sample hypothesis calculator

Hypothesis Testing 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.

t-test Calculator

Table of contents

Welcome to our t-test calculator! Here you can not only easily perform one-sample t-tests , but also two-sample t-tests , as well as paired t-tests .

Do you prefer to find the p-value from t-test, or would you rather find the t-test critical values? Well, this t-test calculator can do both! 😊

What does a t-test tell you? Take a look at the text below, where we explain what actually gets tested when various types of t-tests are performed. Also, we explain when to use t-tests (in particular, whether to use the z-test vs. t-test) and what assumptions your data should satisfy for the results of a t-test to be valid. If you've ever wanted to know how to do a t-test by hand, we provide the necessary t-test formula, as well as tell you how to determine the number of degrees of freedom in a t-test.

When to use a t-test?

A t-test is one of the most popular statistical tests for location , i.e., it deals with the population(s) mean value(s).

There are different types of t-tests that you can perform:

• A one-sample t-test;
• A two-sample t-test; and
• A paired t-test.

In the next section , we explain when to use which. Remember that a t-test can only be used for one or two groups . If you need to compare three (or more) means, use the analysis of variance ( ANOVA ) method.

The t-test is a parametric test, meaning that your data has to fulfill some assumptions :

• The data points are independent; AND
• The data, at least approximately, follow a normal distribution .

If your sample doesn't fit these assumptions, you can resort to nonparametric alternatives. Visit our Mann–Whitney U test calculator or the Wilcoxon rank-sum test calculator to learn more. Other possibilities include the Wilcoxon signed-rank test or the sign test.

Which t-test?

Your choice of t-test depends on whether you are studying one group or two groups:

One sample t-test

Choose the one-sample t-test to check if the mean of a population is equal to some pre-set hypothesized value .

The average volume of a drink sold in 0.33 l cans — is it really equal to 330 ml?

The average weight of people from a specific city — is it different from the national average?

Two-sample t-test

Choose the two-sample t-test to check if the difference between the means of two populations is equal to some pre-determined value when the two samples have been chosen independently of each other.

In particular, you can use this test to check whether the two groups are different from one another .

The average difference in weight gain in two groups of people: one group was on a high-carb diet and the other on a high-fat diet.

The average difference in the results of a math test from students at two different universities.

This test is sometimes referred to as an independent samples t-test , or an unpaired samples t-test .

Paired t-test

A paired t-test is used to investigate the change in the mean of a population before and after some experimental intervention , based on a paired sample, i.e., when each subject has been measured twice: before and after treatment.

In particular, you can use this test to check whether, on average, the treatment has had any effect on the population .

The change in student test performance before and after taking a course.

The change in blood pressure in patients before and after administering some drug.

How to do a t-test?

So, you've decided which t-test to perform. These next steps will tell you how to calculate the p-value from t-test or its critical values, and then which decision to make about the null hypothesis.

Decide on the alternative hypothesis :

Use a two-tailed t-test if you only care whether the population's mean (or, in the case of two populations, the difference between the populations' means) agrees or disagrees with the pre-set value.

Use a one-tailed t-test if you want to test whether this mean (or difference in means) is greater/less than the pre-set value.

Compute your T-score value :

Formulas for the test statistic in t-tests include the sample size , as well as its mean and standard deviation . The exact formula depends on the t-test type — check the sections dedicated to each particular test for more details.

Determine the degrees of freedom for the t-test:

The degrees of freedom are the number of observations in a sample that are free to vary as we estimate statistical parameters. In the simplest case, the number of degrees of freedom equals your sample size minus the number of parameters you need to estimate . Again, the exact formula depends on the t-test you want to perform — check the sections below for details.

The degrees of freedom are essential, as they determine the distribution followed by your T-score (under the null hypothesis). If there are d degrees of freedom, then the distribution of the test statistics is the t-Student distribution with d degrees of freedom . This distribution has a shape similar to N(0,1) (bell-shaped and symmetric) but has heavier tails . If the number of degrees of freedom is large (>30), which generically happens for large samples, the t-Student distribution is practically indistinguishable from N(0,1).

💡 The t-Student distribution owes its name to William Sealy Gosset, who, in 1908, published his paper on the t-test under the pseudonym "Student". Gosset worked at the famous Guinness Brewery in Dublin, Ireland, and devised the t-test as an economical way to monitor the quality of beer. Cheers! 🍺🍺🍺

p-value from t-test

Recall that the p-value is the probability (calculated under the assumption that the null hypothesis is true) that the test statistic will produce values at least as extreme as the T-score produced for your sample . As probabilities correspond to areas under the density function, p-value from t-test can be nicely illustrated with the help of the following pictures:

The following formulae say how to calculate p-value from t-test. By cdf t,d we denote the cumulative distribution function of the t-Student distribution with d degrees of freedom:

p-value from left-tailed t-test:

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

p-value from right-tailed t-test:

p-value = 1 − cdf t,d (t score )

p-value from two-tailed t-test:

p-value = 2 × cdf t,d (−|t score |)

or, equivalently: p-value = 2 − 2 × cdf t,d (|t score |)

However, the cdf of the t-distribution is given by a somewhat complicated formula. To find the p-value by hand, you would need to resort to statistical tables, where approximate cdf values are collected, or to specialized statistical software. Fortunately, our t-test calculator determines the p-value from t-test for you in the blink of an eye!

t-test critical values

Recall, that in the critical values approach to hypothesis testing, you need to set a significance level, α, before computing the critical values , which in turn give rise to critical regions (a.k.a. rejection regions).

Formulas for critical values employ the quantile function of t-distribution, i.e., the inverse of the cdf :

Critical value for left-tailed t-test: cdf t,d -1 (α)

critical region:

(-∞, cdf t,d -1 (α)]

Critical value for right-tailed t-test: cdf t,d -1 (1-α)

[cdf t,d -1 (1-α), ∞)

Critical values for two-tailed t-test: ±cdf t,d -1 (1-α/2)

(-∞, -cdf t,d -1 (1-α/2)] ∪ [cdf t,d -1 (1-α/2), ∞)

To decide the fate of the null hypothesis, just check if your T-score lies within the critical region:

If your T-score belongs to the critical region , reject the null hypothesis and accept the alternative hypothesis.

If your T-score is outside the critical region , then you don't have enough evidence to reject the null hypothesis.

How to use our t-test calculator

Choose the type of t-test you wish to perform:

A one-sample t-test (to test the mean of a single group against a hypothesized mean);

A two-sample t-test (to compare the means for two groups); or

A paired t-test (to check how the mean from the same group changes after some intervention).

Two-tailed;

Left-tailed; or

Right-tailed.

This t-test calculator allows you to use either the p-value approach or the critical regions approach to hypothesis testing!

Enter your T-score and the number of degrees of freedom . If you don't know them, provide some data about your sample(s): sample size, mean, and standard deviation, and our t-test calculator will compute the T-score and degrees of freedom for you .

Once all the parameters are present, the p-value, or critical region, will immediately appear underneath the t-test calculator, along with an interpretation!

One-sample t-test

The null hypothesis is that the population mean is equal to some value μ 0 \mu_0 μ 0 ​ .

The alternative hypothesis is that the population mean is:

• different from μ 0 \mu_0 μ 0 ​ ;
• smaller than μ 0 \mu_0 μ 0 ​ ; or
• greater than μ 0 \mu_0 μ 0 ​ .

One-sample t-test formula :

• μ 0 \mu_0 μ 0 ​ — Mean postulated in the null hypothesis;
• n n n — Sample size;
• x ˉ \bar{x} x ˉ — Sample mean; and
• s s s — Sample standard deviation.

