Hypothesis Testing Calculator
| $H_o$: | |||
| $H_a$: | μ | ≠ | μ₀ |
| $n$ | = | $\bar{x}$ | = | = |
| $\text{Test Statistic: }$ | = |
| $\text{Degrees of Freedom: } $ | $df$ | = |
| $ \text{Level of Significance: } $ | $\alpha$ | = |
Type II Error
| $H_o$: | $\mu$ | ||
| $H_a$: | $\mu$ | ≠ | $\mu_0$ |
| $n$ | = | σ | = | $\mu$ | = |
| $\text{Level of Significance: }$ | $\alpha$ | = |
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 known as a t test and we use the t distribution. Use of the t distribution relies on the degrees of freedom, which is equal to the sample size minus one. Furthermore, if the population standard deviation σ is unknown, the sample standard deviation s is used instead. To switch from σ known to σ unknown, click on $\boxed{\sigma}$ and select $\boxed{s}$ in the Hypothesis Testing Calculator.
| $\sigma$ Known | $\sigma$ Unknown | |
| Test Statistic | $ z = \dfrac{\bar{x}-\mu_0}{\sigma/\sqrt{{\color{Black} n}}} $ | $ t = \dfrac{\bar{x}-\mu_0}{s/\sqrt{n}} $ |
Next, the test statistic is used to conduct the test using either the p-value approach or critical value approach. The particular steps taken in each approach largely depend on the form of the hypothesis test: lower tail, upper tail or two-tailed. The form can easily be identified by looking at the alternative hypothesis (H a ). If there is a less than sign in the alternative hypothesis then it is a lower tail test, greater than sign is an upper tail test and inequality is a two-tailed test. To switch from a lower tail test to an upper tail or two-tailed test, click on $\boxed{\geq}$ and select $\boxed{\leq}$ or $\boxed{=}$, respectively.
| Lower Tail Test | Upper Tail Test | Two-Tailed Test |
| $H_0 \colon \mu \geq \mu_0$ | $H_0 \colon \mu \leq \mu_0$ | $H_0 \colon \mu = \mu_0$ |
| $H_a \colon \mu | $H_a \colon \mu \neq \mu_0$ |
In the p-value approach, the test statistic is used to calculate a p-value. If the test is a lower tail test, the p-value is the probability of getting a value for the test statistic at least as small as the value from the sample. If the test is an upper tail test, the p-value is the probability of getting a value for the test statistic at least as large as the value from the sample. In a two-tailed test, the p-value is the probability of getting a value for the test statistic at least as unlikely as the value from the sample.
To test the hypothesis in the p-value approach, compare the p-value to the level of significance. If the p-value is less than or equal to the level of signifance, reject the null hypothesis. If the p-value is greater than the level of significance, do not reject the null hypothesis. This method remains unchanged regardless of whether it's a lower tail, upper tail or two-tailed test. To change the level of significance, click on $\boxed{.05}$. Note that if the test statistic is given, you can calculate the p-value from the test statistic by clicking on the switch symbol twice.
In the critical value approach, the level of significance ($\alpha$) is used to calculate the critical value. In a lower tail test, the critical value is the value of the test statistic providing an area of $\alpha$ in the lower tail of the sampling distribution of the test statistic. In an upper tail test, the critical value is the value of the test statistic providing an area of $\alpha$ in the upper tail of the sampling distribution of the test statistic. In a two-tailed test, the critical values are the values of the test statistic providing areas of $\alpha / 2$ in the lower and upper tail of the sampling distribution of the test statistic.
To test the hypothesis in the critical value approach, compare the critical value to the test statistic. Unlike the p-value approach, the method we use to decide whether to reject the null hypothesis depends on the form of the hypothesis test. In a lower tail test, if the test statistic is less than or equal to the critical value, reject the null hypothesis. In an upper tail test, if the test statistic is greater than or equal to the critical value, reject the null hypothesis. In a two-tailed test, if the test statistic is less than or equal the lower critical value or greater than or equal to the upper critical value, reject the null hypothesis.
| Lower Tail Test | Upper Tail Test | Two-Tailed Test |
| If $z \leq -z_\alpha$, reject $H_0$. | If $z \geq z_\alpha$, reject $H_0$. | If $z \leq -z_{\alpha/2}$ or $z \geq z_{\alpha/2}$, reject $H_0$. |
| If $t \leq -t_\alpha$, reject $H_0$. | If $t \geq t_\alpha$, reject $H_0$. | If $t \leq -t_{\alpha/2}$ or $t \geq t_{\alpha/2}$, reject $H_0$. |
When conducting a hypothesis test, there is always a chance that you come to the wrong conclusion. There are two types of errors you can make: Type I Error and Type II Error. A Type I Error is committed if you reject the null hypothesis when the null hypothesis is true. Ideally, we'd like to accept the null hypothesis when the null hypothesis is true. A Type II Error is committed if you accept the null hypothesis when the alternative hypothesis is true. Ideally, we'd like to reject the null hypothesis when the alternative hypothesis is true.
