example of hypothesis test calculator

Hypothesis Testing Calculator

Related: confidence interval calculator, type ii error.

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.

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.

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.

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.

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.

Hypothesis Testing Calculator Online

Hypothesis testing is a foundational method used in statistics to infer the validity of a hypothesis about a population parameter. The Hypothesis Testing Calculator facilitates this process by automating the computations necessary for the t-test , a method used to compare sample means against a hypothesized mean or against each other. Let’s delve into the formulas this calculator uses to execute one-sample and two-sample t-tests.

One-Sample t-Test

This test is used to determine if the mean (x̄) of your sample is statistically different from a hypothesized population mean (μ₀).

• t is the test statistic
• x̄ is the sample mean
• μ₀ is the hypothesized population mean
• s is the sample standard deviation
• n is the sample size (number of observations)

Two-Sample t-Test

This formula compares the means of two independent groups (x̄₁ and x̄₂), with variations for equal and unequal variances.

Equal Variances:

• x̄₁ and x̄₂ are the means of the first and second sample, respectively
• s pooled is the pooled standard deviation (calculated from both samples)
• n₁ and n₂ are the sizes of the first and second sample, respectively

Unequal Variances (Welch’s t-Test):

t = (x̄₁ – x̄₂) / (√((s₁² / n₁) + (s₂² / n₂)))

• x̄₁ is the mean of the first sample
• x̄₂ is the mean of the second sample
• s₁ and s₂ are the standard deviations of the first and second sample, respectively

Table of Critical t-Values

The following table provides critical t-values for different confidence levels and degrees of freedom which are typically use to determine the cutoff points at which the null hypothesis is reject.

These values are crucial in hypothesis testing as they help define the threshold for significance, assisting users of the calculator in interpreting their results accurately.

Consider a scenario where a school administrator wants to test if the average test score of the students this semester differs from a hypothesized mean of 70%. Using the one-sample t-test:

• Sample mean (x̄) = 74%
• Hypothesized mean (μ₀) = 70%
• Sample standard deviation (s) = 8%
• Sample size (n) = 36

Using the one-sample t-test formula:

t = (74 – 70) / (8 / √36) = (4 / 1.333) = 3.00

The calculated t-value is 3.00. Using the critical t-values table, at 95% confidence level and 35 degrees of freedom, the critical value is approximately 2.030. Since 3.00 > 2.030, the null hypothesis is reject, indicating a significant difference from the hypothesize mean.

Most Common FAQs

The p-value represents the probability of obtaining test results at least as extreme as the results observed, under the assumption that the null hypothesis is correct. A low p-value (typically below 0.05) indicates strong evidence against the null hypothesis, hence it is usually rejected.

Use a one-sample t-test when comparing the mean of a single sample to a known standard or hypothesized mean. Use a two-sample t-test when comparing the means of two independent groups to see if there is a statistically significant difference between them.

Yes, while the t-test is specifically design for means, the principles of hypothesis testing apply to other parameters such as proportions and variances. Which can also be tested using appropriate versions of hypothesis tests such as the z-test and F-test.

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• One sample t test

A one sample t test compares the mean with a hypothetical value. In most cases, the hypothetical value comes from theory. For example, if you express your data as 'percent of control', you can test whether the average differs significantly from 100. The hypothetical value can also come from previous data. For example, compare whether the mean systolic blood pressure differs from 135, a value determined in a previous study.

1. Choose data entry format

Caution: Changing format will erase your data.

2. Specify the hypothetical mean value

3. enter data, 4. view the results, learn more about the one sample t test.

In this article you will learn the requirements and assumptions of a one sample t test, how to format and interpret the results of a one sample t test, and when to use different types of t tests.

One sample t test: Overview

The one sample t test, also referred to as a single sample t test, is a statistical hypothesis test used to determine whether the mean calculated from sample data collected from a single group is different from a designated value specified by the researcher. This designated value does not come from the data itself, but is an external value chosen for scientific reasons. Often, this designated value is a mean previously established in a population, a standard value of interest, or a mean concluded from other studies. Like all hypothesis testing, the one sample t test determines if there is enough evidence reject the null hypothesis (H0) in favor of an alternative hypothesis (H1). The null hypothesis for a one sample t test can be stated as: "The population mean equals the specified mean value." The alternative hypothesis for a one sample t test can be stated as: "The population mean is different from the specified mean value."

The one sample t test differs from most statistical hypothesis tests because it does not compare two separate groups or look at a relationship between two variables. It is a straightforward comparison between data gathered on a single variable from one population and a specified value defined by the researcher. The one sample t test can be used to look for a difference in only one direction from the standard value (a one-tailed t test ) or can be used to look for a difference in either direction from the standard value (a two-tailed t test ).

Requirements and Assumptions for a one sample t test

A one sample t test should be used only when data has been collected on one variable for a single population and there is no comparison being made between groups. For a valid one sample t test analysis, data values must be all of the following:

The one sample t test assumes that all "errors" in the data are independent. The term "error" refers to the difference between each value and the group mean. The results of a t test only make sense when the scatter is random - that whatever factor caused a value to be too high or too low affects only that one value. Prism cannot test this assumption, but there are graphical ways to explore data to verify this assumption is met.

A t test is only appropriate to apply in situations where data represent variables that are continuous measurements. As they rely on the calculation of a mean value, variables that are categorical should not be analyzed using a t test.

The results of a t test should be based on a random sample and only be generalized to the larger population from which samples were drawn.

As with all parametric hypothesis testing, the one sample t test assumes that you have sampled your data from a population that follows a normal (or Gaussian) distribution. While this assumption is not as important with large samples, it is important with small sample sizes, especially less than 10. If your data do not come from a Gaussian distribution , there are three options to accommodate this. One option is to transform the values to make the distribution more Gaussian, perhaps by transforming all values to their reciprocals or logarithms. Another choice is to use the Wilcoxon signed rank nonparametric test instead of the t test. A final option is to use the t test anyway, knowing that the t test is fairly robust to departures from a Gaussian distribution with large samples.

How to format a one sample t test

Ideally, data for a one sample t test should be collected and entered as a single column from which a mean value can be easily calculated. If data is entered on a table with multiple subcolumns, Prism requires one of the following choices to be selected to perform the analysis:

• Each subcolumn of data can be analyzed separately
• An average of the values in the columns across each row can be calculated, and the analysis conducted on this new stack of means, or
• All values in all columns can be treated as one sample of data (paying no attention to which row or column any values are in).