Number of degrees of freedom in t-test (one-sample) = n − 1 n-1 n − 1 .

The null hypothesis is that the actual difference between these groups' means, μ 1 \mu_1 μ 1 ​ , and μ 2 \mu_2 μ 2 ​ , is equal to some pre-set value, Δ \Delta Δ .

The alternative hypothesis is that the difference μ 1 − μ 2 \mu_1 - \mu_2 μ 1 ​ − μ 2 ​ is:

• Different from Δ \Delta Δ ;
• Smaller than Δ \Delta Δ ; or
• Greater than Δ \Delta Δ .

In particular, if this pre-determined difference is zero ( Δ = 0 \Delta = 0 Δ = 0 ):

The null hypothesis is that the population means are equal.

The alternate hypothesis is that the population means are:

• μ 1 \mu_1 μ 1 ​ and μ 2 \mu_2 μ 2 ​ are different from one another;
• μ 1 \mu_1 μ 1 ​ is smaller than μ 2 \mu_2 μ 2 ​ ; and
• μ 1 \mu_1 μ 1 ​ is greater than μ 2 \mu_2 μ 2 ​ .

Formally, to perform a t-test, we should additionally assume that the variances of the two populations are equal (this assumption is called the homogeneity of variance ).

There is a version of a t-test that can be applied without the assumption of homogeneity of variance: it is called a Welch's t-test . For your convenience, we describe both versions.

Two-sample t-test if variances are equal

Use this test if you know that the two populations' variances are the same (or very similar).

Two-sample t-test formula (with equal variances) :

where s p s_p s p ​ is the so-called pooled standard deviation , which we compute as:

• Δ \Delta Δ — Mean difference postulated in the null hypothesis;
• n 1 n_1 n 1 ​ — First sample size;
• x ˉ 1 \bar{x}_1 x ˉ 1 ​ — Mean for the first sample;
• s 1 s_1 s 1 ​ — Standard deviation in the first sample;
• n 2 n_2 n 2 ​ — Second sample size;
• x ˉ 2 \bar{x}_2 x ˉ 2 ​ — Mean for the second sample; and
• s 2 s_2 s 2 ​ — Standard deviation in the second sample.

Number of degrees of freedom in t-test (two samples, equal variances) = n 1 + n 2 − 2 n_1 + n_2 - 2 n 1 ​ + n 2 ​ − 2 .

Two-sample t-test if variances are unequal (Welch's t-test)

Use this test if the variances of your populations are different.

Two-sample Welch's t-test formula if variances are unequal:

• s 1 s_1 s 1 ​ — Standard deviation in the first sample;
• s 2 s_2 s 2 ​ — Standard deviation in the second sample.

The number of degrees of freedom in a Welch's t-test (two-sample t-test with unequal variances) is very difficult to count. We can approximate it with the help of the following Satterthwaite formula :

Alternatively, you can take the smaller of n 1 − 1 n_1 - 1 n 1 ​ − 1 and n 2 − 1 n_2 - 1 n 2 ​ − 1 as a conservative estimate for the number of degrees of freedom.

🔎 The Satterthwaite formula for the degrees of freedom can be rewritten as a scaled weighted harmonic mean of the degrees of freedom of the respective samples: n 1 − 1 n_1 - 1 n 1 ​ − 1 and n 2 − 1 n_2 - 1 n 2 ​ − 1 , and the weights are proportional to the standard deviations of the corresponding samples.

As we commonly perform a paired t-test when we have data about the same subjects measured twice (before and after some treatment), let us adopt the convention of referring to the samples as the pre-group and post-group.

The null hypothesis is that the true difference between the means of pre- and post-populations is equal to some pre-set value, Δ \Delta Δ .

The alternative hypothesis is that the actual difference between these means is:

Typically, this pre-determined difference is zero. We can then reformulate the hypotheses as follows:

The null hypothesis is that the pre- and post-means are the same, i.e., the treatment has no impact on the population .

The alternative hypothesis:

• The pre- and post-means are different from one another (treatment has some effect);
• The pre-mean is smaller than the post-mean (treatment increases the result); or
• The pre-mean is greater than the post-mean (treatment decreases the result).

Paired t-test formula

In fact, a paired t-test is technically the same as a one-sample t-test! Let us see why it is so. Let x 1 , . . . , x n x_1, ... , x_n x 1 ​ , ... , x n ​ be the pre observations and y 1 , . . . , y n y_1, ... , y_n y 1 ​ , ... , y n ​ the respective post observations. That is, x i , y i x_i, y_i x i ​ , y i ​ are the before and after measurements of the i -th subject.

For each subject, compute the difference, d i : = x i − y i d_i := x_i - y_i d i ​ := x i ​ − y i ​ . All that happens next is just a one-sample t-test performed on the sample of differences d 1 , . . . , d n d_1, ... , d_n d 1 ​ , ... , d n ​ . Take a look at the formula for the T-score :

Δ \Delta Δ — Mean difference postulated in the null hypothesis;

n n n — Size of the sample of differences, i.e., the number of pairs;

x ˉ \bar{x} x ˉ — Mean of the sample of differences; and

s s s  — Standard deviation of the sample of differences.

Number of degrees of freedom in t-test (paired): n − 1 n - 1 n − 1

t-test vs Z-test

We use a Z-test when we want to test the population mean of a normally distributed dataset, which has a known population variance . If the number of degrees of freedom is large, then the t-Student distribution is very close to N(0,1).

Hence, if there are many data points (at least 30), you may swap a t-test for a Z-test, and the results will be almost identical. However, for small samples with unknown variance, remember to use the t-test because, in such cases, the t-Student distribution differs significantly from the N(0,1)!

🙋 Have you concluded you need to perform the z-test? Head straight to our z-test calculator !

What is a t-test?

A t-test is a widely used statistical test that analyzes the means of one or two groups of data. For instance, a t-test is performed on medical data to determine whether a new drug really helps.

What are different types of t-tests?

Different types of t-tests are:

• One-sample t-test;
• Two-sample t-test; and
• Paired t-test.

How to find the t value in a one sample t-test?

To find the t-value:

• Subtract the null hypothesis mean from the sample mean value.
• Divide the difference by the standard deviation of the sample.
• Multiply the resultant with the square root of the sample size.

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ol{padding-top:0;}.css-63uqft ul:not(:first-child),.css-63uqft ol:not(:first-child){padding-top:4px;} Test setup

Choose test type

t-test for the population mean, μ, based on one independent sample . Null hypothesis H 0 : μ = μ 0  

Alternative hypothesis H 1

Test details

Significance level α

The probability that we reject a true H 0 (type I error).

Degrees of freedom

Calculated as sample size minus one.

Test results

One sample t test calculator

The One Sample t Test Calculator allows you to determine p-values, critical values, test statistics, and conclusions using the one-sample t test method.

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How to Use the One Sample t Test Calculator

• Select Data Type: Choose whether to input summary statistics directly or provide a data set.
• Input Your Data: Enter the required values such as population mean, sample size, sample mean, and sample standard deviation.
• Set Hypotheses: Specify the null and alternative hypotheses.
• Calculate: Click the "Calculate" button to see the test statistic, p-value, and other relevant results.

Interpreting the Results

To interpret the results of a one-sample t-test, you must first grasp several crucial components: the test statistic (t-value), degrees of freedom, p-value, and confidence interval. Here's a step-by-step way to interpreting these findings:

Components of One-Sample t-Test Results

Test Statistic (t-value) :

This result indicates the amount of standard deviations your sample mean is from the population mean under the null hypothesis. A larger absolute value of t indicates a wider divergence between the sample mean and the population average.

