## Hypothesis Testing Calculator

$H_o$: | |||

$H_a$: | μ | ≠ | μ₀ |

$n$ | = | $\bar{x}$ | = | = |

$\text{Test Statistic: }$ | = |

$\text{Degrees of Freedom: } $ | $df$ | = |

$ \text{Level of Significance: } $ | $\alpha$ | = |

## Type II Error

$H_o$: | $\mu$ | ||

$H_a$: | $\mu$ | ≠ | $\mu_0$ |

$n$ | = | σ | = | $\mu$ | = |

$\text{Level of Significance: }$ | $\alpha$ | = |

The first step in hypothesis testing is to calculate the test statistic. The formula for the test statistic depends on whether the population standard deviation (σ) is known or unknown. If σ is known, our hypothesis test is known as a z test and we use the z distribution. If σ is unknown, our hypothesis test is known as a t test and we use the t distribution. Use of the t distribution relies on the degrees of freedom, which is equal to the sample size minus one. Furthermore, if the population standard deviation σ is unknown, the sample standard deviation s is used instead. To switch from σ known to σ unknown, click on $\boxed{\sigma}$ and select $\boxed{s}$ in the Hypothesis Testing Calculator.

$\sigma$ Known | $\sigma$ Unknown | |

Test Statistic | $ z = \dfrac{\bar{x}-\mu_0}{\sigma/\sqrt{{\color{Black} n}}} $ | $ t = \dfrac{\bar{x}-\mu_0}{s/\sqrt{n}} $ |

Next, the test statistic is used to conduct the test using either the p-value approach or critical value approach. The particular steps taken in each approach largely depend on the form of the hypothesis test: lower tail, upper tail or two-tailed. The form can easily be identified by looking at the alternative hypothesis (H a ). If there is a less than sign in the alternative hypothesis then it is a lower tail test, greater than sign is an upper tail test and inequality is a two-tailed test. To switch from a lower tail test to an upper tail or two-tailed test, click on $\boxed{\geq}$ and select $\boxed{\leq}$ or $\boxed{=}$, respectively.

Lower Tail Test | Upper Tail Test | Two-Tailed Test |

$H_0 \colon \mu \geq \mu_0$ | $H_0 \colon \mu \leq \mu_0$ | $H_0 \colon \mu = \mu_0$ |

$H_a \colon \mu | $H_a \colon \mu \neq \mu_0$ |

In the p-value approach, the test statistic is used to calculate a p-value. If the test is a lower tail test, the p-value is the probability of getting a value for the test statistic at least as small as the value from the sample. If the test is an upper tail test, the p-value is the probability of getting a value for the test statistic at least as large as the value from the sample. In a two-tailed test, the p-value is the probability of getting a value for the test statistic at least as unlikely as the value from the sample.

To test the hypothesis in the p-value approach, compare the p-value to the level of significance. If the p-value is less than or equal to the level of signifance, reject the null hypothesis. If the p-value is greater than the level of significance, do not reject the null hypothesis. This method remains unchanged regardless of whether it's a lower tail, upper tail or two-tailed test. To change the level of significance, click on $\boxed{.05}$. Note that if the test statistic is given, you can calculate the p-value from the test statistic by clicking on the switch symbol twice.

In the critical value approach, the level of significance ($\alpha$) is used to calculate the critical value. In a lower tail test, the critical value is the value of the test statistic providing an area of $\alpha$ in the lower tail of the sampling distribution of the test statistic. In an upper tail test, the critical value is the value of the test statistic providing an area of $\alpha$ in the upper tail of the sampling distribution of the test statistic. In a two-tailed test, the critical values are the values of the test statistic providing areas of $\alpha / 2$ in the lower and upper tail of the sampling distribution of the test statistic.

To test the hypothesis in the critical value approach, compare the critical value to the test statistic. Unlike the p-value approach, the method we use to decide whether to reject the null hypothesis depends on the form of the hypothesis test. In a lower tail test, if the test statistic is less than or equal to the critical value, reject the null hypothesis. In an upper tail test, if the test statistic is greater than or equal to the critical value, reject the null hypothesis. In a two-tailed test, if the test statistic is less than or equal the lower critical value or greater than or equal to the upper critical value, reject the null hypothesis.