How the one sample t test calculator works

Prism calculates the t ratio by dividing the difference between the actual and hypothetical means by the standard error of the actual mean. The equation is written as follows, where x is the calculated mean, μ is the hypothetical mean (specified value), S is the standard deviation of the sample, and n is the sample size:

A p value is computed based on the calculated t ratio and the numbers of degrees of freedom present (which equals sample size minus 1). The one sample t test calculator assumes it is a two-tailed one sample t test, meaning you are testing for a difference in either direction from the specified value.

How to interpret results of a one sample t test

As discussed, a one sample t test compares the mean of a single column of numbers against a hypothetical mean. This hypothetical mean can be based upon a specific standard or other external prediction. The test produces a P value which requires careful interpretation.

The p value answers this question: If the data were sampled from a Gaussian population with a mean equal to the hypothetical value you entered, what is the chance of randomly selecting N data points and finding a mean as far (or further) from the hypothetical value as observed here?

If the p value is large (usually defined to mean greater than 0.05), the data do not give you any reason to conclude that the population mean differs from the designated value to which it has been compared. This is not the same as saying that the true mean equals the hypothetical value, but rather states that there is no evidence of a difference. Thus, we cannot reject the null hypothesis (H0).

If the p value is small (usually defined to mean less than or equal to 0.05), then it is unlikely that the discrepancy observed between the sample mean and hypothetical mean is due to a coincidence arising from random sampling. There is evidence to reject the idea that the difference is coincidental and conclude instead that the population has a mean that is different from the hypothetical value to which it has been compared. The difference is statistically significant, and the null hypothesis is therefore rejected.

If the null hypothesis is rejected, the question of whether the difference is scientifically important still remains. The confidence interval can be a useful tool in answering this question. Prism reports the 95% confidence interval for the difference between the actual and hypothetical mean. In interpreting these results, one can be 95% sure that this range includes the true difference. It requires scientific judgment to determine if this difference is truly meaningful.

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When to use different types of t tests

There are three types of t tests which can be used for hypothesis testing:

• Independent two-sample (or unpaired) t test
• Paired sample t test

As described, a one sample t test should be used only when data has been collected on one variable for a single population and there is no comparison being made between groups. It only applies when the mean value for data is intended to be compared to a fixed and defined number.

In most cases involving data analysis, however, there are multiple groups of data either representing different populations being compared, or the same population being compared at different times or conditions. For these situations, it is not appropriate to use a one sample t test. Other types of t tests are appropriate for these specific circumstances:

Independent Two-Sample t test (Unpaired t test)

The independent sample t test, also referred to as the unpaired t test, is used to compare the means of two different samples. The independent two-sample t test comes in two different forms:

• the standard Student's t test, which assumes that the variance of the two groups are equal.
• the Welch's t test , which is less restrictive compared to the original Student's test. This is the test where you do not assume that the variance is the same in the two groups, which results in fractional degrees of freedom.

The two methods give very similar results when the sample sizes are equal and the variances are similar.

Paired Sample t test

The paired sample t test is used to compare the means of two related groups of samples. Put into other words, it is used in a situation where you have two values (i.e., a pair of values) for the same group of samples. Often these two values are measured from the same samples either at two different times, under two different conditions, or after a specific intervention.

You can perform multiple independent two-sample comparison tests simultaneously in Prism. Select from parametric and nonparametric tests and specify if the data are unpaired or paired. Try performing a t test with a 30-day free trial of Prism .

Watch this video to learn how to choose between a paired and unpaired t test.

Example of how to apply the appropriate t test

"Alkaline" labeled bottled drinking water has become fashionable over the past several years. Imagine we have collected a random sample of 30 bottles of "alkaline" drinking water from a number of different stores to represent the population of "alkaline" bottled water for a particular brand available to the general consumer. The labels on each of the bottles claim that the pH of the "alkaline" water is 8.5. A laboratory then proceeds to measure the exact pH of the water in each bottle.

Table 1: pH of water in random sample of "alkaline bottled water"

If you look at the table above, you see that some bottles have a pH measured to be lower than 8.5, while other bottles have a pH measured to be higher. What can the data tell us about the actual pH levels found in this brand of "alkaline" water bottles marketed to the public as having a pH of 8.5? Statistical hypothesis testing provides a sound method to evaluate this question. Which specific test to use, however, depends on the specific question being asked.

Is a t test appropriate to apply to this data?

Let's start by asking: Is a t test an appropriate method to analyze this set of pH data? The following list reviews the requirements and assumptions for using a t test:

• Independent sampling : In an independent sample t test, the data values are independent. The pH of one bottle of water does not depend on the pH of any other water bottle. (An example of dependent values would be if you collected water bottles from a single production lot. A sample from a single lot is representative only of that lot, not of alkaline bottled water in general).
• Continuous variable : The data values are pH levels, which are numerical measurements that are continuous.
• Random sample : We assume the water bottles are a simple random sample from the population of "alkaline" water bottles produced by this brand as they are a mix of many production lots.
• Normal distribution : We assume the population from which we collected our samples has pH levels that are normally distributed. To verify this, we should visualize the data graphically. The figure below shows a histogram for the pH measurements of the water bottles. From a quick look at the histogram, we see that there are no unusual points, or outliers. The data look roughly bell-shaped, so our assumption of a normal distribution seems reasonable. The QQ plot can also be used to graphically assess normality and is the preferred choice when the sample size is small.

Based upon these features and assumptions being met, we can conclude that a t test is an appropriate method to be applied to this set of data.

Which t test is appropriate to use?

The next decision is which t test to apply, and this depends on the exact question we would like our analysis to answer. This example illustrates how each type of t test could be chosen for a specific analysis, and why the one sample t test is the correct choice to determine if the measured pH of the bottled water samples match the advertised pH of 8.5.

We could be interested in determining whether a certain characteristic of a water bottle is associated with having a higher or lower pH, such as whether bottles are glass or plastic. For this questions, we would effectively be dividing the bottles into 2 separate groups and comparing the means of the pH between the 2 groups. For this analysis, we would elect to use a two sample t test because we are comparing the means of two independent groups.

We could also be interested in learning if pH is affected by a water bottle being opened and exposed to the air for a week. In this case, each original sample would be tested for pH level after a week had elapsed and the water had been exposed to the air, creating a second set of sample data. To evaluate whether this exposure affected pH, we would again be comparing two different groups of data, but this time the data are in paired samples each having an original pH measurement and a second measurement from after the week of exposure to the open air. For this analysis, it is appropriate to use a paired t test so that data for each bottle is assembled in rows, and the change in pH is considered bottle by bottle.

Returning to the original question we set out to answer-whether bottled water that is advertised to have a pH of 8.5 actually meets this claim-it is now clear that neither an independent two sample t test or a paired t test would be appropriate. In this case, all 30 pH measurements are sampled from one group representing bottled drinking water labeled "alkaline" available to the general consumer. We wish to compare this measured mean with an expected advertised value of 8.5. This is the exact situation for which one should employ a one sample t test!

