## Hypothesis Testing Calculator

Navigating hypothesis testing: unveiling the potential of the hypothesis testing calculator.

Embarking on the journey of statistical exploration, hypothesis testing stands out as an indispensable method for informed decision-making and drawing meaningful conclusions from data. Whether you find yourself in the academic realm, engaged in research endeavors, or navigating the professional landscape, having a trustworthy Hypothesis Testing Calculator in your statistical toolkit can prove to be a game-changer. Let’s delve into the intricacies of hypothesis testing and uncover how this calculator can be your ally in statistical analyses.

## Demystifying Hypothesis Testing:

Null Hypothesis (H0): Positioned as the default assumption, the null hypothesis asserts the absence of any significant difference or effect and is commonly represented as H0.

Alternative Hypothesis (Ha): In direct contradiction to the null hypothesis, the alternative hypothesis posits the existence of a noteworthy difference or effect, denoted as Ha.

Significance Level (α): Acting as the predetermined threshold, typically set at 0.05 or 5%, the significance level plays a pivotal role in determining statistical significance. Should the calculated p-value fall below α, the null hypothesis is rejected.

p-value: Representing the likelihood of observing the results, or more extreme outcomes, under the assumption of the null hypothesis being true, a smaller p-value suggests the unlikelihood of the results occurring by chance.

## Features that Define the Hypothesis Testing Calculator:

Input Parameters: The calculator demands input of sample data, selection of the test type (e.g., t-test, chi-square test), specification of null and alternative hypotheses, and determination of the significance level.

Calculations: Once armed with the requisite data and parameters, the calculator diligently executes statistical tests and computations. The output encompasses crucial details like the test statistic, degrees of freedom, and the all-important p-value.

Interpretation: Armed with the results, the calculator aids in the decision-making process, guiding whether to reject or accept the null hypothesis. An interpretation of the findings is provided, playing a pivotal role in drawing insightful conclusions.

Visual Representation: Some calculators go the extra mile by offering visual aids such as graphs or charts, facilitating a deeper understanding of data distribution and test outcomes.

## Unveiling the Significance of the Hypothesis Testing Calculator:

In Scientific Research: Researchers spanning diverse fields leverage hypothesis testing to validate their hypotheses, thereby extracting meaningful insights from data.

In Quality Control: Industries rely on hypothesis testing as a quality assurance mechanism, ensuring the consistency and excellence of products and processes.

In Medical Studies: Within the realm of medical research, hypothesis testing serves as a critical tool for evaluating the effectiveness of treatments or interventions.

In Academics: Both students and educators find value in hypothesis testing as an educational tool, enabling the comprehension of statistical concepts and the conduct of experiments.

In Data-Driven Decision-Making: Businesses, keen on making decisions grounded in data, turn to hypothesis testing to navigate choices such as launching a new product based on comprehensive market research.

## Concluding Insights:

The Hypothesis Testing Calculator emerges as a formidable ally, simplifying intricate statistical analyses and fostering data-driven decision-making. Whether you are in the midst of experimental undertakings, scrutinizing survey data, or overseeing quality control protocols, a solid understanding of hypothesis testing coupled with the use of this calculator empowers you to make well-informed choices. In doing so, you not only contribute to evidence-based research but also play a pivotal role in shaping decision-making processes across various domains.

## Hypothesis Testing Calculator

Understanding Hypothesis Testing: A Guide to the Hypothesis Testing Calculator

Hypothesis testing is a crucial statistical method used to make informed decisions about data and draw conclusions. Whether you’re a student, researcher, or professional, a Hypothesis Testing Calculator can be an invaluable tool in your statistical toolkit. Let’s explore what hypothesis testing is and how this calculator can assist you:

Hypothesis Testing Basics:

- Null Hypothesis (H0): This is the default assumption or claim that there is no significant difference or effect. It’s often denoted as H0.
- Alternative Hypothesis (Ha): This is the statement that contradicts the null hypothesis. It suggests that there is a significant difference or effect. It’s denoted as Ha.
- Significance Level (α): This is the predetermined threshold (e.g., 0.05 or 5%) used to determine statistical significance. If the calculated p-value is less than α, you reject the null hypothesis.
- p-value: This is the probability of observing the results (or more extreme results) if the null hypothesis is true. A small p-value suggests that the results are unlikely under the null hypothesis.