| Condition | ||||
| $H_0$ True | $H_a$ True | |||
| Conclusion | Accept $H_0$ | Correct | Type II Error | |
| Reject $H_0$ | Type I Error | Correct | ||
Hypothesis testing is closely related to the statistical area of confidence intervals. If the hypothesized value of the population mean is outside of the confidence interval, we can reject the null hypothesis. Confidence intervals can be found using the Confidence Interval Calculator . The calculator on this page does hypothesis tests for one population mean. Sometimes we're interest in hypothesis tests about two population means. These can be solved using the Two Population Calculator . The probability of a Type II Error can be calculated by clicking on the link at the bottom of the page.
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Hypothesis testing calculator
Welcome to our complete hypothesis testing calculator, the ideal tool for doing accurate and trustworthy statistical studies. Our calculator is meant to fulfill the demands of students, researchers, and professionals while also simplifying the hypothesis testing procedure.
- One sample z test hypothesis calculator
- One sample t test hypothesis calculator
- One proportion z test hypothesis calculator
- Two sample z test hypothesis calculator
- Two sample t test hypothesis calculator ( equal and unequal variance )
- Two proportion z test hypothesis calculator
Why Use Our Hypothesis Testing Calculator?
Hypothesis testing is an important part of statistical analysis because it allows you to make population-level inferences based on sample data. Our calculator makes this process easier by providing user-friendly interfaces and step-by-step directions for performing different tests. Here are several significant advantages:
- Accuracy: Our calculators are designed to provide precise calculations, ensuring your results are reliable.
- Variety: With a range of calculators available, you can perform different types of hypothesis tests as needed.
- User-Friendly: Easy-to-use interfaces make it simple for anyone to perform complex statistical analyses.
- Free Access: Our tools are available for free, making high-quality statistical analysis accessible to everyone.
How to Use Our Hypothesis Testing Calculator
Using our hypothesis testing calculator is straightforward. Simply select the type of test you need from the list above, input your data, and follow the prompts. Our calculators will guide you through each step, ensuring you understand the process and obtain accurate results.
What is a Null Hypothesis (H 0 )?
The null hypothesis, denoted as H 0 , is the default assumption in hypothesis testing. It posits that there is no significant effect or difference between groups or conditions. Essentially, it represents the status quo or the idea that any observed differences are due to random chance.
Examples of Null Hypotheses:
- In a clinical trial: H 0 : "There is no difference in the effectiveness of Drug A and Drug B."
- In a manufacturing process: H 0 : "The mean length of the produced parts is equal to the specified length."
- In a survey: H 0 : "The proportion of voters who support Candidate X is 50%."
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- List of all calculator
- P-value calculator
- Critical value Calculator
What is an Alternative Hypothesis (H a )?
The alternative hypothesis, denoted as H a , is the statement that contradicts the null hypothesis. It suggests that there is a significant effect or difference. The alternative hypothesis represents what the researcher aims to prove or the presence of an effect they are testing for.
Examples of Alternative Hypotheses:
- In a clinical trial: H a : "There is a difference in the effectiveness of Drug A and Drug B."
- In a manufacturing process: H a : "The mean length of the produced parts is not equal to the specified length."
- In a survey: H a : "The proportion of voters who support Candidate X is not 50%."
The Importance of Hypothesis Testing
Hypothesis testing is fundamental in statistical analysis as it allows researchers to make data-driven decisions. By comparing the null and alternative hypotheses, researchers can determine the likelihood that their observations are due to chance or if there is evidence to support a significant effect.
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.
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- Null and Alternative Hypotheses | Definitions & Examples
Null & Alternative Hypotheses | Definitions, Templates & Examples
Published on May 6, 2022 by Shaun Turney . Revised on June 22, 2023.
The null and alternative hypotheses are two competing claims that researchers weigh evidence for and against using a statistical test :
- Null hypothesis ( H 0 ): There’s no effect in the population .
- Alternative hypothesis ( H a or H 1 ) : There’s an effect in the population.
Table of contents
Answering your research question with hypotheses, what is a null hypothesis, what is an alternative hypothesis, similarities and differences between null and alternative hypotheses, how to write null and alternative hypotheses, other interesting articles, frequently asked questions.
The null and alternative hypotheses offer competing answers to your research question . When the research question asks “Does the independent variable affect the dependent variable?”:
- The null 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 . Often, we infer whether there’s an effect in the population by looking at differences between groups or relationships between variables in the sample. It’s critical for your research to write strong hypotheses .