From a quick look at the descriptive statistics, we see that the mean of the sample measurements is 8.513, slightly above 8.5. Does this average from our sample of 30 bottles validate the advertised claim of pH 8.5? By applying Prism's one sample t test analysis to this data set, we will get results by which we can evaluate whether the null hypothesis (that there is no difference between the mean pH level in the water bottles and the pH level advertised on the bottles) should be accepted or rejected.

How to Perform a One Sample T Test in Prism

In prior versions of Prism, the one sample t test and the Wilcoxon rank sum tests were computed as part of Prism's Column Statistics analysis. Now, starting with Prism 8, performing one sample t tests is even easier with a separate analysis in Prism.

Steps to perform a one sample t test in Prism

• Create a Column data table.
• Enter each data set in a single Y column so all values from each group are stacked into a column. Prism will perform a one sample t test (or Wilcoxon rank sum test) on each column you enter.
• Click Analyze, look in the list of Column analyses, and choose one sample t test and Wilcoxon test.

It's that simple! Prism streamlines your t test analysis so you can make more accurate and more informed data interpretations. Start your 30-day free trial of Prism and try performing your first one sample t test in Prism.

Watch this video for a step-by-step tutorial on how to perform a t test in Prism.

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T-Test Calculator for 2 Independent Means

This simple t -test calculator, provides full details of the t-test calculation, including sample mean, sum of squares and standard deviation.

Further Information

A t -test is used when you're looking at a numerical variable - for example, height - and then comparing the averages of two separate populations or groups (e.g., males and females).

Requirements

• Two independent samples
• Data should be normally distributed
• The two samples should have the same variance

Null Hypothesis

H0: u1 - u2 = 0, where u1 is the mean of first population and u2 the mean of the second.

As above, the null hypothesis tends to be that there is no difference between the means of the two populations; or, more formally, that the difference is zero (so, for example, that there is no difference between the average heights of two populations of males and females).

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.

P-value Calculator

Statistical significance calculator to easily calculate the p-value and determine whether the difference between two proportions or means (independent groups) is statistically significant. T-test calculator & z-test calculator to compute the Z-score or T-score for inference about absolute or relative difference (percentage change, percent effect). Suitable for analysis of simple A/B tests.

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• Using the p-value calculator
• What is "p-value" and "significance level"
• P-value formula
• Why do we need a p-value?
• How to interpret a statistically significant result / low p-value
• P-value and significance for relative difference in means or proportions

    Using the p-value calculator

This statistical significance calculator allows you to perform a post-hoc statistical evaluation of a set of data when the outcome of interest is difference of two proportions (binomial data, e.g. conversion rate or event rate) or difference of two means (continuous data, e.g. height, weight, speed, time, revenue, etc.). You can use a Z-test (recommended) or a T-test to find the observed significance level (p-value statistic). The Student's T-test is recommended mostly for very small sample sizes, e.g. n < 30. In order to avoid type I error inflation which might occur with unequal variances the calculator automatically applies the Welch's T-test instead of Student's T-test if the sample sizes differ significantly or if one of them is less than 30 and the sampling ratio is different than one.

If entering proportions data, you need to know the sample sizes of the two groups as well as the number or rate of events. These can be entered as proportions (e.g. 0.10), percentages (e.g. 10%) or just raw numbers of events (e.g. 50).

If entering means data, simply copy/paste or type in the raw data, each observation separated by comma, space, new line or tab. Copy-pasting from a Google or Excel spreadsheet works fine.

The p-value calculator will output : p-value, significance level, T-score or Z-score (depending on the choice of statistical hypothesis test), degrees of freedom, and the observed difference. For means data it will also output the sample sizes, means, and pooled standard error of the mean. The p-value is for a one-sided hypothesis (one-tailed test), allowing you to infer the direction of the effect (more on one vs. two-tailed tests ). However, the probability value for the two-sided hypothesis (two-tailed p-value) is also calculated and displayed, although it should see little to no practical applications.

Warning: You must have fixed the sample size / stopping time of your experiment in advance, otherwise you will be guilty of optional stopping (fishing for significance) which will inflate the type I error of the test rendering the statistical significance level unusable. Also, you should not use this significance calculator for comparisons of more than two means or proportions, or for comparisons of two groups based on more than one metric. If a test involves more than one treatment group or more than one outcome variable you need a more advanced tool which corrects for multiple comparisons and multiple testing. This statistical calculator might help.

    What is "p-value" and "significance level"

The p-value is a heavily used test statistic that quantifies the uncertainty of a given measurement, usually as a part of an experiment, medical trial, as well as in observational studies. By definition, it is inseparable from inference through a Null-Hypothesis Statistical Test (NHST) . In it we pose a null hypothesis reflecting the currently established theory or a model of the world we don't want to dismiss without solid evidence (the tested hypothesis), and an alternative hypothesis: an alternative model of the world. For example, the statistical null hypothesis could be that exposure to ultraviolet light for prolonged periods of time has positive or neutral effects regarding developing skin cancer, while the alternative hypothesis can be that it has a negative effect on development of skin cancer.

In this framework a p-value is defined as the probability of observing the result which was observed, or a more extreme one, assuming the null hypothesis is true . In notation this is expressed as:

p(x 0 ) = Pr(d(X) > d(x 0 ); H 0 )

where x 0 is the observed data (x 1 ,x 2 ...x n ), d is a special function (statistic, e.g. calculating a Z-score), X is a random sample (X 1 ,X 2 ...X n ) from the sampling distribution of the null hypothesis. This equation is used in this p-value calculator and can be visualized as such:

Therefore the p-value expresses the probability of committing a type I error : rejecting the null hypothesis if it is in fact true. See below for a full proper interpretation of the p-value statistic .

Another way to think of the p-value is as a more user-friendly expression of how many standard deviations away from the normal a given observation is. For example, in a one-tailed test of significance for a normally-distributed variable like the difference of two means, a result which is 1.6448 standard deviations away (1.6448σ) results in a p-value of 0.05.

The term "statistical significance" or "significance level" is often used in conjunction to the p-value, either to say that a result is "statistically significant", which has a specific meaning in statistical inference ( see interpretation below ), or to refer to the percentage representation the level of significance: (1 - p value), e.g. a p-value of 0.05 is equivalent to significance level of 95% (1 - 0.05 * 100). A significance level can also be expressed as a T-score or Z-score, e.g. a result would be considered significant only if the Z-score is in the critical region above 1.96 (equivalent to a p-value of 0.025).