Key Features of the Hypothesis Testing Calculator:

- Input Parameters: The calculator typically requires you to input sample data, choose the type of test (e.g., t-test, chi-square test), specify the null and alternative hypotheses, and set the significance level.
- Calculations: Once you input the data and parameters, the calculator performs the necessary statistical tests and calculations. It generates results such as the test statistic, degrees of freedom, and the p-value.
- Interpretation: Based on the results, the calculator helps you determine whether to reject or fail to reject the null hypothesis. It provides an interpretation of the findings, which is crucial for drawing conclusions.
- Visual Representation: Some calculators may offer visual aids like graphs or charts to help you better understand the data distribution and test results.

Significance of the Hypothesis Testing Calculator:

- Scientific Research: Researchers across various fields use hypothesis testing to validate their hypotheses and draw meaningful conclusions from data.
- Quality Control: Industries use hypothesis testing to ensure the quality and consistency of products and processes.
- Medical Studies: In medical research, hypothesis testing helps assess the effectiveness of treatments or interventions.
- Academics: Students and educators use hypothesis testing to teach and learn statistical concepts and conduct experiments.
- Data-Driven Decisions: Businesses use hypothesis testing to make data-driven decisions, such as whether to launch a new product based on market research.

Conclusion:

The Hypothesis Testing Calculator is a powerful tool that simplifies complex statistical analysis and enables data-driven decision-making. Whether you’re conducting experiments, analyzing survey data, or performing quality control, understanding hypothesis testing and using this calculator can help you make informed choices and contribute to evidence-based research and decision-making.

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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.

## Cite this Scribbr article

If you want to cite this source, you can copy and paste the citation or click the ‘Cite this Scribbr article’ button to automatically add the citation to our free Reference Generator.

Turney, S. (2022, December 06). Null and Alternative Hypotheses | Definitions & Examples. Scribbr. Retrieved 6 May 2024, from https://www.scribbr.co.uk/stats/null-and-alternative-hypothesis/

## Is this article helpful?

## Shaun Turney

Other students also liked, levels of measurement: nominal, ordinal, interval, ratio, the standard normal distribution | calculator, examples & uses, types of variables in research | definitions & examples.

## p-value Calculator

What is p-value, how do i calculate p-value from test statistic, how to interpret p-value, how to use the p-value calculator to find p-value from test statistic, how do i find p-value from z-score, how do i find p-value from t, p-value from chi-square score (χ² score), p-value from f-score.

Welcome to our p-value calculator! You will never again have to wonder how to find the p-value, as here you can determine the one-sided and two-sided p-values from test statistics, following all the most popular distributions: normal, t-Student, chi-squared, and Snedecor's F.

P-values appear all over science, yet many people find the concept a bit intimidating. Don't worry – in this article, we will explain not only what the p-value is but also how to interpret p-values correctly . Have you ever been curious about how to calculate the p-value by hand? We provide you with all the necessary formulae as well!

🙋 If you want to revise some basics from statistics, our normal distribution calculator is an excellent place to start.

Formally, the p-value is the probability that the test statistic will produce values at least as extreme as the value it produced for your sample . It is crucial to remember that this probability is calculated under the assumption that the null hypothesis H 0 is true !

More intuitively, p-value answers the question:

Assuming that I live in a world where the null hypothesis holds, how probable is it that, for another sample, the test I'm performing will generate a value at least as extreme as the one I observed for the sample I already have?

It is the alternative hypothesis that determines what "extreme" actually means , so the p-value depends on the alternative hypothesis that you state: left-tailed, right-tailed, or two-tailed. In the formulas below, S stands for a test statistic, x for the value it produced for a given sample, and Pr(event | H 0 ) is the probability of an event, calculated under the assumption that H 0 is true:

Left-tailed test: p-value = Pr(S ≤ x | H 0 )

Right-tailed test: p-value = Pr(S ≥ x | H 0 )

Two-tailed test:

p-value = 2 × min{Pr(S ≤ x | H 0 ), Pr(S ≥ x | H 0 )}

(By min{a,b} , we denote the smaller number out of a and b .)

If the distribution of the test statistic under H 0 is symmetric about 0 , then: p-value = 2 × Pr(S ≥ |x| | H 0 )

or, equivalently: p-value = 2 × Pr(S ≤ -|x| | H 0 )

As a picture is worth a thousand words, let us illustrate these definitions. Here, we use the fact that the probability can be neatly depicted as the area under the density curve for a given distribution. We give two sets of pictures: one for a symmetric distribution and the other for a skewed (non-symmetric) distribution.