Degrees of Freedom (df):

This is normally the sample size less one (n-1). The degrees of freedom are utilized to calculate the critical value of t using the t-distribution table.

This represents the likelihood of receiving a test statistic as extreme as the one observed, assuming the null hypothesis is correct. A low p-value ( less than 0.05) indicates that the observed data are unlikely to support the null hypothesis, resulting in its rejection.

Confidence Interval:

This defines a range of values within which the genuine population mean is expected to fall. If the confidence interval excludes the population mean stated in the null hypothesis, it supports the conclusion that the sample mean differs considerably from the population mean.

Steps to Interpret the Results

State the Hypotheses:

Null Hypothesis (H₀): The population mean is equal to a specified value (e.g. \( \mu=\mu_0 \) ).

Alternative Hypothesis (H₁): The population mean is different from the specified value (e.g. \( \mu \ne \mu_0 \) ).

Check the t-value:

Compare the t-value to the crucial value from the t-distribution table based on the specified significance level (α, typically 0.05) and degrees of freedom.

If |t-value| > critical value, reject the null hypothesis.

Examine the P-value:

• If the p-value is less than the chosen significance level \( \alpha \), reject the null hypothesis.
• A p-value less than \(0.05\) typically indicates strong evidence against the null hypothesis.

Review the Confidence Interval:

Check to see if the confidence interval for the sample mean includes the population mean under the null hypothesis.

If the interval excludes the population mean, it indicates that the sample mean is significantly different than the population mean.

Requirements and Assumptions for a one sample t test

To effectively execute and interpret a one-sample t-test, you must first grasp the test's requirements and assumptions. Meeting these parameters assures that the test results are legitimate and reliable.

Requirements for a One-Sample t-Test

You will need a sample of the population you are studying.

To compare the sample mean to the population mean (μ₀), you must have a known population mean.

A suitable sample size is necessary. While the t-test performs well with small sample sizes, higher sample sizes yield more accurate results.

Assumptions of a One-Sample t-Test

Random Sampling:

Data should be acquired from the population using random sampling. This guarantees that the sample is representative of the population and minimises bias.

Scale of Measurement:

The data must be continuous (interval or ratio scale). This means that the data points may be meaningfully arranged, and the differences between them are consistent and observable.

The data should follow a roughly normal distribution. This assumption is particularly essential for small sample sizes (n < 30). For bigger samples, the Central Limit Theorem predicts that the sample mean distribution will be essentially normal, regardless of the distribution of the data.

Independence :

Observations in the sample must be independent of one another. This means that the value of one observation should not affect or predict the value of another.

Unknown Population Standard Deviation: :

The test assumes that the population standard deviation \(\alpha\) is unknown and needs to be determined from the sample.

Applications of the One Sample t Test

This test is widely used in various fields including:

• Education: Assessing whether the average test scores of a class differ from the national average.
• Healthcare: Comparing the mean blood pressure level of a group of patients to a known population mean.
• Business: Evaluating if the average sales of a product differ from the company's historical sales data.

Frequently Asked Questions (FAQs)

What is a one-sample t test.

A one-sample t test is a statistical approach for determining whether a sample's mean differs significantly from a known population mean.

When should I use a one-sample t test?

Use this test when the population standard deviation is unknown and the sample size is small (typically n < 30).

How do I interpret the p-value in a t test?

The p-value indicates the probability of observing the test results under the null hypothesis. A low p-value less than the significance level equal to \( \alpha \) suggests rejecting the null hypothesis.

Why Use Our One Sample t Test Calculator?

Our calculator offers several advantages:

• Accuracy: Provides precise calculations for your hypothesis testing needs.
• User-Friendly Interface: Easy to navigate and input your data.
• Educational Value: Offers detailed explanations and step-by-step guides to help you understand the results.
• Time-Saving: Quickly computes results, allowing you to focus on analysis and interpretation.
• Versatility: Suitable for various fields such as education, healthcare, and business, making it a valuable tool for a wide range of users.

Start your hypothesis testing today with our One Sample t Test Calculator and achieve accurate and reliable results effortlessly.

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T test calculator

A t test compares the means of two groups. There are several types of two sample t tests and this calculator focuses on the three most common: unpaired, welch's, and paired t tests. Directions for using the calculator are listed below, along with more information about two sample t tests and help on which is appropriate for your analysis. NOTE: This is not the same as a one sample t test; for that, you need this One sample t test calculator .

1. Choose data entry format

Caution: Changing format will erase your data.

2. Choose a test

Help me choose

3. Enter data

Help me arrange the data

4. View the results

What is a t test.

A t test is used to measure the difference between exactly two means. Its focus is on the same numeric data variable rather than counts or correlations between multiple variables. If you are taking the average of a sample of measurements, t tests are the most commonly used method to evaluate that data. It is particularly useful for small samples of less than 30 observations. For example, you might compare whether systolic blood pressure differs between a control and treated group, between men and women, or any other two groups.

This calculator uses a two-sample t test, which compares two datasets to see if their means are statistically different. That is different from a one sample t test , which compares the mean of your sample to some proposed theoretical value.

The most general formula for a t test is composed of two means (M1 and M2) and the overall standard error (SE) of the two samples:

See our video on How to Perform a Two-sample t test for an intuitive explanation of t tests and an example.

How to use the t test calculator

• Choose your data entry format . This will change how section 3 on the page looks. The first two options are for entering your data points themselves, either manually or by copy & paste. The last two are for entering the means for each group, along with the number of observations (N) and either the standard error of that mean (SEM) or standard deviation of the dataset (SD) standard error. If you have already calculated these summary statistics, the latter options will save you time.
• Choose a test from the three options: Unpaired t test, Welch's unpaired t test, or Paired t test. Use our Ultimate Guide to t tests if you are unsure which is appropriate, as it includes a section on "How do I know which t test to use?". Notice not all options are available if you enter means only.
• Enter data for the test, based on the format you chose in Step 1.
• Click Calculate Now and View the results. All options will perform a two-tailed test .

Performing t tests? We can help.

Sign up for more information on how to perform t tests and other common statistical analyses.

Common t test confusion

In addition to the number of t test options, t tests are often confused with completely different techniques as well. Here's how to keep them all straight.

Correlation and regression are used to measure how much two factors move together. While t tests are part of regression analysis, they are focused on only one factor by comparing means in different samples.

ANOVA is used for comparing means across three or more total groups. In contrast, t tests compare means between exactly two groups.

Finally, contingency tables compare counts of observations within groups rather than a calculated average. Since t tests compare means of continuous variable between groups, contingency tables use methods such as chi square instead of t tests.

Assumptions of t tests

Because there are several versions of t tests, it's important to check the assumptions to figure out which is best suited for your project. Here are our analysis checklists for unpaired t tests and paired t tests , which are the two most common. These (and the ultimate guide to t tests ) go into detail on the basic assumptions underlying any t test:

• Exactly two groups
• Sample is normally distributed
• Independent observations
• Unequal or equal variance?
• Paired or unpaired data?

Interpreting results

The three different options for t tests have slightly different interpretations, but they all hinge on hypothesis testing and P values. You need to select a significance threshold for your P value (often 0.05) before doing the test.

While P values can be easy to misinterpret , they are the most commonly used method to evaluate whether there is evidence of a difference between the sample of data collected and the null hypothesis. Once you have run the correct t test, look at the resulting P value. If the test result is less than your threshold, you have enough evidence to conclude that the data are significantly different.