    P-value formula

There are different ways to arrive at a p-value depending on the assumption about the underlying distribution. This tool supports two such distributions: the Student's T-distribution and the normal Z-distribution (Gaussian) resulting in a T test and a Z test, respectively.

In both cases, to find the p-value start by estimating the variance and standard deviation, then derive the standard error of the mean, after which a standard score is found using the formula [2] :

X (read "X bar") is the arithmetic mean of the population baseline or the control, μ 0 is the observed mean / treatment group mean, while σ x is the standard error of the mean (SEM, or standard deviation of the error of the mean).

When calculating a p-value using the Z-distribution the formula is Φ(Z) or Φ(-Z) for lower and upper-tailed tests, respectively. Φ is the standard normal cumulative distribution function and a Z-score is computed. In this mode the tool functions as a Z score calculator.

When using the T-distribution the formula is T n (Z) or T n (-Z) for lower and upper-tailed tests, respectively. T n is the cumulative distribution function for a T-distribution with n degrees of freedom and so a T-score is computed. Selecting this mode makes the tool behave as a T test calculator.

The population standard deviation is often unknown and is thus estimated from the samples, usually from the pooled samples variance. Knowing or estimating the standard deviation is a prerequisite for using a significance calculator. Note that differences in means or proportions are normally distributed according to the Central Limit Theorem (CLT) hence a Z-score is the relevant statistic for such a test.

    Why do we need a p-value?

If you are in the sciences, it is often a requirement by scientific journals. If you apply in business experiments (e.g. A/B testing) it is reported alongside confidence intervals and other estimates. However, what is the utility of p-values and by extension that of significance levels?

First, let us define the problem the p-value is intended to solve. People need to share information about the evidential strength of data that can be easily understood and easily compared between experiments. The picture below represents, albeit imperfectly, the results of two simple experiments, each ending up with the control with 10% event rate treatment group at 12% event rate.

However, it is obvious that the evidential input of the data is not the same, demonstrating that communicating just the observed proportions or their difference (effect size) is not enough to estimate and communicate the evidential strength of the experiment. In order to fully describe the evidence and associated uncertainty , several statistics need to be communicated, for example, the sample size, sample proportions and the shape of the error distribution. Their interaction is not trivial to understand, so communicating them separately makes it very difficult for one to grasp what information is present in the data. What would you infer if told that the observed proportions are 0.1 and 0.12 (e.g. conversion rate of 10% and 12%), the sample sizes are 10,000 users each, and the error distribution is binomial?

Instead of communicating several statistics, a single statistic was developed that communicates all the necessary information in one piece: the p-value . A p-value was first derived in the late 18-th century by Pierre-Simon Laplace, when he observed data about a million births that showed an excess of boys, compared to girls. Using the calculation of significance he argued that the effect was real but unexplained at the time. We know this now to be true and there are several explanations for the phenomena coming from evolutionary biology. Statistical significance calculations were formally introduced in the early 20-th century by Pearson and popularized by Sir Ronald Fisher in his work, most notably "The Design of Experiments" (1935) [1] in which p-values were featured extensively. In business settings significance levels and p-values see widespread use in process control and various business experiments (such as online A/B tests, i.e. as part of conversion rate optimization, marketing optimization, etc.).

    How to interpret a statistically significant result / low p-value

Saying that a result is statistically significant means that the p-value is below the evidential threshold (significance level) decided for the statistical test before it was conducted. For example, if observing something which would only happen 1 out of 20 times if the null hypothesis is true is considered sufficient evidence to reject the null hypothesis, the threshold will be 0.05. In such case, observing a p-value of 0.025 would mean that the result is interpreted as statistically significant.

But what does that really mean? What inference can we make from seeing a result which was quite improbable if the null was true?

Observing any given low p-value can mean one of three things [3] :

• There is a true effect from the tested treatment or intervention.
• There is no true effect, but we happened to observe a rare outcome. The lower the p-value, the rarer (less likely, less probable) the outcome.
• The statistical model is invalid (does not reflect reality).

Obviously, one can't simply jump to conclusion 1.) and claim it with one hundred percent certainty, as this would go against the whole idea of the p-value and statistical significance. In order to use p-values as a part of a decision process external factors part of the experimental design process need to be considered which includes deciding on the significance level (threshold), sample size and power (power analysis), and the expected effect size, among other things. If you are happy going forward with this much (or this little) uncertainty as is indicated by the p-value calculation suggests, then you have some quantifiable guarantees related to the effect and future performance of whatever you are testing, e.g. the efficacy of a vaccine or the conversion rate of an online shopping cart.

Note that it is incorrect to state that a Z-score or a p-value obtained from any statistical significance calculator tells how likely it is that the observation is "due to chance" or conversely - how unlikely it is to observe such an outcome due to "chance alone". P-values are calculated under specified statistical models hence 'chance' can be used only in reference to that specific data generating mechanism and has a technical meaning quite different from the colloquial one. For a deeper take on the p-value meaning and interpretation, including common misinterpretations, see: definition and interpretation of the p-value in statistics .

    P-value and significance for relative difference in means or proportions

When comparing two independent groups and the variable of interest is the relative (a.k.a. relative change, relative difference, percent change, percentage difference), as opposed to the absolute difference between the two means or proportions, the standard deviation of the variable is different which compels a different way of calculating p-values [5] . The need for a different statistical test is due to the fact that in calculating relative difference involves performing an additional division by a random variable: the event rate of the control during the experiment which adds more variance to the estimation and the resulting statistical significance is usually higher (the result will be less statistically significant). What this means is that p-values from a statistical hypothesis test for absolute difference in means would nominally meet the significance level, but they will be inadequate given the statistical inference for the hypothesis at hand.

In simulations I performed the difference in p-values was about 50% of nominal: a 0.05 p-value for absolute difference corresponded to probability of about 0.075 of observing the relative difference corresponding to the observed absolute difference. Therefore, if you are using p-values calculated for absolute difference when making an inference about percentage difference, you are likely reporting error rates which are about 50% of the actual, thus significantly overstating the statistical significance of your results and underestimating the uncertainty attached to them.

In short - switching from absolute to relative difference requires a different statistical hypothesis test. With this calculator you can avoid the mistake of using the wrong test simply by indicating the inference you want to make.

    References

1 Fisher R.A. (1935) – "The Design of Experiments", Edinburgh: Oliver & Boyd

2 Mayo D.G., Spanos A. (2010) – "Error Statistics", in P. S. Bandyopadhyay & M. R. Forster (Eds.), Philosophy of Statistics, (7, 152–198). Handbook of the Philosophy of Science . The Netherlands: Elsevier.