- Symmetric case: normal distribution:

- Non-symmetric case: chi-squared distribution:

In the last picture (two-tailed p-value for skewed distribution), the area of the left-hand side is equal to the area of the right-hand side.

To determine the p-value, you need to know the distribution of your test statistic under the assumption that the null hypothesis is true . Then, with the help of the cumulative distribution function ( cdf ) of this distribution, we can express the probability of the test statistics being at least as extreme as its value x for the sample:

Left-tailed test:

p-value = cdf(x) .

Right-tailed test:

p-value = 1 - cdf(x) .

p-value = 2 × min{cdf(x) , 1 - cdf(x)} .

If the distribution of the test statistic under H 0 is symmetric about 0 , then a two-sided p-value can be simplified to p-value = 2 × cdf(-|x|) , or, equivalently, as p-value = 2 - 2 × cdf(|x|) .

The probability distributions that are most widespread in hypothesis testing tend to have complicated cdf formulae, and finding the p-value by hand may not be possible. You'll likely need to resort to a computer or to a statistical table, where people have gathered approximate cdf values.

Well, you now know how to calculate the p-value, but… why do you need to calculate this number in the first place? In hypothesis testing, the p-value approach is an alternative to the critical value approach . Recall that the latter requires researchers to pre-set the significance level, α, which is the probability of rejecting the null hypothesis when it is true (so of type I error ). Once you have your p-value, you just need to compare it with any given α to quickly decide whether or not to reject the null hypothesis at that significance level, α. For details, check the next section, where we explain how to interpret p-values.

As we have mentioned above, the p-value is the answer to the following question:

What does that mean for you? Well, you've got two options:

- A high p-value means that your data is highly compatible with the null hypothesis; and
- A small p-value provides evidence against the null hypothesis , as it means that your result would be very improbable if the null hypothesis were true.

However, it may happen that the null hypothesis is true, but your sample is highly unusual! For example, imagine we studied the effect of a new drug and got a p-value of 0.03 . This means that in 3% of similar studies, random chance alone would still be able to produce the value of the test statistic that we obtained, or a value even more extreme, even if the drug had no effect at all!

The question "what is p-value" can also be answered as follows: p-value is the smallest level of significance at which the null hypothesis would be rejected. So, if you now want to make a decision on the null hypothesis at some significance level α , just compare your p-value with α :

- If p-value ≤ α , then you reject the null hypothesis and accept the alternative hypothesis; and
- If p-value ≥ α , then you don't have enough evidence to reject the null hypothesis.

Obviously, the fate of the null hypothesis depends on α . For instance, if the p-value was 0.03 , we would reject the null hypothesis at a significance level of 0.05 , but not at a level of 0.01 . That's why the significance level should be stated in advance and not adapted conveniently after the p-value has been established! A significance level of 0.05 is the most common value, but there's nothing magical about it. Here, you can see what too strong a faith in the 0.05 threshold can lead to. It's always best to report the p-value, and allow the reader to make their own conclusions.

Also, bear in mind that subject area expertise (and common reason) is crucial. Otherwise, mindlessly applying statistical principles, you can easily arrive at statistically significant, despite the conclusion being 100% untrue.

As our p-value calculator is here at your service, you no longer need to wonder how to find p-value from all those complicated test statistics! Here are the steps you need to follow:

Pick the alternative hypothesis : two-tailed, right-tailed, or left-tailed.

Tell us the distribution of your test statistic under the null hypothesis: is it N(0,1), t-Student, chi-squared, or Snedecor's F? If you are unsure, check the sections below, as they are devoted to these distributions.

If needed, specify the degrees of freedom of the test statistic's distribution.

Enter the value of test statistic computed for your data sample.

Our calculator determines the p-value from the test statistic and provides the decision to be made about the null hypothesis. The standard significance level is 0.05 by default.

Go to the advanced mode if you need to increase the precision with which the calculations are performed or change the significance level .

In terms of the cumulative distribution function (cdf) of the standard normal distribution, which is traditionally denoted by Φ , the p-value is given by the following formulae:

Left-tailed z-test:

p-value = Φ(Z score )

Right-tailed z-test:

p-value = 1 - Φ(Z score )

Two-tailed z-test:

p-value = 2 × Φ(−|Z score |)

p-value = 2 - 2 × Φ(|Z score |)

🙋 To learn more about Z-tests, head to Omni's Z-test calculator .

