p > 0.066\)

Exercise \(\PageIndex{4}\)

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 (\(=, \neq, \geq, <, \leq, >\)) for the null and alternative hypotheses.

• \(H_{0}: p \_ 0.40\)
• \(H_{a}: p \_ 0.40\)
• \(H_{0}: p = 0.40\)
• \(H_{a}: p > 0.40\)

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.

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 \((=, \leq \text{or} \geq)\)
• Always write the alternative hypothesis , typically denoted with \(H_{a}\) or \(H_{1}\), using less than, greater than, or not equals symbols, i.e., \((\neq, >, \text{or} <)\).
• 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.

• If \(\alpha \leq p\)-value, then do not reject \(H_{0}\).
• If\(\alpha > p\)-value, then reject \(H_{0}\).

\(\alpha\) is preconceived. Its value is set before the hypothesis test starts. The \(p\)-value is calculated from the data.References

Data from the National Institute of Mental Health. Available online at http://www.nimh.nih.gov/publicat/depression.cfm .

What is the difference between P-value and alpha?

Table of Contents

P-value and alpha are two statistical terms that are commonly used in hypothesis testing. P-value, also known as the probability value, is a measure of the strength of evidence against the null hypothesis. It represents the probability of obtaining a result at least as extreme as the observed result, assuming the null hypothesis is true. A lower P-value indicates stronger evidence against the null hypothesis.

On the other hand, alpha, also known as the level of significance, is the predetermined threshold or significance level used to determine the statistical significance of a test. It is typically set at 0.05 or 0.01, indicating the acceptable margin of error in rejecting the null hypothesis. If the P-value is less than or equal to alpha, the null hypothesis is rejected, and the results are considered statistically significant.

In summary, the main difference between P-value and alpha is that P-value is a calculated statistic, while alpha is a predetermined value used to make a decision in hypothesis testing. P-value provides a more precise measurement of the strength of evidence, while alpha determines the level at which the evidence is considered significant.

P-Value vs. Alpha: What’s the Difference?

Two terms that students often get confused in statistics are  p-value and  alpha . 

Both terms are used in , which are formal statistical tests we use to reject or fail to reject some hypothesis.

For example, suppose we hypothesize that a new pill reduces blood pressure in patients more than the current standard pill.

To test this, we can conduct a hypothesis test where we define the following null and alternative hypotheses:

Null hypothesis:  There is no difference between the new pill and the standard pill.

Alternative hypothesis:  There is a difference between the new pill and the standard pill.

If we assume the null hypothesis is true, the p-value of the test tells us the probability of obtaining an effect at least as large as the one we actually observed in the sample data. 

For example, suppose we find that the p-value of the hypothesis test is 0.02.

Here’s how to interpret this p-value: If there truly was no difference between the new pill and the standard pill, then 2% of the times that we perform this hypothesis test we would obtain the effect observed in the sample data, or larger, simply due to random sample error.

This tells us that obtaining the sample data that we actually did would be pretty rare if indeed there was no difference between the new pill and the standard pill.

Thus, we would be inclined to reject the statement in the null hypothesis and conclude that there  is a difference between the new pill and the standard pill.

But what threshold should we use to determine if our p-value is low enough to reject the null hypothesis?

This is where alpha comes in!

The Alpha Level

The  alpha level of a hypothesis test is the threshold we use to determine whether or not our p-value is low enough to reject the null hypothesis. It is often set at 0.05 but it is sometimes set as low as 0.01 or as high as 0.10.

For example, if we set the alpha level of a hypothesis test at 0.05 and we get a p-value of 0.02, then we would reject the null hypothesis since the p-value is less than the alpha level. Thus, we would conclude that we have sufficient evidence to say the alternative hypothesis is true.

It’s important to note that the alpha level also defines the probability of incorrectly rejecting a true null hypothesis.