You can use a statistical test to decide whether the evidence favors the null or alternative hypothesis. Each type of statistical test comes with a specific way of phrasing the null and alternative hypothesis. However, the hypotheses can also be phrased in a general way that applies to any test.
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The null hypothesis is the claim that there’s no effect in the population.
If the sample provides enough evidence against the claim that there’s no effect in the population ( p ≤ α), then we can reject the null hypothesis . Otherwise, we fail to reject the null hypothesis.
Although “fail to reject” may sound awkward, it’s the only wording that statisticians accept . Be careful not to say you “prove” or “accept” the null hypothesis.
Null hypotheses often include phrases such as “no effect,” “no difference,” or “no relationship.” When written in mathematical terms, they always include an equality (usually =, but sometimes ≥ or ≤).
You can never know with complete certainty whether there is an effect in the population. Some percentage of the time, your inference about the population will be incorrect. When you incorrectly reject the null hypothesis, it’s called a type I error . When you incorrectly fail to reject it, it’s a type II error.
Examples of null hypotheses
The table below gives examples of research questions and null hypotheses. There’s always more than one way to answer a research question, but these null hypotheses can help you get started.
| ( ) | ||
| Does tooth flossing affect the number of cavities? | Tooth flossing has on the number of cavities. | test: The mean number of cavities per person does not differ between the flossing group (µ ) and the non-flossing group (µ ) in the population; µ = µ . |
| Does the amount of text highlighted in the textbook affect exam scores? | The amount of text highlighted in the textbook has on exam scores. | : There is no relationship between the amount of text highlighted and exam scores in the population; β = 0. |
| Does daily meditation decrease the incidence of depression? | Daily meditation the incidence of depression.* | test: The proportion of people with depression in the daily-meditation group ( ) is greater than or equal to the no-meditation group ( ) in the population; ≥ . |
*Note that some researchers prefer to always write the null hypothesis in terms of “no effect” and “=”. It would be fine to say that daily meditation has no effect on the incidence of depression and p 1 = p 2 .
The alternative hypothesis ( H a ) is the other answer to your research question . It claims that there’s an effect in the population.
Often, your alternative hypothesis is the same as your research hypothesis. In other words, it’s the claim that you expect or hope will be true.
The alternative hypothesis is the complement to the null hypothesis. Null and alternative hypotheses are exhaustive, meaning that together they cover every possible outcome. They are also mutually exclusive, meaning that only one can be true at a time.
Alternative hypotheses often include phrases such as “an effect,” “a difference,” or “a relationship.” When alternative hypotheses are written in mathematical terms, they always include an inequality (usually ≠, but sometimes < or >). As with null hypotheses, there are many acceptable ways to phrase an alternative hypothesis.
Examples of alternative hypotheses
The table below gives examples of research questions and alternative hypotheses to help you get started with formulating your own.
| Does tooth flossing affect the number of cavities? | Tooth flossing has an on the number of cavities. | test: The mean number of cavities per person differs between the flossing group (µ ) and the non-flossing group (µ ) in the population; µ ≠ µ . |
| Does the amount of text highlighted in a textbook affect exam scores? | The amount of text highlighted in the textbook has an on exam scores. | : There is a relationship between the amount of text highlighted and exam scores in the population; β ≠ 0. |
| Does daily meditation decrease the incidence of depression? | Daily meditation the incidence of depression. | test: The proportion of people with depression in the daily-meditation group ( ) is less than the no-meditation group ( ) in the population; < . |
Null and alternative hypotheses are similar in some ways:
- They’re both answers to the research question.
- They both make claims about the population.
- They’re both evaluated by statistical tests.
However, there are important differences between the two types of hypotheses, summarized in the following table.
| A claim that there is in the population. | A claim that there is in the population. | |
|
| ||
| Equality symbol (=, ≥, or ≤) | Inequality symbol (≠, <, or >) | |
| Rejected | Supported | |
| Failed to reject | Not supported |
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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.