If the test result is larger or equal to your threshold, you cannot conclude that there is a difference. However, you cannot conclude that there was definitively no difference either. It's possible that a dataset with more observations would have resulted in a different conclusion.

Depending on the test you run, you may see other statistics that were used to calculate the P value, including the mean difference, t statistic, degrees of freedom, and standard error. The confidence interval and a review of your dataset is given as well on the results page.

Graphing t tests

This calculator does not provide a chart or graph of t tests, however, graphing is an important part of analysis because it can help explain the results of the t test and highlight any potential outliers. See our Prism guide for some graphing tips for both unpaired and paired t tests.

Prism is built for customized, publication quality graphics and charts. For t tests we recommend simply plotting the datapoints themselves and the mean, or an estimation plot . Another popular approach is to use a violin plot, like those available in Prism.

For more information

Our ultimate guide to t tests includes examples, links, and intuitive explanations on the subject. It is quite simply the best place to start if you're looking for more about t tests!

If you enjoyed this calculator, you will love using Prism for analysis. Take a free 30-day trial to do more with your data, such as:

• Clear guidance to pick the right t test and detailed results summaries
• Custom, publication quality t test graphics, violin plots, and more
• More t test options, including normality testing as well as nested and multiple t tests
• Non-parametric test alternatives such as Wilcoxon, Mann-Whitney, and Kolmogorov-Smirnov

Check out our video on how to perform a t test in Prism , for an example from start to finish!

Remember, this page is just for two sample t tests. If you only have one sample, you need to use this calculator instead.

We Recommend:

Analyze, graph and present your scientific work easily with GraphPad Prism. No coding required.

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).

Two-Sample t-Test

Equal variances:, unequal variances (welch’s t-test):.

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

Table of Critical t-Values

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.

Using the one-sample t-test formula:

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

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.

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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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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T-Test calculator

The Student's t-test is used to determine if means of two data sets differ significantly. This calculator will generate a step by step explanation on how to apply t – test.