If we set the alpha level of a hypothesis test at 0.05 then this means that if we repeated the process of performing the hypothesis test many times, we would expect to incorrectly reject the null hypothesis in about 5% of the tests.

How to Choose the Alpha Level

As mentioned earlier, the most common choice for the alpha level of a hypothesis test is 0.05. However, in some situations where there are serious consequences for making incorrect conclusions, we may set the alpha level to be even lower, perhaps at 0.01.

For example, in the medical field it’s common for researchers to set the alpha level at 0.01 because they want to be highly confident that the results of a hypothesis test are reliable.

Conversely, in fields like marketing it may be more common to set the alpha level at a higher level like 0.10 because the consequences for being wrong aren’t life or death.

It’s worth noting that increasing the alpha level of a test will increase the chances of finding a significance test result, but it also increases the chances that we incorrectly reject a true null hypothesis.

Here’s what we learned in this article:

1.  A  p-value tells us the probability of obtaining an effect at least as large as the one we actually observed in the sample data. 

2. An alpha level is the probability of incorrectly rejecting a true null hypothesis.

3. If the p-value of a hypothesis test is less than the alpha level, then we can reject the null hypothesis.

4. Increasing the alpha level of a test increases the chances that we can find a significant test result, but it also increases the chances that we incorrectly reject a true null hypothesis.

Additional Resources

Related terms:.

• What is difference between P-Value and Alpha?
• What is the difference between Within-Group and Between Group Variation in ANOVA?
• What is the value of Z Alpha/2, also known as za/2?
• What is the difference between CDF and PDF?
• What is the difference between confidentiality and anonymity?
• What is the difference between a statistic and a parameter?
• What is the difference between Qualitative and Quantitative Variables?
• What is the difference between Normal Distribution and t-Distribution?
• What is the difference between correlation and regression?
• How do I calculate the difference between two dates in Excel?

Z-test vs T-test: the differences and when to use each

What is hypothesis testing, what is a z-test, examples of a z-test, what is a t-test, examples of a t-test, how to know when to use z-test vs t-test, difference between z-test and t-test: a comparative table.

The EPAM Anywhere Editorial Team is an international collective of senior software engineers, managers and communications professionals who create, review and share their insights on technology, career, remote work, and the daily life here at Anywhere.

Testing is how you determine effectiveness. Whether you work as a data scientist , statistician, or software developer, to ensure quality, you must measure performance. Without tests, you could deploy flawed code, features, or data points.

With that in mind, the use cases of testing are endless. Machine learning models need statistical tests. Data analysis involves statistical tests to validate assumptions. Optimization of any kind requires evaluation. You even need to test the strength of your hypothesis before you begin an inquiry.

Let's explore two inferential statistics: the Z-test vs the T-test. That way you can understand their differences, their unique purposes, and when to use a Z-test vs T-test.

To start, imagine you have a good idea. At the moment of inception, you have no data to back up your idea. It is an unformed thought. But the idea is an excellent starting point that can launch a full investigation. We consider this starting point a hypothesis.

But what if your hypothesis is off-base? You don’t want to dive into a full-scale search if it is a pointless chase with no reward. That is a waste of resources. You need to determine if you have a workable hypothesis.

Enter hypothesis testing. It is a statistical act used to assess the viability of a hypothesis. The method discovers whether there is sufficient data to support your idea. If there is next to no significance, you do not have a very plausible hypothesis.

To confirm the validity of a hypothesis, you compare it against the status quo (also known as the null hypothesis). Your idea is something new, opposite from normal conditions (also known as the alternative hypothesis). It is zero sum: only one hypothesis between the null and alternate hypothesis can be true in a given set of parameters.

In such a comparison test, you can now determine validity. You can compare and contrast conditions to find meaningful conclusions. Whichever conditions become statistically apparent determines which hypothesis is plausible.

A Z-test is a test statistic. It works with two groups: the sample mean and the population mean. It will test whether these two groups are different.