3 Georgiev G.Z. (2017) "Statistical Significance in A/B Testing – a Complete Guide", [online] https://blog.analytics-toolkit.com/2017/statistical-significance-ab-testing-complete-guide/ (accessed Apr 27, 2018)

4 Mayo D.G., Spanos A. (2006) – "Severe Testing as a Basic Concept in a Neyman–Pearson Philosophy of Induction", British Society for the Philosophy of Science , 57:323-357

5 Georgiev G.Z. (2018) "Confidence Intervals & P-values for Percent Change / Relative Difference", [online] https://blog.analytics-toolkit.com/2018/confidence-intervals-p-values-percent-change-relative-difference/ (accessed May 20, 2018)

Cite this calculator & page

If you'd like to cite this online calculator resource and information as provided on the page, you can use the following citation: Georgiev G.Z., "P-value Calculator" , [online] Available at: https://www.gigacalculator.com/calculators/p-value-significance-calculator.php URL [Accessed Date: 30 Apr, 2024].

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27: Hypothesis Test for a Population Mean Given Statistics Calculator

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• Page ID 8349

• Larry Green
• Lake Tahoe Community College

hypothesis test for a population mean given statistics calculator

Select if the population standard deviation, \(\sigma\), is known or unknown. Then fill in the standard deviation, the sample mean, \(\bar{x}\), the sample size, \(n\), the hypothesized population mean \(\mu_0\), and indicate if the test is left tailed, <, right tailed, >, or two tailed, \(\neq\).  Then hit "Calculate" and the test statistic and p-Value will be calculated for you.

Critical Value Calculator

How to use critical value calculator, what is a critical value, critical value definition, how to calculate critical values, z critical values, t critical values, chi-square critical values (χ²), f critical values, behind the scenes of the critical value calculator.

Welcome to the critical value calculator! Here you can quickly determine the critical value(s) for two-tailed tests, as well as for one-tailed tests. It works for most common distributions in statistical testing: the standard normal distribution N(0,1) (that is when you have a Z-score), t-Student, chi-square, and F-distribution .

What is a critical value? And what is the critical value formula? Scroll down – we provide you with the critical value definition and explain how to calculate critical values in order to use them to construct rejection regions (also known as critical regions).

The critical value calculator is your go-to tool for swiftly determining critical values in statistical tests, be it one-tailed or two-tailed. To effectively use the calculator, follow these steps:

In the first field, input the distribution of your test statistic under the null hypothesis: is it a standard normal N (0,1), t-Student, chi-squared, or Snedecor's F? If you are not sure, check the sections below devoted to those distributions, and try to localize the test you need to perform.

In the field What type of test? choose the alternative hypothesis : two-tailed, right-tailed, or left-tailed.

If needed, specify the degrees of freedom of the test statistic's distribution. If you need more clarification, check the description of the test you are performing. You can learn more about the meaning of this quantity in statistics from the degrees of freedom calculator .

Set the significance level, α \alpha α . By default, we pre-set it to the most common value, 0.05, but you can adjust it to your needs.

The critical value calculator will display your critical value(s) and the rejection region(s).

Click the advanced mode if you need to increase the precision with which the critical values are computed.

For example, let's envision a scenario where you are conducting a one-tailed hypothesis test using a t-Student distribution with 15 degrees of freedom. You have opted for a right-tailed test and set a significance level (α) of 0.05. The results indicate that the critical value is 1.7531, and the critical region is (1.7531, ∞). This implies that if your test statistic exceeds 1.7531, you will reject the null hypothesis at the 0.05 significance level.

👩‍🏫 Want to learn more about critical values? Keep reading!

In hypothesis testing, critical values are one of the two approaches which allow you to decide whether to retain or reject the null hypothesis. The other approach is to calculate the p-value (for example, using the p-value calculator ).

The critical value approach consists of checking if the value of the test statistic generated by your sample belongs to the so-called rejection region , or critical region , which is the region where the test statistic is highly improbable to lie . A critical value is a cut-off value (or two cut-off values in the case of a two-tailed test) that constitutes the boundary of the rejection region(s). In other words, critical values divide the scale of your test statistic into the rejection region and the non-rejection region.

Once you have found the rejection region, check if the value of the test statistic generated by your sample belongs to it :

• If so, it means that you can reject the null hypothesis and accept the alternative hypothesis; and
• If not, then there is not enough evidence to reject H 0 .

But how to calculate critical values? First of all, you need to set a significance level , α \alpha α , which quantifies the probability of rejecting the null hypothesis when it is actually correct. The choice of α is arbitrary; in practice, we most often use a value of 0.05 or 0.01. Critical values also depend on the alternative hypothesis you choose for your test , elucidated in the next section .

To determine critical values, you need to know the distribution of your test statistic under the assumption that the null hypothesis holds. Critical values are then points with the property that the probability of your test statistic assuming values at least as extreme at those critical values is equal to the significance level α . Wow, quite a definition, isn't it? Don't worry, we'll explain what it all means.

First, let us point out it is the alternative hypothesis that determines what "extreme" means. In particular, if the test is one-sided, then there will be just one critical value; if it is two-sided, then there will be two of them: one to the left and the other to the right of the median value of the distribution.

Critical values can be conveniently depicted as the points with the property that the area under the density curve of the test statistic from those points to the tails is equal to α \alpha α :

Left-tailed test: the area under the density curve from the critical value to the left is equal to α \alpha α ;

Right-tailed test: the area under the density curve from the critical value to the right is equal to α \alpha α ; and

Two-tailed test: the area under the density curve from the left critical value to the left is equal to α / 2 \alpha/2 α /2 , and the area under the curve from the right critical value to the right is equal to α / 2 \alpha/2 α /2 as well; thus, total area equals α \alpha α .

As you can see, finding the critical values for a two-tailed test with significance α \alpha α boils down to finding both one-tailed critical values with a significance level of α / 2 \alpha/2 α /2 .

The formulae for the critical values involve the quantile function , Q Q Q , which is the inverse of the cumulative distribution function ( c d f \mathrm{cdf} cdf ) for the test statistic distribution (calculated under the assumption that H 0 holds!): Q = c d f − 1 Q = \mathrm{cdf}^{-1} Q = cdf − 1 .

Once we have agreed upon the value of α \alpha α , the critical value formulae are the following:

• Left-tailed test :
• Right-tailed test :
• Two-tailed test :

In the case of a distribution symmetric about 0 , the critical values for the two-tailed test are symmetric as well:

Unfortunately, the probability distributions that are the most widespread in hypothesis testing have somewhat complicated c d f \mathrm{cdf} cdf formulae. To find critical values by hand, you would need to use specialized software or statistical tables. In these cases, the best option is, of course, our critical value calculator! 😁

Use the Z (standard normal) option if your test statistic follows (at least approximately) the standard normal distribution N(0,1) .