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 template sentences
Once you know the statistical test you’ll be using, you can write your hypotheses in a more precise and mathematical way specific to the test you chose. The table below provides template sentences for common statistical tests.
| ( ) | ||
| test
with two groups | The mean dependent variable does not differ between group 1 (µ ) and group 2 (µ ) in the population; µ = µ . | The mean dependent variable differs between group 1 (µ ) and group 2 (µ ) in the population; µ ≠ µ . |
| with three groups | The mean dependent variable does not differ between group 1 (µ ), group 2 (µ ), and group 3 (µ ) in the population; µ = µ = µ . | The mean dependent variable of group 1 (µ ), group 2 (µ ), and group 3 (µ ) are not all equal in the population. |
| There is no correlation between independent variable and dependent variable in the population; ρ = 0. | There is a correlation between independent variable and dependent variable in the population; ρ ≠ 0. | |
| There is no relationship between independent variable and dependent variable in the population; β = 0. | There is a relationship between independent variable and dependent variable in the population; β ≠ 0. | |
| Two-proportions test | The dependent variable expressed as a proportion does not differ between group 1 ( ) and group 2 ( ) in the population; = . | The dependent variable expressed as a proportion differs between group 1 ( ) and group 2 ( ) in the population; ≠ . |
Note: The template sentences above assume that you’re performing one-tailed tests . One-tailed tests are appropriate for most studies.
If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.
- Normal distribution
- Descriptive statistics
- Measures of central tendency
- Correlation coefficient
Methodology
- Cluster sampling
- Stratified sampling
- Types of interviews
- Cohort study
- Thematic analysis
Research bias
- Implicit bias
- Cognitive bias
- Survivorship bias
- Availability heuristic
- Nonresponse bias
- Regression to the mean
Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.
Null and alternative hypotheses are used in statistical hypothesis testing . The null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship.
The null hypothesis is often abbreviated as H 0 . When the null hypothesis is written using mathematical symbols, it always includes an equality symbol (usually =, but sometimes ≥ or ≤).
The alternative hypothesis is often abbreviated as H a or H 1 . When the alternative hypothesis is written using mathematical symbols, it always includes an inequality symbol (usually ≠, but sometimes < or >).
A research hypothesis is your proposed answer to your research question. The research hypothesis usually includes an explanation (“ x affects y because …”).
A statistical hypothesis, on the other hand, is a mathematical statement about a population parameter. Statistical hypotheses always come in pairs: the null and alternative hypotheses . In a well-designed study , the statistical hypotheses correspond logically to the research hypothesis.
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Hypothesis Testing Calculator
Enhanced hypothesis test calculator, hypothesis test calculator: your essential tool for statistical analysis.
The Hypothesis Test Calculator is an intuitive, web-based tool crafted to simplify the process of conducting statistical hypothesis tests. Whether you’re a researcher, student, or educator, this tool streamlines the process, providing accurate results and clear visualizations with minimal effort.
How to Use the Hypothesis Test Calculator
- Step 1: Start by entering the required data into the respective input fields. You’ll need to provide the sample mean, sample size, population mean, population standard deviation, and the significance level (alpha). If you have raw data, you can also upload a CSV file directly.
- Step 2: Choose whether you’re conducting a one-tailed or two-tailed test. If you’re interested in knowing the confidence interval, select the option to calculate it.
- Step 3: Click the “Perform Test” button. The tool will quickly process the information, performing the hypothesis test and generating a detailed report.
- Step 4: Review the results. The calculator provides the calculated t-value, critical value, and the test’s conclusion. You can also visualize the sample distribution with a dynamic chart that compares it to the population mean.
Understanding Hypothesis Testing
Hypothesis testing is a core statistical method used to determine whether there is enough evidence in a sample to infer that a certain condition is true for the entire population. The test compares the sample data against the null hypothesis, which typically represents the status quo or a baseline condition.
In this context, the Hypothesis Test Calculator is particularly useful for t-tests, which assess whether the sample mean significantly differs from a known population mean. By inputting your data, the tool computes the necessary values, helping you determine whether to reject the null hypothesis.
Hypothesis Test Formula:
$$ t = \frac{\text{Sample Mean} – \text{Population Mean}}{\text{Standard Error}} $$
Features and Advantages
- User-Friendly Design: The calculator is built with simplicity in mind, allowing users to quickly enter data and obtain results without any hassle.
- Rapid Calculations: With just a few clicks, the tool performs complex statistical calculations, delivering results almost instantaneously.
- Comprehensive Results: Not only does the calculator provide the final conclusion, but it also breaks down the steps involved in the calculation, aiding in understanding the statistical process.
- Visual Representation: The tool includes a dynamic chart that illustrates the sample distribution against the population mean, helping users visualize the data and results.
- Versatile Input Options: You can input data manually or upload a CSV file, making it easy to work with real datasets.
- Educational Utility: Perfect for both students and educators, the Hypothesis Test Calculator serves as an excellent learning tool, offering insights into the mechanics of hypothesis testing.
Enhancing Your Statistical Toolkit
The Hypothesis Test Calculator is an invaluable resource for anyone involved in data analysis. It simplifies the hypothesis testing process, delivering clear, accurate results and visual aids that enhance your understanding of statistical concepts.
Try the Hypothesis Test Calculator today and take your statistical analysis to the next level!