We use the Z-score if the test statistic approximately follows the standard normal distribution N(0,1) . Thanks to the central limit theorem, you can count on the approximation if you have a large sample (say at least 50 data points) and treat your distribution as normal.

A Z-test most often refers to testing the population mean , or the difference between two population means, in particular between two proportions. You can also find Z-tests in maximum likelihood estimations.

The p-value from the t-score is given by the following formulae, in which cdf t,d stands for the cumulative distribution function of the t-Student distribution with d degrees of freedom:

Left-tailed t-test:

p-value = cdf t,d (t score )

Right-tailed t-test:

p-value = 1 - cdf t,d (t score )

Two-tailed t-test:

p-value = 2 × cdf t,d (−|t score |)

p-value = 2 - 2 × cdf t,d (|t score |)

Use the t-score option if your test statistic follows the t-Student distribution . This distribution has a shape similar to N(0,1) (bell-shaped and symmetric) but has heavier tails – the exact shape depends on the parameter called the degrees of freedom . If the number of degrees of freedom is large (>30), which generically happens for large samples, the t-Student distribution is practically indistinguishable from the normal distribution N(0,1).

The most common t-tests are those for population means with an unknown population standard deviation, or for the difference between means of two populations , with either equal or unequal yet unknown population standard deviations. There's also a t-test for paired (dependent) samples .

🙋 To get more insights into t-statistics, we recommend using our t-test calculator .

Use the χ²-score option when performing a test in which the test statistic follows the χ²-distribution .

This distribution arises if, for example, you take the sum of squared variables, each following the normal distribution N(0,1). Remember to check the number of degrees of freedom of the χ²-distribution of your test statistic!

How to find the p-value from chi-square-score ? You can do it with the help of the following formulae, in which cdf χ²,d denotes the cumulative distribution function of the χ²-distribution with d degrees of freedom:

Left-tailed χ²-test:

p-value = cdf χ²,d (χ² score )

Right-tailed χ²-test:

p-value = 1 - cdf χ²,d (χ² score )

Remember that χ²-tests for goodness-of-fit and independence are right-tailed tests! (see below)

Two-tailed χ²-test:

p-value = 2 × min{cdf χ²,d (χ² score ), 1 - cdf χ²,d (χ² score )}

(By min{a,b} , we denote the smaller of the numbers a and b .)

The most popular tests which lead to a χ²-score are the following:

Testing whether the variance of normally distributed data has some pre-determined value. In this case, the test statistic has the χ²-distribution with n - 1 degrees of freedom, where n is the sample size. This can be a one-tailed or two-tailed test .

Goodness-of-fit test checks whether the empirical (sample) distribution agrees with some expected probability distribution. In this case, the test statistic follows the χ²-distribution with k - 1 degrees of freedom, where k is the number of classes into which the sample is divided. This is a right-tailed test .

Independence test is used to determine if there is a statistically significant relationship between two variables. In this case, its test statistic is based on the contingency table and follows the χ²-distribution with (r - 1)(c - 1) degrees of freedom, where r is the number of rows, and c is the number of columns in this contingency table. This also is a right-tailed test .

Finally, the F-score option should be used when you perform a test in which the test statistic follows the F-distribution , also known as the Fisher–Snedecor distribution. The exact shape of an F-distribution depends on two degrees of freedom .

To see where those degrees of freedom come from, consider the independent random variables X and Y , which both follow the χ²-distributions with d 1 and d 2 degrees of freedom, respectively. In that case, the ratio (X/d 1 )/(Y/d 2 ) follows the F-distribution, with (d 1 , d 2 ) -degrees of freedom. For this reason, the two parameters d 1 and d 2 are also called the numerator and denominator degrees of freedom .

The p-value from F-score is given by the following formulae, where we let cdf F,d1,d2 denote the cumulative distribution function of the F-distribution, with (d 1 , d 2 ) -degrees of freedom:

Left-tailed F-test:

p-value = cdf F,d1,d2 (F score )

Right-tailed F-test:

p-value = 1 - cdf F,d1,d2 (F score )

Two-tailed F-test:

p-value = 2 × min{cdf F,d1,d2 (F score ), 1 - cdf F,d1,d2 (F score )}

Below we list the most important tests that produce F-scores. All of them are right-tailed tests .

A test for the equality of variances in two normally distributed populations . Its test statistic follows the F-distribution with (n - 1, m - 1) -degrees of freedom, where n and m are the respective sample sizes.