In the formulae below, u u u denotes the quantile function of the standard normal distribution N(0,1):

Left-tailed Z critical value: u ( α ) u(\alpha) u ( α )

Right-tailed Z critical value: u ( 1 − α ) u(1-\alpha) u ( 1 − α )

Two-tailed Z critical value: ± u ( 1 − α / 2 ) \pm u(1- \alpha/2) ± u ( 1 − α /2 )

Check out Z-test calculator to learn more about the most common Z-test used on the population mean. There are also Z-tests for the difference between two population means, in particular, one between two proportions.

Use the t-Student option if your test statistic follows the t-Student distribution . This distribution is similar to N(0,1) , but its tails are fatter – the exact shape depends on the number of degrees of freedom . If this number is large (>30), which generically happens for large samples, then the t-Student distribution is practically indistinguishable from N(0,1). Check our t-statistic calculator to compute the related test statistic.

In the formulae below, Q t , d Q_{\text{t}, d} Q t , d ​ is the quantile function of the t-Student distribution with d d d degrees of freedom:

Left-tailed t critical value: Q t , d ( α ) Q_{\text{t}, d}(\alpha) Q t , d ​ ( α )

Right-tailed t critical value: Q t , d ( 1 − α ) Q_{\text{t}, d}(1 - \alpha) Q t , d ​ ( 1 − α )

Two-tailed t critical values: ± Q t , d ( 1 − α / 2 ) \pm Q_{\text{t}, d}(1 - \alpha/2) ± Q t , d ​ ( 1 − α /2 )

Visit the t-test calculator to learn more about various t-tests: the one for a population mean with an unknown population standard deviation , those for the difference between the means of two populations (with either equal or unequal population standard deviations), as well as about the t-test for paired samples .

Use the χ² (chi-square) option when performing a test in which the test statistic follows the χ²-distribution .

You need to determine the number of degrees of freedom of the χ²-distribution of your test statistic – below, we list them for the most commonly used χ²-tests.

Here we give the formulae for chi square critical values; Q χ 2 , d Q_{\chi^2, d} Q χ 2 , d ​ is the quantile function of the χ²-distribution with d d d degrees of freedom:

Left-tailed χ² critical value: Q χ 2 , d ( α ) Q_{\chi^2, d}(\alpha) Q χ 2 , d ​ ( α )

Right-tailed χ² critical value: Q χ 2 , d ( 1 − α ) Q_{\chi^2, d}(1 - \alpha) Q χ 2 , d ​ ( 1 − α )

Two-tailed χ² critical values: Q χ 2 , d ( α / 2 ) Q_{\chi^2, d}(\alpha/2) Q χ 2 , d ​ ( α /2 ) and Q χ 2 , d ( 1 − α / 2 ) Q_{\chi^2, d}(1 - \alpha/2) Q χ 2 , d ​ ( 1 − α /2 )

Several different tests lead to a χ²-score:

Goodness-of-fit test : does the empirical distribution agree with the expected distribution?

This test is right-tailed . Its test statistic follows the χ²-distribution with k − 1 k - 1 k − 1 degrees of freedom, where k k k is the number of classes into which the sample is divided.

Independence test : is there a statistically significant relationship between two variables?

This test is also right-tailed , and its test statistic is computed from the contingency table. There are ( r − 1 ) ( c − 1 ) (r - 1)(c - 1) ( r − 1 ) ( c − 1 ) degrees of freedom, where r r r is the number of rows, and c c c is the number of columns in the contingency table.

Test for the variance of normally distributed data : does this variance have some pre-determined value?

This test can be one- or two-tailed! Its test statistic has the χ²-distribution with n − 1 n - 1 n − 1 degrees of freedom, where n n n is the sample size.

Finally, choose F (Fisher-Snedecor) if your test statistic follows the F-distribution . This distribution has a pair of degrees of freedom .

Let us see how those degrees of freedom arise. Assume that you have two independent random variables, X X X and Y Y Y , that follow χ²-distributions with d 1 d_1 d 1 ​ and d 2 d_2 d 2 ​ degrees of freedom, respectively. If you now consider the ratio ( X d 1 ) : ( Y d 2 ) (\frac{X}{d_1}):(\frac{Y}{d_2}) ( d 1 ​ X ​ ) : ( d 2 ​ Y ​ ) , it turns out it follows the F-distribution with ( d 1 , d 2 ) (d_1, d_2) ( d 1 ​ , d 2 ​ ) degrees of freedom. That's the reason why we call d 1 d_1 d 1 ​ and d 2 d_2 d 2 ​ the numerator and denominator degrees of freedom , respectively.

In the formulae below, Q F , d 1 , d 2 Q_{\text{F}, d_1, d_2} Q F , d 1 ​ , d 2 ​ ​ stands for the quantile function of the F-distribution with ( d 1 , d 2 ) (d_1, d_2) ( d 1 ​ , d 2 ​ ) degrees of freedom:

Left-tailed F critical value: Q F , d 1 , d 2 ( α ) Q_{\text{F}, d_1, d_2}(\alpha) Q F , d 1 ​ , d 2 ​ ​ ( α )

Right-tailed F critical value: Q F , d 1 , d 2 ( 1 − α ) Q_{\text{F}, d_1, d_2}(1 - \alpha) Q F , d 1 ​ , d 2 ​ ​ ( 1 − α )

Two-tailed F critical values: Q F , d 1 , d 2 ( α / 2 ) Q_{\text{F}, d_1, d_2}(\alpha/2) Q F , d 1 ​ , d 2 ​ ​ ( α /2 ) and Q F , d 1 , d 2 ( 1 − α / 2 ) Q_{\text{F}, d_1, d_2}(1 -\alpha/2) Q F , d 1 ​ , d 2 ​ ​ ( 1 − α /2 )

Here we list the most important tests that produce F-scores: each of them is right-tailed .

ANOVA : tests the equality of means in three or more groups that come from normally distributed populations with equal variances. There are ( k − 1 , n − k ) (k - 1, n - k) ( k − 1 , n − k ) degrees of freedom, where k k k is the number of groups, and n n n is the total sample size (across every group).

Overall significance in regression analysis . The test statistic has ( k − 1 , n − k ) (k - 1, n - k) ( k − 1 , n − k ) degrees of freedom, where n n n is the sample size, and k k k is the number of variables (including the intercept).

Compare two nested regression models . The test statistic follows the F-distribution with ( k 2 − k 1 , n − k 2 ) (k_2 - k_1, n - k_2) ( k 2 ​ − k 1 ​ , n − k 2 ​ ) degrees of freedom, where k 1 k_1 k 1 ​ and k 2 k_2 k 2 ​ are the number of variables in the smaller and bigger models, respectively, and n n n is the sample size.