Lower Tail Test | Upper Tail Test | Two-Tailed Test |

If $z \leq -z_\alpha$, reject $H_0$. | If $z \geq z_\alpha$, reject $H_0$. | If $z \leq -z_{\alpha/2}$ or $z \geq z_{\alpha/2}$, reject $H_0$. |

If $t \leq -t_\alpha$, reject $H_0$. | If $t \geq t_\alpha$, reject $H_0$. | If $t \leq -t_{\alpha/2}$ or $t \geq t_{\alpha/2}$, reject $H_0$. |

When conducting a hypothesis test, there is always a chance that you come to the wrong conclusion. There are two types of errors you can make: Type I Error and Type II Error. A Type I Error is committed if you reject the null hypothesis when the null hypothesis is true. Ideally, we'd like to accept the null hypothesis when the null hypothesis is true. A Type II Error is committed if you accept the null hypothesis when the alternative hypothesis is true. Ideally, we'd like to reject the null hypothesis when the alternative hypothesis is true.

Condition | ||||

$H_0$ True | $H_a$ True | |||

Conclusion | Accept $H_0$ | Correct | Type II Error | |

Reject $H_0$ | Type I Error | Correct |

Hypothesis testing is closely related to the statistical area of confidence intervals. If the hypothesized value of the population mean is outside of the confidence interval, we can reject the null hypothesis. Confidence intervals can be found using the Confidence Interval Calculator . The calculator on this page does hypothesis tests for one population mean. Sometimes we're interest in hypothesis tests about two population means. These can be solved using the Two Population Calculator . The probability of a Type II Error can be calculated by clicking on the link at the bottom of the page.

## 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

## Two-Sample t-Test

Equal variances:, unequal variances (welch’s t-test):, table of critical t-values.

Confidence Level (%) | df=10 | df=30 | df=50 | df=100 |
---|---|---|---|---|

90 | 1.812 | 1.697 | 1.676 | 1.660 |

95 | 2.228 | 2.042 | 2.009 | 1.984 |

99 | 3.169 | 2.750 | 2.678 | 2.626 |

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.

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

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.

## Related Calculators

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## 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-1dtpypy ul:not(:first-child),.css-1dtpypy 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.

## Related Calculators :

- List of all calculator
- P-value calculator
- Critical value Calculator
- One sample z test calculator

## 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.

For optimal use, please visit DATAtab on your desktop PC!

## Metric Variables:

Ordinal variables:, nominal variables:, hypothesis test calculator.

Do you want to calculate a hypothesis test such as a t-test , Chi Square test or an ANOVA ? You can do that easily here in the browser.

If you want to use your own data just clear the upper table

- 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.

In order to use the hypothesis test calculator, you must first formulate your hypothesis and collect your data. DATAtab will then suggest the hypothesis test you need based on the data entered into the statistics calculator.

## p value calculator

With the p value calculator you can calculate the p value for different tests. There is a wide range of methods for this. Just click on the variables you want to evaluate above and DATAtab will give you the tests you can use.

For example, if you select a metric and a categorical variable, the Independent t-Test calculator is automatically selected. If your data is not normally distributed, simply use the Mann-Whitney U-test calculator.

## H0 and H1 calculator

With the h0 and h1 calculator for the different hypothesis test you can calculate the p-value which gives you an indication if you can reject the H0 or not.

Cite DATAtab: DATAtab Team (2024). DATAtab: Online Statistics Calculator. DATAtab e.U. Graz, Austria. URL https://datatab.net

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Test Statistic Calculator

Choose the method, enter the values into the test statistic calculator, and click on the “Calculate” button to calculate the statistical value for hypothesis evaluation.

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This test statistic calculator helps to find the static value for hypothesis testing. The calculated test value shows if there’s enough evidence to reject a null hypothesis. Also, this calculator performs calculations of either for one population mean, comparing two means, single population proportion, and two population proportions.

Our tool is highly useful in various fields like research, experimentation, quality control, and data analysis.

## What is Test Statistics?

A test statistic is a numerical value obtained from the sample data set. It summarizes the differences between what you observe within your sample and what would be expected if a hypothesis were true.

The t-test statistic also shows how closely your data matches the predicted distribution among the sample tests you perform.