ANOVA is used to test the equality of means in three or more groups that come from normally distributed populations with equal variances. We arrive at the F-distribution with (k - 1, n - k) -degrees of freedom, where k is the number of groups, and n is the total sample size (in all groups together).

A test for overall significance of regression analysis . The test statistic has an F-distribution with (k - 1, n - k) -degrees of freedom, where n is the sample size, and k is the number of variables (including the intercept).

With the presence of the linear relationship having been established in your data sample with the above test, you can calculate the coefficient of determination, R 2 , which indicates the strength of this relationship . You can do it by hand or use our coefficient of determination calculator .

A test to compare two nested regression models . The test statistic follows the F-distribution with (k 2 - k 1 , n - k 2 ) -degrees of freedom, where k 1 and k 2 are the numbers of variables in the smaller and bigger models, respectively, and n is the sample size.

You may notice that the F-test of an overall significance is a particular form of the F-test for comparing two nested models: it tests whether our model does significantly better than the model with no predictors (i.e., the intercept-only model).

## Can p-value be negative?

No, the p-value cannot be negative. This is because probabilities cannot be negative, and the p-value is the probability of the test statistic satisfying certain conditions.

## What does a high p-value mean?

A high p-value means that under the null hypothesis, there's a high probability that for another sample, the test statistic will generate a value at least as extreme as the one observed in the sample you already have. A high p-value doesn't allow you to reject the null hypothesis.

## What does a low p-value mean?

A low p-value means that under the null hypothesis, there's little probability that for another sample, the test statistic will generate a value at least as extreme as the one observed for the sample you already have. A low p-value is evidence in favor of the alternative hypothesis – it allows you to reject the null 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 .

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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.

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

Statistics Made Easy

## How to Write a Null Hypothesis (5 Examples)

A hypothesis test uses sample data to determine whether or not some claim about a population parameter is true.

Whenever we perform a hypothesis test, we always write a null hypothesis and an alternative hypothesis, which take the following forms:

H 0 (Null Hypothesis): Population parameter =, ≤, ≥ some value

H A (Alternative Hypothesis): Population parameter <, >, ≠ some value

Note that the null hypothesis always contains the equal sign .

We interpret the hypotheses as follows:

Null hypothesis: The sample data provides no evidence to support some claim being made by an individual.

Alternative hypothesis: The sample data does provide sufficient evidence to support the claim being made by an individual.

For example, suppose it’s assumed that the average height of a certain species of plant is 20 inches tall. However, one botanist claims the true average height is greater than 20 inches.

To test this claim, she may go out and collect a random sample of plants. She can then use this sample data to perform a hypothesis test using the following two hypotheses:

H 0 : μ ≤ 20 (the true mean height of plants is equal to or even less than 20 inches)

H A : μ > 20 (the true mean height of plants is greater than 20 inches)

If the sample data gathered by the botanist shows that the mean height of this species of plants is significantly greater than 20 inches, she can reject the null hypothesis and conclude that the mean height is greater than 20 inches.

Read through the following examples to gain a better understanding of how to write a null hypothesis in different situations.

## Example 1: Weight of Turtles

A biologist wants to test whether or not the true mean weight of a certain species of turtles is 300 pounds. To test this, he goes out and measures the weight of a random sample of 40 turtles.

Here is how to write the null and alternative hypotheses for this scenario:

H 0 : μ = 300 (the true mean weight is equal to 300 pounds)

H A : μ ≠ 300 (the true mean weight is not equal to 300 pounds)

## Example 2: Height of Males

It’s assumed that the mean height of males in a certain city is 68 inches. However, an independent researcher believes the true mean height is greater than 68 inches. To test this, he goes out and collects the height of 50 males in the city.

H 0 : μ ≤ 68 (the true mean height is equal to or even less than 68 inches)

H A : μ > 68 (the true mean height is greater than 68 inches)

## Example 3: Graduation Rates

A university states that 80% of all students graduate on time. However, an independent researcher believes that less than 80% of all students graduate on time. To test this, she collects data on the proportion of students who graduated on time last year at the university.

H 0 : p ≥ 0.80 (the true proportion of students who graduate on time is 80% or higher)

H A : μ < 0.80 (the true proportion of students who graduate on time is less than 80%)

## Example 4: Burger Weights

A food researcher wants to test whether or not the true mean weight of a burger at a certain restaurant is 7 ounces. To test this, he goes out and measures the weight of a random sample of 20 burgers from this restaurant.