With a Z-test, you know the population standard deviation. That is to ensure statistical accuracy as you compare one group (the sample mean) vs the second group (the population mean). In other words, you can minimize external confounding factors with a normal distribution. In addition, a defining characteristic of a Z-test is that it works with large sample sizes (typically more than 30, so we achieve normal distribution as defined by the central limit theorem). These are two crucial criteria for using a Z-test.

Within hypothesis testing, your null hypothesis states there is no difference between the two groups your Z-test will compare. Your alternative hypothesis will state there is a difference that your Z-test will expose.

How to perform a Z-test

A Z-test occurs in the following standard format:

• Formulate your hypothesis: First, define the parameters of your alternative and null hypothesis.
• Choose a critical value: Second, determine what you consider a viable difference between your two groups. This threshold determines when you can say the null hypothesis should be rejected. Common levels are 0.01 (1%) or 0.05 (5%) , values found to best balance Type I and Type II errors .
• Collect samples: Obtain the needed data. The data must be large enough and random.
• Calculate for a Z-score: Input your data into the standard Z-test statistics formula, shown below, where Z = standard score, x = observed value, mu = mean of the sample, sigma = standard deviation of the sample .
• Compare: If the statistical test is greater than the critical value, you have achieved statistical significance. The sample mean is so different so you can reject the null hypothesis. Your alternative hypothesis (something other than the status quo) is at work, and that's worth investigating.

There are different variations of a Z-test. Let's explore examples of one-sample and two-sample Z-tests.

One-sample Z-test

A one-sample Z-test looks for either an increase or a decrease. There is one sample group involved, taken from a population. We want to see if there is a difference between those two means.

For example, consider a school principal who believes their students' IQ is higher than the state average. The state average is 100 points (population mean), give or take 20 (the population standard deviation). To prove this hypothesis, the principal takes 50 students (the sample size) and finds their IQ scores. To their delight, they earn an average of 110.

But does the difference offer any statistical value? The principal then plugs the numbers into a Z-test. Any Z-score greater than the critical value would state there is sufficient significance. The claim that the students have an above-average IQ is valid.

Two-sample Z-test

A two-sample test involves comparing the average of two sample groups against the population means. It is to determine a difference between two independent samples.

For example, our principal wants to compare their students' IQ scores to the school across the street. They believe their students' average IQ is higher. They don’t need to know the exact numerical increase or decrease. All they want is proof that their student's average scores are higher than the other group.

To confirm the validity of this hypothesis, the principal will search for statistical significance. They can take a 50-student sample size from their school and a 50-student sample size from the rival school. Now in possession of both sample group's average IQ (and the sample standard deviation), they hope to find a number value that is not equal. And they need them to be unequal by a significant amount.

If the test statistic comes in less than the critical value, the differences are negligible. There is not enough evidence to say the hypothesis is worth exploring, the null hypothesis is maintained. He would not have enough proof that the IQ levels between the two schools are different.

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A T-test performs the same crucial function as a Z-test: determine if there is a difference between the means of two groups. If there is a significant difference, you have achieved statistical validity for your hypothesis.

However, a T-test involves a different set of factors. Most importantly, a T-test applies when you do not know the sample variance of your values. You must generalize the normal distribution (or T-distribution). Plus, there is an expectation that you do not possess all the data in a given scenario.

These conditions better match reality, as it is often hard to collect data from entire populations or always obtain a standard normal distribution. That is why T-tests are more widely applicable than Z-tests, though they operate with less precision.

How to perform a T-test

A T-test occurs in the following standard format:

• Formulate your hypothesis: First, define the parameters of your null and alternative hypothesis.
• Choose a critical value: Like a Z-test, determine what you consider a viable difference between your two groups.
• Collect data: Obtain the needed data. One of the key differences is degrees of freedom in the samples of a T-test, so try to define the typical values and range of values in each group.
• Calculate your T-score: Input your data into the T-test formula you chose. Here is a one-sample formula:
• Compare: If the statistical test is greater than the critical value, you have achieved statistical significance. The sample mean is so far from the population mean that you likely have a useful hypothesis.