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Equations Probability Calculator Probability Distributions Descriptive Statistics Standard Deviation Z - score Calculator Normal Distribution T-Test Calculator Correlation & Regression Simple Interest Compound Interest Amortization Calculator Annuity Calculator Work Problems Hire MATHPORTAL experts to do math homework for you. Prices start at $3 per problem. Twelve younger adults and twelve older adults conducted a life satisfaction test. The data are presented in the table below. Compute the appropriate t-test. Are the means between two data sets are significantly different at level $\alpha < 0.05$. Welcome to MathPortal. This website's owner is mathematician Miloš Petrović. I designed this website and wrote all the calculators, lessons, and formulas . If you want to contact me, probably have some questions, write me using the contact form or email me on [email protected] Email (optional) Calculator: One-Sample t-Test One-Sample t-Test Calculator This calculator will conduct a complete one-sample t-test, given the sample mean, the sample size, the hypothesized mean, and the sample standard deviation. The results generated by the calculator include the t-statistic, the degrees of freedom, the critical t-values for both one-tailed (directional) and two-tailed (non-directional) hypotheses, and the one-tailed and two-tailed probability values associated with the test. Please enter the necessary parameter values, and then click 'Calculate'. Sample mean (x): Sample size: Sample standard deviation: One Sample T Test Calculator Enter sample data Reporting results in APA style One sample t-test, what is a one sample t-test, how to use the one sample t test calculator, calculators. Calculators Descriptive Statistics Merchandise Which Statistics Test? 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). Two sample t test Kruskal wallis Pearson product moment Two sample f test One way anova Bartlett test Chi square goodness of fit Fligner killeen Hypothesis Test Calculator Upload your data set below to get started Or input your data as csv Sharing helps us build more free tools “extremely user friendly” “truly amazing!” “so easy to use!” Statistics Calculator You want to analyze your data effortlessly? DATAtab makes it easy and online. Online Statistics Calculator What do you want to calculate online? The online statistics calculator is simple and uncomplicated! Here you can find a list of all implemented methods! Create charts online with DATAtab Create your charts for your data directly online and uncomplicated. To do this, insert your data into the table under Charts and select which chart you want. The advantages of DATAtab Statistics, as simple as never before.. DATAtab is a modern statistics software, with unique user-friendliness. Statistical analyses are done with just a few clicks, so DATAtab is perfect for statistics beginners and for professionals who want more flow in the user experience. Directly in the browser, fully flexible. Directly in the browser, fully flexible. DATAtab works directly in your web browser. You have no installation and maintenance effort whatsoever. Wherever and whenever you want to use DATAtab, just go to the website and get started. All the statistical methods you need. DATAtab offers you a wide range of statistical methods. We have selected the most central and best known statistical methods for you and do not overwhelm you with special cases. Data security is a top priority. All data that you insert and evaluate on DATAtab always remain on your end device. The data is not sent to any server or stored by us (not even temporarily). Furthermore, we do not pass on your data to third parties in order to analyze your user behavior. Many tutorials with simple examples. In order to facilitate the introduction, DATAtab offers a large number of free tutorials with focused explanations in simple language. We explain the statistical background of the methods and give step-by-step explanations for performing the analyses in the statistics calculator. Practical Auto-Assistant. DATAtab takes you by the hand in the world of statistics. When making statistical decisions, such as the choice of scale or measurement level or the selection of suitable methods, Auto-Assistants ensure that you get correct results quickly. Charts, simple and clear. With DATAtab data visualization is fun! Here you can easily create meaningful charts that optimally illustrate your results. New in the world of statistics? DATAtab was primarily designed for people for whom statistics is new territory. Beginners are not overwhelmed with a lot of complicated options and checkboxes, but are encouraged to perform their analyses step by step. Online survey very simple. DATAtab offers you the possibility to easily create an online survey, which you can then evaluate immediately with DATAtab. Our references Alternative to statistical software like SPSS and STATA DATAtab was designed for ease of use and is a compelling alternative to statistical programs such as SPSS and STATA. On datatab.net, data can be statistically evaluated directly online and very easily (e.g. t-test, regression, correlation etc.). DATAtab's goal is to make the world of statistical data analysis as simple as possible, no installation and easy to use. Of course, we would also be pleased if you take a look at our second project Statisty . Extensive tutorials Descriptive statistics. Here you can find out everything about location parameters and dispersion parameters and how you can describe and clearly present your data using characteristic values. Hypothesis Test Here you will find everything about hypothesis testing: One sample t-test , Unpaired t-test , Paired t-test and Chi-square test . You will also find tutorials for non-parametric statistical procedures such as the Mann-Whitney u-Test and Wilcoxon-Test . mann-whitney-u-test and the Wilcoxon test The regression provides information about the influence of one or more independent variables on the dependent variable. Here are simple explanations of linear regression and logistic regression . Correlation Correlation analyses allow you to analyze the linear association between variables. Learn when to use Pearson correlation or Spearman rank correlation . With partial correlation , you can calculate the correlation between two variables to the exclusion of a third variable. Partial Correlation The partial correlation shows you the correlation between two variables to the exclusion of a third variable. Levene Test The Levene Test checks your data for variance equality. Thus, the levene test is used as a prerequisite test for many hypothesis tests . The p-value is needed for every hypothesis test to be able to make a statement whether the null hypothesis is accepted or rejected. Distributions DATAtab provides you with tables with distributions and helpful explanations of the distribution functions. These include the Table of t-distribution and the Table of chi-squared distribution Contingency table With a contingency table you can get an overview of two categorical variables in the statistics. Equivalence and non-inferiority In an equivalence trial, the statistical test aims at showing that two treatments are not too different in characteristics and a non-inferiority trial wants to show that an experimental treatment is not worse than an established treatment. If there is a clear cause-effect relationship between two variables, then we can speak of causality. Learn more about causality in our tutorial. Multicollinearity Multicollinearity is when two or more independent variables have a high correlation. Effect size for independent t-test Learn how to calculate the effect size for the t-test for independent samples. Reliability analysis calculator On DATAtab, Cohen's Kappa can be easily calculated online in the Cohen’s Kappa Calculator . there is also the Fleiss Kappa Calculator . Of course, the Cronbach's alpha can also be calculated in the Cronbach's Alpha Calculator . Analysis of variance with repeated measurement Repeated measures ANOVA tests whether there are statistically significant differences in three or more dependent samples. Cite DATAtab: DATAtab Team (2024). DATAtab: Online Statistics Calculator. DATAtab e.U. Graz, Austria. URL https://datatab.net Two Sample T-Test Calculator Two sample t-test is used to check whether the means of two groups are significantly different from each other. For example, if you want to see if mean weight of males and females have statistically significant difference between them. Independent T-test assumes that the two samples have equal variances. Welch's t-test is used if you have unequal variances. Either enter raw data or summary information to calculate two sample t-test. You can directly paste data from MS Excel. Enter Raw Data Enter Summary Data Scores are normally distributed within each of the two groups Each score is sampled independently and randomly. Data must be continuous Find Us On Facebook Follow on Twitter Subscribe using RSS Two-Sample t-test Use this calculator to test whether samples from two independent populations provide evidence that the populations have different means. For example, based on blood pressures measurements taken from a sample of women and a sample of men, can we conclude that women and men have different mean blood pressures? This test is known as an a two sample (or unpaired) t-test. It produces a “p-value”, which can be used to decide whether there is evidence of a difference between the two population means. The p-value is the probability that the difference between the sample means is at least as large as what has been observed, under the assumption that the population means are equal. The smaller the p-value, the more surprised we would be by the observed difference in sample means if there really was no difference between the population means. Therefore, the smaller the p-value, the stronger the evidence is that the two populations have different means. Typically a threshold (known as the significance level) is chosen, and a p-value less than the threshold is interpreted as indicating evidence of a difference between the population means. The most common choice of significance level is 0.05, but other values, such as 0.1 or 0.01 are also used. This calculator should be used when the sampling units (e.g. the sampled individuals) in the two groups are independent. If you are comparing two measurements taken on the same sampling unit (e.g. blood pressure of an individual before and after a drug is administered) then the appropriate test is the paired t-test. The p-value is With sample means of The p-value would be With sample standard deviations of The p-value would be With sample sizes of The p-value would be More Information Worked example. A study compares the average capillary density in the feet of individuals with and without ulcers. A sample of 10 patients with ulcers has mean capillary density of 29, with standard deviation 7.5. A control sample of 10 individuals without ulcers has mean capillary density of 34, with standard deviation 8.0. (All measurements are in capillaries per square mm.) Using this information, the p-value is calculated as 0.167. Since this p-value is greater than 0.05, it would conventionally be interpreted as meaning that the data do not provide strong evidence of a difference in capillary density between individuals with and without ulcers. If both sample sizes were increased