H 0 : μ = 7 (the true mean weight is equal to 7 ounces)

H A : μ ≠ 7 (the true mean weight is not equal to 7 ounces)

## Example 5: Citizen Support

A politician claims that less than 30% of citizens in a certain town support a certain law. To test this, he goes out and surveys 200 citizens on whether or not they support the law.

H 0 : p ≥ .30 (the true proportion of citizens who support the law is greater than or equal to 30%)

H A : μ < 0.30 (the true proportion of citizens who support the law is less than 30%)

## Additional Resources

Introduction to Hypothesis Testing Introduction to Confidence Intervals An Explanation of P-Values and Statistical Significance

## Featured Posts

Hey there. My name is Zach Bobbitt. I have a Masters of Science degree in Applied Statistics and I’ve worked on machine learning algorithms for professional businesses in both healthcare and retail. I’m passionate about statistics, machine learning, and data visualization and I created Statology to be a resource for both students and teachers alike. My goal with this site is to help you learn statistics through using simple terms, plenty of real-world examples, and helpful illustrations.

## 2 Replies to “How to Write a Null Hypothesis (5 Examples)”

you are amazing, thank you so much

Say I am a botanist hypothesizing the average height of daisies is 20 inches, or not? Does T = (ave – 20 inches) / √ variance / (80 / 4)? … This assumes 40 real measures + 40 fake = 80 n, but that seems questionable. Please advise.

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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 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.

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.

## 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

## Example 9.4

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 ≤ 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

## Collaborative Exercise

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

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Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.

Access for free at https://openstax.org/books/introductory-statistics-2e/pages/1-introduction

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## Single Proportion Hypothesis Test Calculator

Use the calculator below to analyze the results of a single proportion hypothesis test. Enter your null hypothesis's proportion, sample proportion, sample size, test type, and significance level.

You will find a description of how to conduct a hypothesis test of a proportion below the calculator.

## Define the z-test

Sample proportion under the null distribution, conducting single proportion hypothesis tests.

A hypothesis test of a sample proportion can help you make inferences about the population from which you drew it. It is a tool to determine what is probably true about an event or phenomena.

## Testing a Proportion

For the results of a hypothesis test to be valid, you should follow these steps:

## Check Your Conditions

State your hypothesis, determine your analysis plan, analyze your sample, interpret your results.

To use the testing procedure described below, you should check the following conditions:

- Binary Outcomes - When conducting a hypothesis test for a proportion, each sample point should consist of only one of two outcomes. We often label one outcome a “success” and one outcome a “failure,” but it does not matter which of the two outcomes gets which label.
- Success-Failure Rate - Your sample size should be large enough that under the null hypothesis proportion you are likely to see at least 10 “success” and 10 “failures.” For example, if you have null hypothesis proportion with a 10% or 0.1 “success” rate, then you would need a sample of 100 [10 = 100 * 10%] to have a large enough sample to meet this condition. This condition helps ensure that the sampling distribution from which you collect your sample reasonably follows the Normal Distribution.
- Simple Random Sampling - You should collect your sample with simple random sampling. This type of sampling requires that every occurrence of a category or event in a population has an equal chance of being selected when taking a sample.
- Sample-to-Population Ratio - The population should be much larger than the sample you collect. As a rule-of-thumb, the sample size should represent no more than 5% of the population.

You must state a null hypothesis and an alternative hypothesis to conduct a hypothesis test for a proportion.

The null hypothesis, is a skeptical claim that you would like to test. It is defined by a hypothesized proportion, which is often labeled P 0 .

The alternative hypothesis represents an alternative claim to the null hypothesis.

Your null hypothesis and alternative hypothesis should be stated in one of three mutually exclusive ways listed in the table below.

Before conducting a hypothesis test, you must determine a reasonable significance level , α, or the probability of rejecting the null hypothesis assuming it is true. The lower your significance level, the more confident you can be of the conclusion of your hypothesis test. Common significance levels are 10%, 5%, and 1%.