## How to Calculate Test Statistics Value?

- Collect the data from the populations
- Use the data to find the standard deviation of the population
- Calculate the mean (μ) of the population using this data
- Determine the z-value or sample size
- Use the suitable test statistic formula and get the results

## Test Statistic For One Population Mean:

Test statistics for a single population mean is calculated when a variable is numeric and involves one population or a group.

x̄ - µ 0 σ / √n

- x̄ = Mean of your sample data
- µ 0 = Hypothesized population mean that you are comparing to your sample mean
- σ = Population standard deviation
- n = number of observations (sample size) in your data set

Suppose we want to test if the average height of adult males in a city is 70 inches. We take a sample of 25 adult males and find the sample mean height to be 71 inches with a sample standard deviation of 3 inches. We use a significance level of 0.05.

t = 70 - 71 3√25

## Test Statistic Comparing Two Population Means:

This test is applied when the numeric value is compared across the various populations or groups. To compute the resulting test statistic, two distinct random samples must be chosen, one from each population.

\(\frac{√x̄ - √ȳ}{\sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}}\)

- ȳ = means of hypothesized population

Suppose we want to test if there is a difference in average test scores between two schools. We take a sample of 30 students from school A with an average score of 85 and a standard deviation of 5, and a sample of 35 students from school B with an average score of 82 and a standard deviation of 6.

t = 85 - 82 √5 2 / 30 + 6 2 / 35

t = 3 √ 25/30 + 36/35

t = 3 √0.833 + 1.029

t = 3 √1.862

## Test Statistic For a Single Population Proportion:

This test is used to determine if a single population's proportion differs from a specified standard. The test stat calculator works for a population proportion when dealing with data by having a limit of P₀ because proportions represent parts of a whole and cannot logically exceed the total or be negative.

\(\frac{\hat{p}-p_{0}}{\sqrt{\frac{p_{0}(1-p_{0})}{n}}}\)

- P̂ = Sample proportion
- P 0 = Population proportion

Suppose we want to test if the proportion of left-handed people in a population is 10%. We take a sample of 100 people and find that 8 are left-handed. We use a significance level of 0.05.

= P̂ - P₀ √0.10 (1 - 0.10)/100

= 0.08 - 0.10 √0.10 (1 - 0.10)/100

= -0.02 √0.10 (0.9)/100

= -0.02 √0.009

= -0.02 0.03

= −0.67

## Test Statistic For Two Population Proportion:

This test identifies the difference in proportions between two independent groups to assess their significance.

\(\frac{\hat{p}_{1}-\hat{p}_{2}}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_{1}}+\frac{1}{n_{2}})}}\)

- P̂ 1 and P̂ 2 = Sample proportions for two groups

Suppose we want to test if the proportion of smokers is different between two cities. We take a sample of 150 people from City A and find that 30 are smokers, and a sample of 200 people from City B and find that 50 are smokers.

- P̂ 1 = 30 / 150 = 0.20
- P̂ 2 = 50 / 200 = 0.25
- P̂ = 30 + 50 / 150 + 200 = 0.229

Calculation:

= 0.20 - 0.25 √0.229 (1 - 0.229) (1 / 150 + 1/200)

= -0.05 √0.229 (0.771) (1 / 150 + 1 / 200)

= -0.05 √0.176 (1/150 + 1/200)

= -0.05 √0.176 (0.0113)

= -0.05 √0.002

= -0.05 0.045

= −1.11

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## P-value Calculator

Please provide any one value below to compute p-value from z-score or vice versa for a normal distribution.

Z-score | ||

P-value (x<Z, left tail) | ||

P-value (x>Z, right tail) | ||

P-value (0 to Z or Z to 0, from center) | ||

P-value (-Z<x<Z, between) | ||

P-value (x<-Z or x>Z, two tails) | ||

A p-value (probability value) is a value used in statistical hypothesis testing that is intended to determine whether the obtained results are significant. In statistical hypothesis testing, the null hypothesis is a type of hypothesis that states a default position, such as there is no association among groups or relationship between two observations. Assuming that the given null hypothesis is correct, a p-value is the probability of obtaining test results in an experiment that are at least as extreme as the observed results. In other words, determining a p-value helps you determine how likely it is that the observed results actually differ from the null hypothesis.