to 20, the p-value would reduce to 0.048 (assuming the sample means and standard deviations remained the same), which we would interpret as strong evidence of a difference. Note that this result is not inconsistent with the previous result: with bigger samples we are able to detect smaller differences between populations. Assumptions This test assumes that the two populations follow normal distributions (otherwise known as Gaussian distributions). Normality of the distributions can be tested using, for example, a Q-Q plot . An alternative test that can be used if you suspect that the data are drawn from non-normal distributions is the Mann-Whitney U test . The version of the test used here also assumes that the two populations have different variances. If you think the populations have the same variance, an alternative version of the two sample t-test (two sample t-test with a pooled variance estimator) can be used. The advantage of the alternative version is that if the populations have the same variance then it has greater statistical power – that is, there is a higher probability of detecting a difference between the population means if such a difference exists. Performing this test assesses the extent to which the difference between the sample means provides evidence of a difference between the population means. The test puts forward a “null” hypothesis that the population means are equal, and measures the probability of observing a difference at least as big as that seen in the data under the null hypothesis (the p-value). If the p-value is large then the observed difference between the sample means is unsurprising and is interpreted as being consistent with hypothesis of equal population means. If on the other hand the p-value is small then we would be surprised about the observed difference if the null hypothesis really was true. Therefore, a small p-value is interpreted as evidence that the null hypothesis is false and that there really is a difference between the population means. Typically a threshold (known as the significance level) is chosen, and a p-value less than the threshold is interpreted as indicating evidence of a difference between the population means. The most common choice of significance level is 0.05, but other values, such as 0.1 or 0.01 are also used. Note that a large p-value (say, larger than 0.05) cannot in itself be interpreted as evidence that the populations have equal means. It may just mean that the sample size is not large enough to detect a difference. To find out how large your sample needs to be in order to detect a difference (if a difference exists), see our sample size calculator . If evidence of a difference in the population means is found, you may wish to quantify that difference. The difference between the sample means is a point estimate of the difference between the population means, but it can be useful to assess how reliable this estimate is using a confidence interval . A confidence interval provides you with a set of limits in which you expect the difference between the population means to lie. The p-value and the confidence interval are related and have a consistent interpretation: if the p-value is less than α then a (1-α)*100% confidence interval will not contain zero. For example, if the p-value is less than 0.05 then a 95% confidence interval will not contain zero. If you wish to calculate a confidence interval, our confidence interval calculator will do the work for you. Definitions Sample mean. The sample mean is your ‘best guess’ for what the true population mean is given your sample of data and is calcuated as: μ = (1/n)* ∑ n i=1 x i , where n is the sample size and x 1 ,…,x n are the n sample observations. Sample standard deviation The sample standard deviation is calcuated as s=√ σ 2 , where: σ 2 = (1/(n-1))* ∑ n i=1 (x i -μ) 2 , μ is the sample mean, n is the sample size and x 1 ,…,x n are the n sample observations. Sample size This is the total number of samples randomly drawn from you population.  The larger the sample size, the more certain you can be that the estimate reflects the population.  Choosing a sample size is an important aspect when desiging your study or survey.  For some further information, see our blog post on The Importance and Effect of Sample Size and for guidance on how to choose your sample size, see our sample size calculator . Tell us what you want to achieve Data Collection & Management Data Mining Innovation & Research Qualitative Analysis Surveys & Sampling Visualisation Agriculture Environment Market Research Public Sector Select Statistical Services Ltd Oxygen House, Grenadier Road, Exeter Business Park, Exeter EX1 3LH t: 01392 440426 e: [email protected] Sign up to our Newsletter Please tick this box to confirm that you are happy for us to store and process the information supplied above for the purpose of managing your subscription to our newsletter. Name This field is for validation purposes and should be left unchanged. Telephone Number By using this form you agree with the storage and handling of your data by this website. Email This field is for validation purposes and should be left unchanged. Enquiry - Jobs Difference in Means Hypothesis Test Calculator Use the calculator below to analyze the results of a difference in sample means hypothesis test. Enter your sample means, sample standard deviations, sample sizes, hypothesized difference in means, test type, and significance level to calculate your results. You will find a description of how to conduct a two sample t-test below the calculator. Define the Two Sample t-test Significance Level Difference in Means t-score Probability The Difference Between the Sample Means Under the Null Distribution Conducting a hypothesis test for the difference in means. When two populations are related, you can compare them by analyzing the difference between their means. A hypothesis test for the difference in samples means can help you make inferences about the relationships between two population means. Testing for a Difference in Means For the results of a hypothesis test to be valid, you should follow these steps: Check Your Conditions State your hypothesis, determine your analysis plan, analyze your sample, interpret your results. To use the testing procedure described below, you should check the following conditions: Independence of Samples - Your samples should be collected independently of one another. Simple Random Sampling - You should collect your samples with simple random sampling. This type of sampling requires that every occurrence of a value in a population has an equal chance of being selected when taking a sample. Normality of Sample Distributions - The sampling distributions for both samples should follow the Normal or a nearly Normal distribution. A sampling distribution will be nearly Normal when the samples are collected independently and when the population distribution is nearly Normal. Generally, the larger the sample size, the more normally distributed the sampling distribution. Additionally, outlier data points can make a distribution less Normal, so if your data contains many outliers, exercise caution when verifying this condition. You must state a null hypothesis and an alternative hypothesis to conduct an hypothesis test of the difference in means. The null hypothesis is a skeptical claim that you would like to test. The alternative hypothesis represents the alternative claim to the null hypothesis. Your null hypothesis and alternative hypothesis should be stated in one of three mutually exclusive ways listed in the table below. Null Hypothesis Alternative Hypothesis Number of Tails Description - μ = D - μ ≠ D Tests whether the sample means come from populations with a difference in means equal to D. If D = 0, then tests if the samples come from populations with means that are different from each other. - μ ≤ D - μ > D Tests whether sample one comes from a population with a mean that is greater than sample two's population mean by a difference of D. If D = 0, then tests if sample one comes from a population with a mean greater than sample two's population mean. - μ ≥ D - μ < D Tests whether sample one comes from a population with a mean that is less than sample two's population mean by a difference of D. If D = 0, then tests if sample one comes from a population with a mean less than sample two's population mean. D is the hypothesized difference between the populations' means that you would like to test. Before conducting a hypothesis test, you must determine a reasonable significance level, α, or the probability of rejecting the null hypothesis assuming it is true. The lower your significance level, the more confident you can be of the conclusion of your hypothesis test. Common significance levels are 10%, 5%, and 1%. To evaluate your hypothesis test at the significance level that you set, consider if you are conducting a one or two tail test: Two-tail tests divide the rejection region, or critical region, evenly above and below the null distribution, i.e. to the tails of the null sampling distribution. For example, in a two-tail test with a 5% significance level, your rejection region would be the upper and lower 2.5% of the null distribution. An alternative hypothesis of μ 1 - μ 2 ≠ D requires a two tail test. One-tail tests place the rejection region entirely on one side of the distribution i.e. to the right or left tail of the null distribution. For example, in a one-tail test evaluating if the actual difference in means, D, is above the null distribution with a 5% significance level, your rejection region would be the upper 5% of the null distribution. μ 1 - μ 2 > D and μ 1 - μ 2 < D alternative hypotheses require one-tail tests. The graphical results section of the calculator above shades rejection regions blue. After checking your conditions, stating your hypothesis, determining your significance level, and collecting your sample, you are ready to analyze your hypothesis. Sample means follow the Normal distribution with the following parameters: The Difference in the Population Means, D - The true difference in the population means is unknown, but we use the hypothesized difference in the means, D, from the null hypothesis in the calculations. The Standard Error, SE - The standard error of the difference in the sample means can be computed as follows:      SE = (s 1 2 /n 1 + s 2 2 /n 2 ) (1/2) with s 1 being the standard deviation of sample one, n 1 being the sample size of sample one, s 2 being the standard deviation of sample one, and n 2 being the sample size of sample two. The standard error defines how differences in sample means are expected to vary around the null difference in means sampling distribution given the sample sizes and under the assumption that the null hypothesis is true. The Degrees of Freedom, DF - The degrees of freedom calculation can be estimated as the smaller of n 1 - 1 or n 2 - 1. For more accurate results, use the following formula for the degrees of freedom (DF):      DF = (s 1 2 /n 1 + s 2 2 /n 2 ) 2 / ((s 1 2 /n 1 ) 2 / (n 1 - 1) + (s 2 2 /n 2 ) 2 / (n 2 - 1)) In a difference in means hypothesis test, we calculate the probability that we would observe the difference in sample means (x̄ 1 - x̄ 2 ), assuming the null hypothesis is true, also known as the p-value . If the p-value is less than the significance level, then we can reject the null hypothesis. You can determine a precise p-value using the calculator above, but we