To evaluate your hypothesis test at the significance level that you set, consider if you are conducting a one or two tail test:

- Two-tail tests divide the rejection region, or critical region, evenly above and below the null distribution, i.e. to the tails of the null sampling distribution. For example, in a two-tail test with a 5% significance level, your rejection region would be the upper and lower 2.5% of the null distribution. An alternative hypothesis of P ≠ P 0 requires a two-tail test.
- One-tail tests place the rejection region entirely on one side of the null distribution i.e. to the right or left tail of the null sampling distribution. For example, in a one-tail test evaluating if the sampling distribution is above the null sampling distribution with a 5% significance level, your rejection region would be the upper 5% of the null distribution. P > P 0 and P < P 0 alternative hypotheses require one-tail tests.

The graphical results section of the calculator above shades rejection regions blue.

After checking your conditions, stating your hypothesis, determining your significance level, α, and collecting your sample, you are ready to analyze your hypothesis.

Sample proportions follow the Normal Distribution with the following parameters (i.e. numbers that define the distribution):

- The Population Proportion, P - The population proportion is assumed to be the proportion given by the null hypothesis in a single proportion hypothesis test.
- The Standard Error, SE - The standard error can be computed as follows: SE = sqrt((P x (1 - P))/ n), with n being the sample size. It defines how sample proportions are expected to vary around the null hypothesis's proportion given the sample size and under the assumption that the null hypothesis is true.

In a single proportion hypothesis test, we calculate the probability that we would observe the sample proportion, p, assuming the null hypothesis is true, also known as the p-value . If the p-value is less than the significance level, then we can reject the null hypothesis.

You can determine a precise p-value using the calculator above, but we can find an estimate of the p-value manually by calculating the z-score as follows: z = (p - P) / SE

The z-score is a test statistic that tells us how far our observation is from the null hypothesis's proportion under the null distribution. Using any z-score table, we can look up the probability of observing the results under the null distribution. You will need to look up the z-score for the type of test you are conducting, i.e. one or two tail. A hypothesis test for a proportion is sometimes known as a z-test because of the use of a z-score in analyzing results.

If we find the probability is below the significance level, we reject the null hypothesis.

The conclusion of a hypothesis test for a proportion is always either:

- Reject the null hypothesis
- Do not reject the null hypothesis

If you reject the null hypothesis, you cannot say that your sample proportion is the true population proportion. If you do not reject the null hypothesis, you cannot say that the null hypothesis is true.

A hypothesis test is simply a way to look at a sample and conclude if it provides sufficient evidence to reject the null hypothesis.

## Example: Hypothesis Test for a Proportion

Let’s say you are the Marketing Director of a software company. You have set up a demo request page on your website, and you believe that 40% of visitors to that page will request a demo.

You decide to test your claim that 40% of visitors to the demo page will request a demo. So, you decide to run a hypothesis test for a proportion with a sample size of 500 visitors. Let’s go through the steps you would take to run the test.

- Check the conditions - Your test consists of binary outcomes (i.e. request demo and not request demo), your sample size is large enough to meet the success-failure condition but not too large to violate the sample-to-population ratio condition, and you collect your sample using simple random sampling . So, your test satisfies the conditions for a z-test of a single proportion.
- State Your Hypothesis - Your null hypothesis is that the true proportion of visitors requesting a demo equals 40%, formally stated P = 40%. Your alternate hypothesis is that the true proportion of vistors requesting a demo does not equal 40%, formally stated P ≠ 40%.
- Determine Your Analysis Plan - You believe that a 5% significance level is reasonable. As your test is two-tail test, you will evaluate if your sample proportion would occur at the upper or lower 2.5% [2.5% = 5%/2] of the null distribution.
- Analyze Your Sample - You collect your samle (which you do after steps 1-3). You find that the proportion of visitors request a demo in your sample is 44%. Using the calculator above, you find that a sample proportion of 44% would results in a z-score of 1.83 under the null distribution, which translates to a p-value of 6.79%.
- Interpret Your Results - Since your p-value of 6.79% is greater than the significance level of 5%, you do not have sufficient evidence to reject the null hypothesis.

In this example, you found that you cannot reject your original claim that 40% of your demo webpage vistors request demos. The test does not guarantee that your 40% figure is correct, but it does give you confidence that you do not have sufficient evidence to say otherwise.

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## 9.1: Null and Alternative Hypotheses

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\(H_0\): 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.

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 \(\PageIndex{1}\)

- \(H_{0}\): 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 .

## IMAGES

## VIDEO

## COMMENTS

Hypothesis Testing Calculator. The first step in hypothesis testing is to calculate the test statistic. The formula for the test statistic depends on whether the population standard deviation (σ) is known or unknown. If σ is known, our hypothesis test is known as a z test and we use the z distribution. If σ is unknown, our hypothesis test is ...