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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.
Null Hypothesis Calculator: Quick & Accurate Statistical Analysis
This null hypothesis calculator helps you determine the statistical significance of your data set by calculating the probability of observing your results under the null hypothesis.
How to Use the Null Hypothesis Calculator
To use the null hypothesis calculator, enter the sample mean, hypothesized population mean, sample standard deviation, and sample size into their respective fields. Then press the “Calculate” button, and the calculator will display the test statistic used to determine if the null hypothesis can be rejected.
How It Calculates the Results
The calculator computes the test statistic by using the formula:
( z = frac{(bar{x} – mu_0)}{(sigma / sqrt{n})} )
- ( bar{x} ) is the sample mean
- ( mu_0 ) is the hypothesized population mean
- ( sigma ) is the standard deviation of the sample
- ( n ) is the sample size
This test statistic is then used in statistical hypothesis testing to determine if there is enough evidence to reject the null hypothesis at a certain significance level.
Limitations
The calculator presumes that the sample data are representative of the population and that the sampling distribution of the mean is approximately normally distributed, especially as the sample size becomes large (Central Limit Theorem). It also assumes that an accurate standard deviation is known and that the sample is drawn randomly.
Be aware that for smaller sample sizes, particularly under 30, this calculator may not provide accurate results because the central limit theorem applies better to larger samples. For small samples, other techniques or adjustments (like t-tests) typically need to be used.
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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: a statement of no difference between sample means or proportions or no difference between a sample mean or proportion and a population mean or proportion. In other words, the difference equals 0.
H a —, the alternative hypothesis: a claim about the population that is contradictory to H 0 and what we conclude when we reject H 0 .
Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data.
After you have determined which hypothesis the sample supports, you make a decision. There are two options for a decision. They are reject H 0 if the sample information favors the alternative hypothesis or do not reject H 0 or decline to reject H 0 if the sample information is insufficient to reject the null hypothesis.
Mathematical Symbols Used in H 0 and H a :
| equal (=) | not equal (≠) greater than (>) less than (<) |
| greater than or equal to (≥) | less than (<) |
| less than or equal to (≤) | more than (>) |
H 0 always has a symbol with an equal in it. H a never has a symbol with an equal in it. The choice of symbol depends on the wording of the hypothesis test. However, be aware that many researchers use = in the null hypothesis, even with > or < as the symbol in the alternative hypothesis. This practice is acceptable because we only make the decision to reject or not reject the null hypothesis.
Example 9.1
H 0 : No more than 30 percent of the registered voters in Santa Clara County voted in the primary election. p ≤ 30 H a : More than 30 percent of the registered voters in Santa Clara County voted in the primary election. p > 30
A medical trial is conducted to test whether or not a new medicine reduces cholesterol by 25 percent. State the null and alternative hypotheses.
Example 9.2
We want to test whether the mean GPA of students in American colleges is different from 2.0 (out of 4.0). The null and alternative hypotheses are the following: H 0 : μ = 2.0 H a : μ ≠ 2.0
We want to test whether the mean height of eighth graders is 66 inches. State the null and alternative hypotheses. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.
- H 0 : μ __ 66
- H a : μ __ 66
Example 9.3
We want to test if college students take fewer than five years to graduate from college, on the average. The null and alternative hypotheses are the following: H 0 : μ ≥ 5 H a : μ < 5
We want to test if it takes fewer than 45 minutes to teach a lesson plan. State the null and alternative hypotheses. Fill in the correct symbol ( =, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.
- H 0 : μ __ 45
- H a : μ __ 45
Example 9.4
An article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third of the students pass. The same article stated that 6.6 percent of U.S. students take advanced placement exams and 4.4 percent pass. Test if the percentage of U.S. students who take advanced placement exams is more than 6.6 percent. State the null and alternative hypotheses. H 0 : p ≤ 0.066 H a : p > 0.066
On a state driver’s test, about 40 percent pass the test on the first try. We want to test if more than 40 percent pass on the first try. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.
- H 0 : p __ 0.40
- H a : p __ 0.40
Collaborative Exercise
Bring to class a newspaper, some news magazines, and some internet articles. In groups, find articles from which your group can write null and alternative hypotheses. Discuss your hypotheses with the rest of the class.
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Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute Texas Education Agency (TEA). The original material is available at: https://www.texasgateway.org/book/tea-statistics . Changes were made to the original material, including updates to art, structure, and other content updates.