can find an estimate of the p-value manually by calculating the t-score, or t-statistic, as follows: t = (x̄ 1 - x̄ 2 - D) / SE The t-score is a test statistic that tells you how far our observation is from the null hypothesis's difference in means under the null distribution. Using any t-score table, you can look up the probability of observing the results under the null distribution. You will need to look up the t-score for the type of test you are conducting, i.e. one or two tail. A hypothesis test for the difference in means is sometimes known as a two sample mean t-test because of the use of a t-score in analyzing results. The conclusion of a hypothesis test for the difference in means is always either: Reject the null hypothesis Do not reject the null hypothesis If you reject the null hypothesis, you cannot say that your sample difference in means is the true difference between the means. If you do not reject the null hypothesis, you cannot say that the hypothesized difference in means is true. A hypothesis test is simply a way to look at evidence and conclude if it provides sufficient evidence to reject the null hypothesis. Example: Hypothesis Test for the Difference in Two Means Let’s say you are a manager at a company that designs batteries for smartphones. One of your engineers believes that she has developed a battery that will last more than two hours longer than your standard battery. Before you can consider if you should replace your standard battery with the new one, you need to test the engineer’s claim. So, you decided to run a difference in means hypothesis test to see if her claim that the new battery will last two hours longer than the standard one is reasonable. You direct your team to run a study. They will take a sample of 100 of the new batteries and compare their performance to 1,000 of the old standard batteries. Check the conditions - Your test consists of independent samples . Your team collects your samples using simple random sampling , and you have reason to believe that all your batteries' performances are always close to normally distributed . So, the conditions are met to conduct a two sample t-test. State Your Hypothesis - Your null hypothesis is that the charge of the new battery lasts at most two hours longer than your standard battery (i.e. μ 1 - μ 2 ≤ 2). Your alternative hypothesis is that the new battery lasts more than two hours longer than the standard battery (i.e. μ 1 - μ 2 > 2). Determine Your Analysis Plan - You believe that a 1% significance level is reasonable. As your test is a one-tail test, you will evaluate if the difference in mean charge between the samples would occur at the upper 1% of the null distribution. Analyze Your Sample - After collecting your samples (which you do after steps 1-3), you find the new battery sample had a mean charge of 10.4 hours, x̄ 1 , with a 0.8 hour standard deviation, s 1 . Your standard battery sample had a mean charge of 8.2 hours, x̄ 2 , with a standard deviation of 0.2 hours, s 2 . Using the calculator above, you find that a difference in sample means of 2.2 hours [2 = 10.4 – 8.2] would results in a t-score of 2.49 under the null distribution, which translates to a p-value of 0.72%. Interpret Your Results - Since your p-value of 0.72% is less than the significance level of 1%, you have sufficient evidence to reject the null hypothesis. In this example, you found that you can reject your null hypothesis that the new battery design does not result in more than 2 hours of extra battery life. The test does not guarantee that your engineer’s new battery lasts two hours longer than your standard battery, but it does give you strong reason to believe her claim. Sample Size Calculator Find out the sample size. This calculator computes the minimum number of necessary samples to meet the desired statistical constraints. Confidence Level:   Margin of Error:   Population Proportion: Use 50% if not sure Population Size: Leave blank if unlimited population size.   Find Out the Margin of Error This calculator gives out the margin of error or confidence interval of observation or survey. Confidence Level:   Sample Size:   Population Proportion:   Population Size: Leave blank if unlimited population size.   Related Standard Deviation Calculator | Probability Calculator In statistics, information is often inferred about a population by studying a finite number of individuals from that population, i.e. the population is sampled, and it is assumed that characteristics of the sample are representative of the overall population. For the following, it is assumed that there is a population of individuals where some proportion, p , of the population is distinguishable from the other 1-p in some way; e.g., p may be the proportion of individuals who have brown hair, while the remaining 1-p have black, blond, red, etc. Thus, to estimate p in the population, a sample of n individuals could be taken from the population, and the sample proportion, p̂ , calculated for sampled individuals who have brown hair. Unfortunately, unless the full population is sampled, the estimate p̂ most likely won't equal the true value p , since p̂ suffers from sampling noise, i.e. it depends on the particular individuals that were sampled. However, sampling statistics can be used to calculate what are called confidence intervals, which are an indication of how close the estimate p̂ is to the true value p . Statistics of a Random Sample The uncertainty in a given random sample (namely that is expected that the proportion estimate, p̂ , is a good, but not perfect, approximation for the true proportion p ) can be summarized by saying that the estimate p̂ is normally distributed with mean p and variance p(1-p)/n . For an explanation of why the sample estimate is normally distributed, study the Central Limit Theorem . As defined below, confidence level, confidence intervals, and sample sizes are all calculated with respect to this sampling distribution. In short, the confidence interval gives an interval around p in which an estimate p̂ is "likely" to be. The confidence level gives just how "likely" this is – e.g., a 95% confidence level indicates that it is expected that an estimate p̂ lies in the confidence interval for 95% of the random samples that could be taken. The confidence interval depends on the sample size, n (the variance of the sample distribution is inversely proportional to n , meaning that the estimate gets closer to the true proportion as n increases); thus, an acceptable error rate in the estimate can also be set, called the margin of error, ε , and solved for the sample size required for the chosen confidence interval to be smaller than e ; a calculation known as "sample size calculation." Confidence Level The confidence level is a measure of certainty regarding how accurately a sample reflects the population being studied within a chosen confidence interval. The most commonly used confidence levels are 90%, 95%, and 99%, which each have their own corresponding z-scores (which can be found using an equation or widely available tables like the one provided below) based on the chosen confidence level. Note that using z-scores assumes that the sampling distribution is normally distributed, as described above in "Statistics of a Random Sample." Given that an experiment or survey is repeated many times, the confidence level essentially indicates the percentage of the time that the resulting interval found from repeated tests will contain the true result. Confidence Level z-score (±) 0.70 1.04 0.75 1.15 0.80 1.28 0.85 1.44 0.92 1.75 0.95 1.96 0.96 2.05 0.98 2.33 0.99 2.58 0.999 3.29 0.9999 3.89 0.99999 4.42 Confidence Interval In statistics, a confidence interval is an estimated range of likely values for a population parameter, for example, 40 ± 2 or 40 ± 5%. Taking the commonly used 95% confidence level as an example, if the same population were sampled multiple times, and interval estimates made on each occasion, in approximately 95% of the cases, the true population parameter would be contained within the interval. Note that the 95% probability refers to the reliability of the estimation procedure and not to a specific interval. Once an interval is calculated, it either contains or does not contain the population parameter of interest. Some factors that affect the width of a confidence interval include: size of the sample, confidence level, and variability within the sample. There are different equations that can be used to calculate confidence intervals depending on factors such as whether the standard deviation is known or smaller samples (n<30) are involved, among others. The calculator provided on this page calculates the confidence interval for a proportion and uses the following equations: where is z score is the population proportion and are sample size is the population size Within statistics, a population is a set of events or elements that have some relevance regarding a given question or experiment. It can refer to an existing group of objects, systems, or even a hypothetical group of objects. Most commonly, however, population is used to refer to a group of people, whether they are the number of employees in a company, number of people within a certain age group of some geographic area, or number of students in a university's library at any given time. It is important to note that the equation needs to be adjusted when considering a finite population, as shown above. The (N-n)/(N-1) term in the finite population equation is referred to as the finite population correction factor, and is necessary because it cannot be assumed that all individuals in a sample are independent. For example, if the study population involves 10 people in a room with ages ranging from 1 to 100, and one of those chosen has an age of 100, the next person chosen is more likely to have a lower age. The finite population correction factor accounts for factors such as these. Refer below for an example of calculating a confidence interval with an unlimited population. EX: Given that 120 people work at Company Q, 85 of which drink coffee daily, find the 99% confidence interval of the true proportion of people who drink coffee at Company Q on a daily basis. Sample Size Calculation Sample size is a statistical concept that involves determining the number of observations or replicates (the repetition of an experimental condition used to estimate the variability of a phenomenon) that should be included in a statistical sample. It is an important aspect of any empirical study requiring that inferences be made about a population based on a sample. Essentially, sample sizes are used to represent parts of a population chosen for any given survey or experiment. To carry out this calculation, set the margin of error, ε , or the maximum distance desired for the sample estimate to deviate from the true value. To do this, use the confidence interval equation above, but set the term to the right of the ± sign equal to the margin of error, and solve for the resulting equation for sample size, n . The equation for calculating sample size is shown below. where is the z score is the margin of error is the population size is the population proportion EX: Determine the sample size necessary to estimate the proportion of people shopping at a supermarket in the U.S. that identify as vegan with 95% confidence, and a margin of error of 5%. Assume a population proportion of 0.5, and unlimited population size. Remember that z