This section answers some common questions about . Use this Hypothesis Test Calculator for quick results in Python and R. Learn the step-by-step hypothesis test process and why hypothesis testing is important.

The null hypothesis is that the true difference between the means of pre- and post-populations is equal to some pre-set value, Δ \Delta Δ. The alternative hypothesis is that the actual difference between these means is: Different from Δ \Delta Δ; Smaller than Δ \Delta Δ; or; Greater than Δ \Delta Δ. Typically, this pre-determined ...

The null hypothesis (H 0) answers "No, there's no effect in the population." 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.

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

This Z-test calculator is a tool that helps you perform a one-sample Z-test on the population's mean. Two forms of this test - a two-tailed Z-test and a one-tailed Z-tests - exist, and can be used depending on your needs. ... It is the alternative hypothesis which determines what more extreme means: ... Let us state the null and alternative ...

Table of contents. Step 1: State your null and alternate hypothesis. Step 2: Collect data. Step 3: Perform a statistical test. Step 4: Decide whether to reject or fail to reject your null hypothesis. Step 5: Present your findings. Other interesting articles. Frequently asked questions about hypothesis testing.

The calculated p-value is used in comparison with a predefined significance level (alpha) to make decisions about the null hypothesis. If the p-value is less than or equal to alpha, typically 0.05, the results are considered statistically significant, leading to the rejection of the null hypothesis in favor of the alternative hypothesis.

Demystifying Hypothesis Testing: Null Hypothesis (H0): Positioned as the default assumption, the null hypothesis asserts the absence of any significant difference or effect and is commonly represented as H0. Alternative Hypothesis (Ha): In direct contradiction to the null hypothesis, the alternative hypothesis posits the existence of a noteworthy difference or effect, denoted as Ha.

A small p-value suggests that the results are unlikely under the null hypothesis. Key Features of the Hypothesis Testing Calculator: Input Parameters: The calculator typically requires you to input sample data, choose the type of test (e.g., t-test, chi-square test), specify the null and alternative hypotheses, and set the significance level.

Result: . This Hypothesis Testing Calculator determines whether an alternative hypothesis is true or not. Based on whether it is true or not determines whether we accept or reject the hypothesis. We accept true hypotheses and reject false hypotheses. The null hypothesis is the hypothesis that is claimed and that we will test against.

The null and alternative hypotheses are two competing claims that researchers weigh evidence for and against using a statistical test: Null hypothesis (H0): There's no effect in the population. Alternative hypothesis (HA): There's an effect in the population. The effect is usually the effect of the independent variable on the dependent ...

If p-value ≥ α, then you don't have enough evidence to reject the null hypothesis. Obviously, the fate of the null hypothesis depends on α. For instance, if the p-value was 0.03, we would reject the null hypothesis at a significance level of 0.05, but not at a level of 0.01. That's why the significance level should be stated in advance and ...

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.

It is the opposite of your research hypothesis. The alternative hypothesis--that is, the research hypothesis--is the idea, phenomenon, observation that you want to prove. If you suspect that girls take longer to get ready for school than boys, then: Alternative: girls time > boys time. Null: girls time <= boys time.

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

There are two options for a decision. They are "reject H0 H 0 " if the sample information favors the alternative hypothesis or "do not reject H0 H 0 " or "decline to reject H0 H 0 " if the sample information is insufficient to reject the null hypothesis. Table 12.3.1.1 12.3.1. 1: Mathematical Symbols Used in H0 H 0 and Ha H a: H0 H 0.

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.

Decision Rule Calculator. In hypothesis testing, we want to know whether we should reject or fail to reject some statistical hypothesis. To make this decision, we compare the p-value of the test statistic to a significance level we have chosen to use for the test. If the p-value is less than the significance level, we reject the null hypothesis.

Whenever we perform a hypothesis test, we always write a null hypothesis and an alternative hypothesis, which take the following forms: H0 (Null Hypothesis): Population parameter =, ≤, ≥ some value. HA (Alternative Hypothesis): Population parameter <, >, ≠ some value. Note that the null hypothesis always contains the equal sign.

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.

You must state a null hypothesis and an alternative hypothesis to conduct a hypothesis test for a proportion. The null hypothesis, is a skeptical claim that you would like to test. It is defined by a hypothesized proportion, which is often labeled P 0. The alternative hypothesis represents an alternative claim to the null hypothesis.

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.