The equality of variances in two normally distributed populations . There are ( n − 1 , m − 1 ) (n - 1, m - 1) ( n − 1 , m − 1 ) degrees of freedom, where n n n and m m m are the respective sample sizes.

I'm Anna, the mastermind behind the critical value calculator and a PhD in mathematics from Jagiellonian University .

The idea for creating the tool originated from my experiences in teaching and research. Recognizing the need for a tool that simplifies the critical value determination process across various statistical distributions, I built a user-friendly calculator accessible to both students and professionals. After publishing the tool, I soon found myself using the calculator in my research and as a teaching aid.

Trust in this calculator is paramount to me. Each tool undergoes a rigorous review process , with peer-reviewed insights from experts and meticulous proofreading by native speakers. This commitment to accuracy and reliability ensures that users can be confident in the content. Please check the Editorial Policies page for more details on our standards.

What is a Z critical value?

A Z critical value is the value that defines the critical region in hypothesis testing when the test statistic follows the standard normal distribution . If the value of the test statistic falls into the critical region, you should reject the null hypothesis and accept the alternative hypothesis.

Get science-backed answers as you write with Paperpal's Research feature

How to Write a Hypothesis? Types and Examples 

All research studies involve the use of the scientific method, which is a mathematical and experimental technique used to conduct experiments by developing and testing a hypothesis or a prediction about an outcome. Simply put, a hypothesis is a suggested solution to a problem. It includes elements that are expressed in terms of relationships with each other to explain a condition or an assumption that hasn’t been verified using facts. 1 The typical steps in a scientific method include developing such a hypothesis, testing it through various methods, and then modifying it based on the outcomes of the experiments.  

A research hypothesis can be defined as a specific, testable prediction about the anticipated results of a study. 2 Hypotheses help guide the research process and supplement the aim of the study. After several rounds of testing, hypotheses can help develop scientific theories. 3 Hypotheses are often written as if-then statements. 

Here are two hypothesis examples: 

Dandelions growing in nitrogen-rich soils for two weeks develop larger leaves than those in nitrogen-poor soils because nitrogen stimulates vegetative growth. 4  

If a company offers flexible work hours, then their employees will be happier at work. 5  

Table of Contents

• What is a hypothesis? 
• Types of hypotheses 
• Characteristics of a hypothesis 
• Functions of a hypothesis 
• How to write a hypothesis 
• Hypothesis examples 
• Frequently asked questions 

What is a hypothesis?

A hypothesis expresses an expected relationship between variables in a study and is developed before conducting any research. Hypotheses are not opinions but rather are expected relationships based on facts and observations. They help support scientific research and expand existing knowledge. An incorrectly formulated hypothesis can affect the entire experiment leading to errors in the results so it’s important to know how to formulate a hypothesis and develop it carefully.

A few sources of a hypothesis include observations from prior studies, current research and experiences, competitors, scientific theories, and general conditions that can influence people. Figure 1 depicts the different steps in a research design and shows where exactly in the process a hypothesis is developed. 4  

There are seven different types of hypotheses—simple, complex, directional, nondirectional, associative and causal, null, and alternative. 

Types of hypotheses

The seven types of hypotheses are listed below: 5 , 6,7  

• Simple : Predicts the relationship between a single dependent variable and a single independent variable. 

Example: Exercising in the morning every day will increase your productivity.  

• Complex : Predicts the relationship between two or more variables. 

Example: Spending three hours or more on social media daily will negatively affect children’s mental health and productivity, more than that of adults.  

• Directional : Specifies the expected direction to be followed and uses terms like increase, decrease, positive, negative, more, or less. 

Example: The inclusion of intervention X decreases infant mortality compared to the original treatment.  

• Non-directional : Does not predict the exact direction, nature, or magnitude of the relationship between two variables but rather states the existence of a relationship. This hypothesis may be used when there is no underlying theory or if findings contradict prior research. 

Example: Cats and dogs differ in the amount of affection they express.  

• Associative and causal : An associative hypothesis suggests an interdependency between variables, that is, how a change in one variable changes the other.  

Example: There is a positive association between physical activity levels and overall health.  

A causal hypothesis, on the other hand, expresses a cause-and-effect association between variables. 

Example: Long-term alcohol use causes liver damage.  

• Null : Claims that the original hypothesis is false by showing that there is no relationship between the variables. 

Example: Sleep duration does not have any effect on productivity.  

• Alternative : States the opposite of the null hypothesis, that is, a relationship exists between two variables. 

Example: Sleep duration affects productivity.  

Characteristics of a hypothesis

So, what makes a good hypothesis? Here are some important characteristics of a hypothesis. 8,9  

• Testable : You must be able to test the hypothesis using scientific methods to either accept or reject the prediction. 
• Falsifiable : It should be possible to collect data that reject rather than support the hypothesis. 
• Logical : Hypotheses shouldn’t be a random guess but rather should be based on previous theories, observations, prior research, and logical reasoning. 
• Positive : The hypothesis statement about the existence of an association should be positive, that is, it should not suggest that an association does not exist. Therefore, the language used and knowing how to phrase a hypothesis is very important. 
• Clear and accurate : The language used should be easily comprehensible and use correct terminology. 
• Relevant : The hypothesis should be relevant and specific to the research question. 
• Structure : Should include all the elements that make a good hypothesis: variables, relationship, and outcome. 

Functions of a hypothesis

The following list mentions some important functions of a hypothesis: 1  

• Maintains the direction and progress of the research. 
• Expresses the important assumptions underlying the proposition in a single statement. 
• Establishes a suitable context for researchers to begin their investigation and for readers who are referring to the final report. 
• Provides an explanation for the occurrence of a specific phenomenon. 
• Ensures selection of appropriate and accurate facts necessary and relevant to the research subject. 

To summarize, a hypothesis provides the conceptual elements that complete the known data, conceptual relationships that systematize unordered elements, and conceptual meanings and interpretations that explain the unknown phenomena. 1  

How to write a hypothesis

Listed below are the main steps explaining how to write a hypothesis. 2,4,5  

• Make an observation and identify variables : Observe the subject in question and try to recognize a pattern or a relationship between the variables involved. This step provides essential background information to begin your research.  

For example, if you notice that an office’s vending machine frequently runs out of a specific snack, you may predict that more people in the office choose that snack over another. 

• Identify the main research question : After identifying a subject and recognizing a pattern, the next step is to ask a question that your hypothesis will answer.  

For example, after observing employees’ break times at work, you could ask “why do more employees take breaks in the morning rather than in the afternoon?” 