Access for free at https://openstax.org/books/statistics/pages/1-introduction
- Authors: Barbara Illowsky, Susan Dean
- Publisher/website: OpenStax
- Book title: Statistics
- Publication date: Mar 27, 2020
- Location: Houston, Texas
- Book URL: https://openstax.org/books/statistics/pages/1-introduction
- Section URL: https://openstax.org/books/statistics/pages/9-1-null-and-alternative-hypotheses
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Test Statistic Calculator
Choose the method, enter the values into the test statistic calculator, and click on the “Calculate” button to calculate the statistical value for hypothesis evaluation.
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This test statistic calculator helps to find the static value for hypothesis testing. The calculated test value shows if there’s enough evidence to reject a null hypothesis. Also, this calculator performs calculations of either for one population mean, comparing two means, single population proportion, and two population proportions.
Our tool is highly useful in various fields like research, experimentation, quality control, and data analysis.
What is Test Statistics?
A test statistic is a numerical value obtained from the sample data set. It summarizes the differences between what you observe within your sample and what would be expected if a hypothesis were true.
The t-test statistic also shows how closely your data matches the predicted distribution among the sample tests you perform.
How to Calculate Test Statistics Value?
- Collect the data from the populations
- Use the data to find the standard deviation of the population
- Calculate the mean (μ) of the population using this data
- Determine the z-value or sample size
- Use the suitable test statistic formula and get the results
Test Statistic For One Population Mean:
Test statistics for a single population mean is calculated when a variable is numeric and involves one population or a group.
x̄ - µ 0 σ / √n
- x̄ = Mean of your sample data
- µ 0 = Hypothesized population mean that you are comparing to your sample mean
- σ = Population standard deviation
- n = number of observations (sample size) in your data set
Suppose we want to test if the average height of adult males in a city is 70 inches. We take a sample of 25 adult males and find the sample mean height to be 71 inches with a sample standard deviation of 3 inches. We use a significance level of 0.05.
t = 70 - 71 3√25
Test Statistic Comparing Two Population Means:
This test is applied when the numeric value is compared across the various populations or groups. To compute the resulting t statistic, two distinct random samples must be chosen, one from each population.
\(\frac{√x̄ - √ȳ}{\sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}}\)
- ȳ = means of hypothesized population
Suppose we want to test if there is a difference in average test scores between two schools. We take a sample of 30 students from school A with an average score of 85 and a standard deviation of 5, and a sample of 35 students from school B with an average score of 82 and a standard deviation of 6.
t = 85 - 82 √5 2 / 30 + 6 2 / 35
t = 3 √ 25/30 + 36/35
t = 3 √0.833 + 1.029
t = 3 √1.862
Test Statistic For a Single Population Proportion:
This test is used to determine if a single population's proportion differs from a specified standard. The t statistic calculator works for a population proportion when dealing with data by having a limit of P₀ because proportions represent parts of a whole and cannot logically exceed the total or be negative.
\(\frac{\hat{p}-p_{0}}{\sqrt{\frac{p_{0}(1-p_{0})}{n}}}\)
- P̂ = Sample proportion
- P 0 = Population proportion
Suppose we want to test if the proportion of left-handed people in a population is 10%. We take a sample of 100 people and find that 8 are left-handed. We use a significance level of 0.05.
= P̂ - P₀ √0.10 (1 - 0.10)/100
= 0.08 - 0.10 √0.10 (1 - 0.10)/100
= -0.02 √0.10 (0.9)/100
= -0.02 √0.009
= -0.02 0.03
= −0.67
Test Statistic For Two Population Proportion:
This test identifies the difference in proportions between two independent groups to assess their significance.
\(\frac{\hat{p}_{1}-\hat{p}_{2}}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_{1}}+\frac{1}{n_{2}})}}\)
- P̂ 1 and P̂ 2 = Sample proportions for two groups
Suppose we want to test if the proportion of smokers is different between two cities. We take a sample of 150 people from City A and find that 30 are smokers, and a sample of 200 people from City B and find that 50 are smokers.
- P̂ 1 = 30 / 150 = 0.20
- P̂ 2 = 50 / 200 = 0.25
- P̂ = 30 + 50 / 150 + 200 = 0.229
Calculation:
= 0.20 - 0.25 √0.229 (1 - 0.229) (1 / 150 + 1/200)
= -0.05 √0.229 (0.771) (1 / 150 + 1 / 200)
= -0.05 √0.176 (1/150 + 1/200)
= -0.05 √0.176 (0.0113)
= -0.05 √0.002
= -0.05 0.045
= −1.11
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p-value Calculator
Table of contents
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.
What is p-value?
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.
How do I calculate p-value from test statistic?
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.
How to interpret p-value
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.
How to use the p-value calculator to find p-value from test statistic
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.
By default, the calculator uses the significance level of 0.05.
Our calculator determines the p-value from the test statistic and provides the decision to be made about the null hypothesis.
How do I find p-value from z-score?
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.