There are several different kinds of T-tests as well. Let's go through the standard one-sample and two-sample T-tests.

One-sample T-test

A one-sample T-test looks for an increase or decrease compared to a population mean.

For example, your company just went through sales training. Now, the manager wants to know if the training helped improve sales.

Previous company sales data shows an average of $100 on each transaction from all workers. The null hypothesis would be no change. The alternative hypothesis (which you hope is significant), is that there is an improvement.

To test if there is significance, you take the sales average of 20 salesmen. That is the only available data, and you have no other data from nationwide stores. The average of that sample of salesmen in the past month is $130. We will also assume that the standard deviation is approximately equal .

With this set of factors, you can calculate your T-score with a T-test. You compare the sample result to the critical value. In addition, you assess it against the number of degrees of freedom. Since we know with smaller sample data sizes there is greater uncertainty, we allow more room for our data to vary.

After comparing, we may find a lot of significance. That means the data possesses enough strength to support our hypothesis that sales training likely impacted sales. Of course, this is an estimate, as we only assessed one factor with a small group. Sales could have risen for numerous other reasons. But with our set of assumptions, our hypothesis is valid.

Two-sample T-test

A two-sample T-test occurs the same as a two-sample Z-test and compares if two groups are equal when compared to a defined population parameter.

For example, consider English and non-native speakers. We want to see the effect of maternal language on test scores inside a country. To do that, we will offer both groups a reading test and compare those scores to the average.

Of course, finding the mean of an entire population of language speakers is impossible to procure. Still, we can make some assumptions and compare them with a smaller size. We take 15 English speakers and 15 non-native speakers and collect their results. We can decide on a critical score value on the reading test as well. If the average score on the test is not crucially different or outside the population standard deviation, our assumption failed. There is no significant difference between the groups, so the impact of maternal language is not worth investigating.

Both a Z-test and a T-test validate a hypothesis. Both are parametric tests that rely on assumptions. The key difference between Z-test and T-test is in their assumptions (e.g. population variance).

Key differences about the data used result in different applications. You want to use the appropriate tool, otherwise you won’t draw valid conclusions from your data.

So when should you use a Z-test vs a T-test? Here are some factors to consider:

• Sample size: If the available sample size is small, opt for a T-test. Small sample sizes are more variable, so the greater spread of distribution and estimation of error involved with T-tests is ideal.
• Knowledge of the population standard deviation: Z-tests are more precise and often simpler to execute. So if you know the standard deviation, use a Z-test.
• Test purpose: If you are assessing the validity of a mean, a T-test is the best choice. If you are working with a hypothesized population proportion, go for a Z-test.
• Assumption of normality: A Z-test assumes a normal distribution. This does not apply to all real-world scenarios. If you hope to validate a hypothesis that is not well-defined, opt for a T-test instead.
• Type of data: You can only work within the constraints of the available data. The more information the better, but that is often not possible given testing and collecting conditions. If you have limited data describing means between groups, opt for a T-test. If you have large data sets comparing means between populations, you can use a Z-test.

Knowing the key differences with each statistical test makes selecting the right tool far easier. Here is a table that can help you compare:

Statistical testing lets you determine the validity of a hypothesis. You discover validity by determining if there is a significant difference between your hypothesis and the status quo. If there is, you have a possible idea worth exploring.

That process has numerous applications in the field of computer science and data analysis . You might want to determine the performance of an app with an A/B test. Or you might need to test if an application fits within the defined limits and compare performance metrics. Z-tests and T-tests can depict whether there is significant evidence in each of these scenarios. With that information, you can take the appropriate measures to fix bugs or optimize processes.

Z-test and T-test are helpful tools, especially for hypothesis testing. For data engineers of the future, knowledge of statistical testing will only help your work and overall career trajectory.

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