for a 95% confidence level is 1.96. Refer to the table provided in the confidence level section for z scores of a range of confidence levels. Thus, for the case above, a sample size of at least 385 people would be necessary. In the above example, some studies estimate that approximately 6% of the U.S. population identify as vegan, so rather than assuming 0.5 for p̂ , 0.06 would be used. If it was known that 40 out of 500 people that entered a particular supermarket on a given day were vegan, p̂ would then be 0.08. Search Inorganic Chemistry Frontiers A sunlight sensitive metal–organic framework film for the environment-friendly self-sterilization application †. * Corresponding authors a School of Chemistry and Pharmaceutical Engineering, Medical Science and Technology Innovation Center, Shandong First Medical University & Shandong Academy of Medical Sciences, Jinan 250021, China E-mail: [email protected] b Key Laboratory of Bionic Engineering, Ministry of Education, Jilin University, Changchun 130022, China Frequent outbreaks of respiratory diseases caused by the spread of pathogenic microorganisms poses a great threat to human health. Disposable surgical masks (SM) can limit the transmission to some extent but the inability to self-sterilize leads to a potential source of cross-contamination as well as causing hazardous impacts on the soil and aquatic ecosystems. Herein, a sunlight-triggered photoactive self-sterilization metal–organic framework (MOF) film modified SM is reported. The MOF film can be prepared easily and quickly by the co-assembly of UiO-66 nanoparticles and metal-phenolic networks (MPNs, the complexation of tannic acid and iron ions). Unlike the traditional MPN only assisted assembly, this MOF film exhibited a dramatic synergetic photodynamic and photothermal self-sterilization ability for protective equipment surfaces under simulated-solar irradiation without any extra photosensitizers or photothermal agents. A study of the mechanism revealed that the light absorption region of UiO-66 can be expanded by the MPNs, demonstrating a unique photocatalytic sensitization, and thus the MOF film can utilize the sunlight extensively to achieve a photodynamic property. Moreover, such an MOF film demonstrated excellent cytocompatibility, haemocompatibility and breathability, indicating the security of the skin-touched application. This dual-modal photocatalytic MOF film are promising for the long-term usage of disposable protective equipment and for mitigating the heavy burden on the environment. This article is part of the themed collection: FOCUS: Design and applications of metal-organic frameworks (MOFs) Supplementary files Supplementary information PDF (514K) Article information Download citation, permissions. A sunlight sensitive metal–organic framework film for the environment-friendly self-sterilization application L. Hao, J. Gao, X. Han, Z. Li, Y. Dong, L. Sun, L. Zhou, Z. Ning, J. Zhao and R. Jiang, Inorg. Chem. Front. , 2024,  11 , 4229 DOI: 10.1039/D4QI00715H To request permission to reproduce material from this article, please go to the Copyright Clearance Center request page . If you are an author contributing to an RSC publication, you do not need to request permission provided correct acknowledgement is given. If you are the author of this article, you do not need to request permission to reproduce figures and diagrams provided correct acknowledgement is given. If you want to reproduce the whole article in a third-party publication (excluding your thesis/dissertation for which permission is not required) please go to the Copyright Clearance Center request page . Read more about how to correctly acknowledge RSC content . Social activity Search articles by author, advertisements. 233 IMAGES Hypothesis Testing Formula Sample size hypothesis test calculator Two Sample Z Hypothesis Test Hypothesis Sample Mean Calculator hypothesis test formula statistics Hypothesis Testing Formula COMMENTS Hypothesis Testing Calculator with Steps Hypothesis Testing Calculator. The first step in hypothesis testing is to calculate the test statistic. The formula for the test statistic depends on whether the population standard deviation (σ) is known or unknown. If σ is known, our hypothesis test is known as a z test and we use the z distribution. If σ is unknown, our hypothesis test is ... Hypothesis Test Calculator Use this Hypothesis Test Calculator for quick results in Python and R. Learn the step-by-step hypothesis test process and why hypothesis testing is important. t-test Calculator This t-test calculator allows you to use either the p-value approach or the critical regions approach to hypothesis testing! Enter your T-score and the number of degrees of freedom . If you don't know them, provide some data about your sample(s): sample size, mean, and standard deviation, and our t-test calculator will compute the T-score and ... 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. Single Sample T-Test Calculator Single Sample T-Test Calculator. A single sample t-test (or one sample t-test) is used to compare the mean of a single sample of scores to a known or hypothetical population mean. So, for example, it could be used to determine whether the mean diastolic blood pressure of a particular group differs from 85, a value determined by a previous study. One sample t test calculator Select Data Type: Choose whether to input summary statistics directly or provide a data set. Input Your Data: Enter the required values such as population mean, sample size, sample mean, and sample standard deviation. Set Hypotheses: Specify the null and alternative hypotheses. Calculate: Click the "Calculate" button to see the test statistic ... T test calculator A t test compares the means of two groups. There are several types of two sample t tests and this calculator focuses on the three most common: unpaired, welch's, and paired t tests. Directions for using the calculator are listed below, along with more information about two sample t tests and help on which is appropriate for your analysis. NOTE: This is not the same as a one sample t test; for ... 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 ... Hypothesis Testing Calculator 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 ... T-Test Calculator T-Test calculator. The Student's t-test is used to determine if means of two data sets differ significantly. This calculator will generate a step by step explanation on how to apply t - test. Two sample t-test One sample t-test. Free One-Sample t-Test Calculator This calculator will conduct a complete one-sample t-test, given the sample mean, the sample size, the hypothesized mean, and the sample standard deviation. The results generated by the calculator include the t-statistic, the degrees of freedom, the critical t-values for both one-tailed (directional) and two-tailed (non-directional) hypotheses, and the one-tailed and two-tailed probability ... One Sample T Test Calculator The one-sample t-test determines if the mean of a single sample is significantly different from a known population mean. The one sample t-test calculator calculates the one sample t-test p-value and the effect size. When you enter the raw data, the one sample t-test calculator provides also the Shapiro-Wilk normality test result and the outliers. Hypothesis Test Calculator: t-test, chi-square, analysis of variance Clear the table in the Hypothesis test calculator. Copy your data into the table. Select the variables. In the hypothesis test calculator you can calculate e.g. a t-test, a chi-square test, a binomial test or an analysis of variance. If you need a more detailed explanation, you can find more information in the tutorials. One Sample t-test Calculator A one sample t-test is used to test whether or not the mean of a population is equal to some value. To perform a one sample t-test, simply fill in the information below and then click the "Calculate" button. Enter raw data Enter summary data. 301, 298, 295, 297, 304, 305, 309, 298, 291, 299, 293, 304. μ0 (hypothesized population mean) t ... T-Test Calculator for 2 Independent Means 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. 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 ... Two Sample t-test Calculator A two sample t-test is used to test whether or not the means of two populations are equal. This type of test assumes that the two samples have equal variances. If this is not the case, you should instead use the Welch's t-test calculator. To perform a two sample t-test, simply fill in the information below and then click the "Calculate ... Quickly Perform Hypothesis Tests Online for Free Hypothesis Test Calculator. Upload your data set below to get started. Upload File. Or input your data as csv. column_one,column_two,column_three 1,2,3 4,5,6 7,8,9. Submit CSV. Sharing helps us build more free tools. Online Statistics Calculator: Hypothesis testing, t-test, chi-square Here you will find everything about hypothesis testing: One sample t-test, Unpaired t-test, Paired t-test and Chi-square test. You will also find tutorials for non-parametric statistical procedures such as the Mann-Whitney u-Test and Wilcoxon-Test . mann-whitney-u-test and the Wilcoxon test Calculators Phi Coefficient Calculator. Hypothesis Tests One Sample t-test Calculator Two Sample t-test Calculator One Sample Z-Test Calculator Two Sample Z-Test Calculator Welch's t-test Calculator Paired Samples t-test Calculator F-Test for Equal Variances Calculator Wilcoxon Signed-Rank Test Calculator Two Sample T-Test Calculator Two Sample T-Test Calculator. Two sample t-test is used to check whether the means of two groups are significantly different from each other. For example, if you want to see if mean weight of males and females have statistically significant difference between them. Independent T-test assumes that the two samples have equal variances. Two-Sample t-test This calculator should be used when the sampling units (e.g. the sampled individuals) in the two groups are independent. If you are comparing two measurements taken on the same sampling unit (e.g. blood pressure of an individual before and after a drug is administered) then the appropriate test is the paired t-test. Difference in Means Hypothesis Test Calculator Calculate the results of your two sample t-test. Use the calculator below to analyze the results of a difference in sample means hypothesis test. Enter your sample means, sample standard deviations, sample sizes, hypothesized difference in means, test type, and significance level to calculate your results. You will find a description of how to ... Sample Size Calculator Sample Size Calculation. Sample size is a statistical concept that involves determining the number of observations or replicates (the repetition of an experimental condition used to estimate the variability of a phenomenon) that should be included in a statistical sample. It is an important aspect of any empirical study requiring that ... A sunlight sensitive metal-organic framework film for the environment Frequent outbreaks of respiratory diseases caused by the spread of pathogenic microorganisms poses a great threat to human health. Disposable surgical masks (SM) can limit the transmission to some extent but the inability to self-sterilize leads to a potential source of cross-contamination as well as causing FOCUS: Design and applications of metal-organic frameworks (MOFs) assignment essay homework paper phd report resume review speech thesis