How do I find p-value from t?
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 .
p-value from chi-square score (χ² score)
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 .
p-value from F-score
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.
What do you want?
What do you know?
Your Z-score
Z-score : the test statistic follows the standard normal distribution N(0,1).
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Significance level α
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- A P-value calculator is used to determine the statistical significance of an observed result in hypothesis testing. It takes as input the observed test statistic, the null hypothesis, and the relevant parameters of the statistical test (such as degrees of freedom), and computes the p-value. The p-value represents the probability of obtaining results as extreme as, or more extreme than, the observed data, assuming the null hypothesis is true. A lower p-value suggests stronger evidence against the null hypothesis, indicating that the observed result is unlikely to have occurred by random chance alone. 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. If the p-value is greater than alpha, there is insufficient evidence to reject the null hypothesis.
- How do I calculate p-value?
- The p-value is calculated by determining the probability of observing a test statistic as extreme as, or more extreme than, the observed one under the assumption of the null hypothesis.
- What is p-value in Z test?
- In a Z-test, the p-value is the probability of observing a Z-statistic as extreme as, or more extreme than, the calculated one, assuming a normal distribution and under the null hypothesis.
- What is the p-value?
- The p-value, or probability value, is a measure in statistics that quantifies the strength of evidence against a null hypothesis. It indicates the likelihood of observing a test statistic as extreme as, or more extreme than, the one obtained from the data, assuming the null hypothesis is true.
- What is the alpha for p-value?
- The alpha (α) for a p-value is the chosen level of significance that determines the threshold for rejecting the null hypothesis. It represents the maximum probability of making a Type I error (incorrectly rejecting a true null hypothesis) and is typically set at common values such as 0.05 or 0.01.
- What does p-value under 0.05 mean?
- A p-value under 0.05 typically suggests that there is statistically significant evidence to reject the null hypothesis in favor of an alternative hypothesis.
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Hypothesis Testing: How to Determine the Null and Alternative Hypotheses?
One thing that can be tricky when attempting to solve a hypothesis testing problem is to establish precisely what the null and alternative hypotheses are. Typically, such information can be easily inferred from the context of the problem, but you need to know what to look for in order to get it right.
HOW TO GET STARTED
First thing to keep in mind is the precise specification of the null and alternative hypotheses can be inferred from the wording on the actual problem. Somewhere in the setting of the problem you will find where the hypotheses are stated.
Second, you need to keep in mind that the null and alternative hypotheses DO NOT OVERLAP. This implies that for the most part you can tell the null hypothesis if you know the alternative hypothesis, and vice versa, with some exceptions as we will see in the next paragraph.
Third, when reading the setting of an hypothesis testing problem, we need to identify any claim made about a population parameter, and express it in mathematical terms, such as \(\mu =2.3\), \(\mu \le 3\), \(\sigma >3.5\), etc. This is VERY IMPORTANT, because once we have expressed the claim(s) provided mathematically, we need to take note of which mathematical sign is used (\(\le\), \(\ge\), = , < or >).
The fourth point to keep in mind is the hypothesis of no effect, and it must contain the "=" sign, which means that the sign in the null hypothesis can be "\(\le\)", "=" or "\(\ge\)". And since the null hypothesis and alternative hypothesis cannot overlap, the only options for the sign of the alternative hypothesis are ">" or "<".
The above information should in fact be sufficient to determine the null and alternative hypothesis with ease.
SOME PRACTICAL EXAMPLES
For example, say that we are examining a hypothesis testing question from our stats homework, and scanning the problem we read something like "and the investigator wants to prove if the average mileage for the new model is greater than 18 mpg". Such statement is a claim about the population mean mileage of the new car model, which we call \(\mu\).
The claim that the investigator is making is that "\(\mu >18\)". Since the mathematical expression of the claim does not contain "=", then the claim must be the alternative hypothesis. So then in this case we have the alternative hypothesis is Ha: \(\mu >18\). What is the null hypothesis then? Well, we know that the null and alternative hypotheses do not overlap, so we can say that the null hypothesis is the COMPLEMENT to what is expressed in the alternative hypothesis, so then in this case the null hypothesis is Ho: \(\mu \le 18\).
Therefore, summarizing, in this case the null and alternative hypotheses would be:
\[\begin{align} {{H}_{0}}:\mu \le 18 \\ {{H}_{A}}:\mu >18 \\ \end{align}\]
Another example : Assume that the setting of the problem reads something like "a sample was collected to assess if the IQ of Stats professors is the same as national average IQ of 102". In that case, there is a claim about the population IQ of all Stats professor, which we shall call \(\mu\). The claim made is \(\mu =102\), and since this statement contains the sign "=", then this MUST be the null hypothesis. Hence, in this example we have that Ho: \(\mu =102\).