• Conduct some preliminary research to ensure originality and novelty : Your initial answer, which is your hypothesis, to the question is based on some pre-existing information about the subject. However, to ensure that your hypothesis has not been asked before or that it has been asked but rejected by other researchers you would need to gather additional information.  

For example, based on your observations you might state a hypothesis that employees work more efficiently when the air conditioning in the office is set at a lower temperature. However, during your preliminary research you find that this hypothesis was proven incorrect by a prior study. 

• Develop a general statement : After your preliminary research has confirmed the originality of your proposed answer, draft a general statement that includes all variables, subjects, and predicted outcome. The statement could be if/then or declarative.  
• Finalize the hypothesis statement : Use the PICOT model, which clarifies how to word a hypothesis effectively, when finalizing the statement. This model lists the important components required to write a hypothesis. 

P opulation: The specific group or individual who is the main subject of the research 

I nterest: The main concern of the study/research question 

C omparison: The main alternative group 

O utcome: The expected results  

T ime: Duration of the experiment 

Once you’ve finalized your hypothesis statement you would need to conduct experiments to test whether the hypothesis is true or false. 

Hypothesis examples

The following table provides examples of different types of hypotheses. 10 ,11  

Key takeaways  

Here’s a summary of all the key points discussed in this article about how to write a hypothesis. 

• A hypothesis is an assumption about an association between variables made based on limited evidence, which should be tested. 
• A hypothesis has four parts—the research question, independent variable, dependent variable, and the proposed relationship between the variables.   
• The statement should be clear, concise, testable, logical, and falsifiable. 
• There are seven types of hypotheses—simple, complex, directional, non-directional, associative and causal, null, and alternative. 
• A hypothesis provides a focus and direction for the research to progress. 
• A hypothesis plays an important role in the scientific method by helping to create an appropriate experimental design. 

Frequently asked questions

Hypotheses and research questions have different objectives and structure. The following table lists some major differences between the two. 9  

Here are a few examples to differentiate between a research question and hypothesis. 

Yes, here’s a simple checklist to help you gauge the effectiveness of your hypothesis. 9   1. When writing a hypothesis statement, check if it:  2. Predicts the relationship between the stated variables and the expected outcome.  3. Uses simple and concise language and is not wordy.  4. Does not assume readers’ knowledge about the subject.  5. Has observable, falsifiable, and testable results. 

As mentioned earlier in this article, a hypothesis is an assumption or prediction about an association between variables based on observations and simple evidence. These statements are usually generic. Research objectives, on the other hand, are more specific and dictated by hypotheses. The same hypothesis can be tested using different methods and the research objectives could be different in each case.     For example, Louis Pasteur observed that food lasts longer at higher altitudes, reasoned that it could be because the air at higher altitudes is cleaner (with fewer or no germs), and tested the hypothesis by exposing food to air cleaned in the laboratory. 12 Thus, a hypothesis is predictive—if the reasoning is correct, X will lead to Y—and research objectives are developed to test these predictions. 

Null hypothesis testing is a method to decide between two assumptions or predictions between variables (null and alternative hypotheses) in a statistical relationship in a sample. The null hypothesis, denoted as H 0 , claims that no relationship exists between variables in a population and any relationship in the sample reflects a sampling error or occurrence by chance. The alternative hypothesis, denoted as H 1 , claims that there is a relationship in the population. In every study, researchers need to decide whether the relationship in a sample occurred by chance or reflects a relationship in the population. This is done by hypothesis testing using the following steps: 13   1. Assume that the null hypothesis is true.  2. Determine how likely the sample relationship would be if the null hypothesis were true. This probability is called the p value.  3. If the sample relationship would be extremely unlikely, reject the null hypothesis and accept the alternative hypothesis. If the relationship would not be unlikely, accept the null hypothesis. 

To summarize, researchers should know how to write a good hypothesis to ensure that their research progresses in the required direction. A hypothesis is a testable prediction about any behavior or relationship between variables, usually based on facts and observation, and states an expected outcome.  

We hope this article has provided you with essential insight into the different types of hypotheses and their functions so that you can use them appropriately in your next research project. 

References  

• Dalen, DVV. The function of hypotheses in research. Proquest website. Accessed April 8, 2024. https://www.proquest.com/docview/1437933010?pq-origsite=gscholar&fromopenview=true&sourcetype=Scholarly%20Journals&imgSeq=1  
• McLeod S. Research hypothesis in psychology: Types & examples. SimplyPsychology website. Updated December 13, 2023. Accessed April 9, 2024. https://www.simplypsychology.org/what-is-a-hypotheses.html  
• Scientific method. Britannica website. Updated March 14, 2024. Accessed April 9, 2024. https://www.britannica.com/science/scientific-method  
• The hypothesis in science writing. Accessed April 10, 2024. https://berks.psu.edu/sites/berks/files/campus/HypothesisHandout_Final.pdf  
• How to develop a hypothesis (with elements, types, and examples). Indeed.com website. Updated February 3, 2023. Accessed April 10, 2024. https://www.indeed.com/career-advice/career-development/how-to-write-a-hypothesis  
• Types of research hypotheses. Excelsior online writing lab. Accessed April 11, 2024. https://owl.excelsior.edu/research/research-hypotheses/types-of-research-hypotheses/  
• What is a research hypothesis: how to write it, types, and examples. Researcher.life website. Published February 8, 2023. Accessed April 11, 2024. https://researcher.life/blog/article/how-to-write-a-research-hypothesis-definition-types-examples/  
• Developing a hypothesis. Pressbooks website. Accessed April 12, 2024. https://opentext.wsu.edu/carriecuttler/chapter/developing-a-hypothesis/  
• What is and how to write a good hypothesis in research. Elsevier author services website. Accessed April 12, 2024. https://scientific-publishing.webshop.elsevier.com/manuscript-preparation/what-how-write-good-hypothesis-research/  
• How to write a great hypothesis. Verywellmind website. Updated March 12, 2023. Accessed April 13, 2024. https://www.verywellmind.com/what-is-a-hypothesis-2795239  
• 15 Hypothesis examples. Helpfulprofessor.com Published September 8, 2023. Accessed March 14, 2024. https://helpfulprofessor.com/hypothesis-examples/ 
• Editage insights. What is the interconnectivity between research objectives and hypothesis? Published February 24, 2021. Accessed April 13, 2024. https://www.editage.com/insights/what-is-the-interconnectivity-between-research-objectives-and-hypothesis  
• Understanding null hypothesis testing. BCCampus open publishing. Accessed April 16, 2024. https://opentextbc.ca/researchmethods/chapter/understanding-null-hypothesis-testing/#:~:text=In%20null%20hypothesis%20testing%2C%20this,said%20to%20be%20statistically%20significant  

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