What is the alternative hypothesis then? Since the null and alternative hypotheses do not overlap, the alternative hypotheses is the complement to the null hypothesis, so in this case the alternative hypothesis would be $\mu \ne 102$.
\[\begin{align} {{H}_{0}}:\mu =102 \\ & {{H}_{A}}:\mu \ne 102 \\ \end{align}\]
Another example: Things are not always that easy. Sometimes, things get a bit more complicated (but only a bit, I promise) when it comes down determining the null and alternative hypothesis from the setting of a question. Indeed, sometimes, there are actually two claims about a population parameter. For example, you start reading a question and you find the following: "it has been claimed that the population mean GPA for some state college is 3.94".
So you think, ok, the parameter is the population mean GPA for the state college, which we call \(\mu\), so then this statement is saying that \(\mu =3.94\), and since this mathematical statement contains the sign "=", then this must be the null hypothesis Ho. So we know for a fact that Ho: \(\mu =3.94\). Then you say, I can say that obviously the alternative hypothesis is Ha: \(\mu \ne 3.94\), right? Not so fast! If NOTHING else is claimed about \(\mu\) in the setting of the problem, then you can go and say that Ha: \(\mu \ne 3.94\).
BUT, sometimes another claimed is made. Indeed, suppose that in this case, you take a close look and you reread the problem, and it says " it has been claimed that the population mean GPA for some state college is 3.94, and a random sample has been collected to test the claim of the dean of the college, who claims that the mean GPA is less than that". Aha! In this case there is ANOTHER claim saying \(\mu <3.94\). And since this claim does NOT contain the sign "=", the it must be the alternative hypothesis. So then in this case, we get that Ha: \(\mu <3.94\) and not Ha: \(\mu \ne 3.94\).
Should you be worried about seeing more than two claims in a problem involving hypothesis testing? The answer is NO. More than two claims will either lead to redundant or contradictory claims, for which reason you will likely not find such situation (unless the problem is wrongly posed, which is always a possibility). So then, when facing a problem, you will find one claim about a population parameter which will determine the null or alternative hypothesis, and you can deduce the other by using getting the complement of the given claim. OR, you will find two claims that do not overlap, which will define the null and alternative hypotheses.
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Hypothesis Testing Calculator with Steps
Hypothesis Test Calculator
What is an Alternative Hypothesis (H a)? The alternative hypothesis, denoted as H a, is the statement that contradicts the null hypothesis. It suggests that there is a significant effect or difference. The alternative hypothesis represents what the researcher aims to prove or the presence of an effect they are testing for. Examples of ...
t-test Calculator | Formula | p-value
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 ...
Null & Alternative Hypothesis
Null & Alternative Hypotheses | Definitions, Templates & ...
Step 3: Click the "Perform Test" button. The tool will quickly process the information, performing the hypothesis test and generating a detailed report. Step 4: Review the results. The calculator provides the calculated t-value, critical value, and the test's conclusion. You can also visualize the sample distribution with a dynamic chart ...
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.
How to Use the Null Hypothesis Calculator. To use the null hypothesis calculator, enter the sample mean, hypothesized population mean, sample standard deviation, and sample size into their respective fields. Then press the "Calculate" button, and the calculator will display the test statistic used to determine if the null hypothesis can be ...
Alternative hypothesis. Alternative hypothesis \(\left(H_{A}\right)\): If we conclude that the null hypothesis is false, or rather and more precisely, we find that we provisionally fail to reject the null hypothesis, then we provisionally accept the alternative hypothesis.The view then is that something other than random chance has influenced the sample observations.
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.
Test of Hypothesis is the technique used in probability & statistics to check if the significance of estimated population parameters by analyzing the samples of population is accepted in statistical experiments. The test of hypothesis in experiments classified as null hypothesis (H 0) and alternative hypothesis (H 1) popularly used to analyze one or two tailed normal distribution, t ...
Z-test Calculator | Definition | Examples
This test statistic calculator helps to find the static value for hypothesis testing. The calculated test value shows if there's enough evidence to reject a null hypothesis. Also, this calculator performs calculations of either for one population mean, comparing two means, single population proportion, and two population proportions.
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.
p-value Calculator | Formula | Interpretation
P-value Calculator
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.
Third, when reading the setting of an hypothesis testing problem, we need to identify any claim made about a population parameter, and express it in mathematical terms, such as \mu =2.3 μ =2.3, \mu \le 3 μ ≤ 3, \sigma >3.5 σ> 3.5, etc. This is VERY IMPORTANT, because once we have expressed the claim (s) provided mathematically, we need to ...
One Sample T Test Calculator
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.
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.