The smaller the p-value, the higher the significance, and the more evidence there is that the null hypothesis should be rejected for an alternative hypothesis. Typically, a p-value of ≤ 0.05 is accepted as significant and the null hypothesis is rejected, while a p-value > 0.05 indicates that there is not enough evidence against the null hypothesis to reject it.

Given that the data being studied follows a normal distribution, a Z-score table can be used to determine p-values, as in this calculator.

Search | |

## Z-test for two Means, with Known Population Standard Deviations

Instructions: This calculator conducts a Z-test for two population means (\(\mu_1\) and \(\mu_2\)), with known population standard deviations ( \(\sigma_1\) and \(\sigma_2\)). Please select the null and alternative hypotheses, type the significance level, the sample means, the population standard deviations, the sample sizes, and the results of the z-test will be displayed for you:

## The Z-test for Two Means

More about the z-test for two means so you can better use the results delivered by this solver: A z-test for two means is a hypothesis test that attempts to make a claim about the population means (\(\mu_1\) and \(\mu_2\)). More specifically, we are interested in assessing whether or not it is reasonable to claim that the two population means the population means \(\mu\) 1 and \(\mu\) 2 are equal, based on the information provided by the samples. The test has two non-overlapping hypotheses, the null and the alternative hypothesis.

The null hypothesis is a statement about the population means, corresponding to the assumption of no effect, and the alternative hypothesis is the complementary hypothesis to the null hypothesis. The main properties of a one sample z-test for two population means are:

- Depending on our knowledge about the "no effect" situation, the z-test can be two-tailed, left-tailed or right-tailed
- The main principle of hypothesis testing is that the null hypothesis is rejected if the test statistic obtained is sufficiently unlikely under the assumption that the null hypothesis is true
- The p-value is the probability of obtaining sample results as extreme or more extreme than the sample results obtained, under the assumption that the null hypothesis is true
- In a hypothesis tests there are two types of errors. Type I error occurs when we reject a true null hypothesis, and the Type II error occurs when we fail to reject a false null hypothesis

How to calculate the test statistic for the two samples? We have that the formula for a z-statistic for two population means is:

The above formula allows you to assess whether or not there is a statistically significant difference between two means. The null hypothesis is rejected when the z-statistic lies on the rejection region, which is determined by the significance level (\(\alpha\)) and the type of tail (two-tailed, left-tailed or right-tailed).

In case that the population standard deviations are not known, you can use a t-test for two sample means calculator .

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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 ...

Calculation Example: There are six steps you would follow in hypothesis testing: Formulate the null and alternative hypotheses in three different ways: H 0: θ = θ 0 v e r s u s H 1: θ ≠ θ 0. H 0: θ ≤ θ 0 v e r s u s H 1: θ > θ 0. H 0: θ ≥ θ 0 v e r s u s H 1: θ < θ 0.

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.

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 ...

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:

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

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 ...

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. In order to use the hypothesis test calculator, you must first formulate ...

A particular statistical calculation that figures out the relationship between the sample and its population is known as the test statistics. It's a score that is used in the hypothesis test and informs about how likely the results are under the assumption that the null hypothesis is true.

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., males and females).

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 ... Statology makes learning statistics easy by explaining topics in simple and straightforward ways. Our team of writers have over 40 years of experience in ...

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. Requirements.

P-value Calculator. Please provide any one value below to compute p-value from z-score or vice versa for a normal distribution. A p-value (probability value) is a value used in statistical hypothesis testing that is intended to determine whether the obtained results are significant. In statistical hypothesis testing, the null hypothesis is a ...

Instructions: This calculator conducts a Z-test for one population mean (\ (\mu\)), with known population standard deviation (\ (\sigma\)). Please select the null and alternative hypotheses, type the hypothesized mean, the significance level, the sample mean, the population standard deviation, and the sample size, and the results of the z-test ...

Instructions: This calculator conducts a Z-test for two population means ( \mu_1 μ1 and \mu_2 μ2 ), with known population standard deviations ( \sigma_1 σ1 and \sigma_2 σ2 ). Please select the null and alternative hypotheses, type the significance level, the sample means, the population standard deviations, the sample sizes, and the results ...

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.

The significance level (α) is a critical concept in hypothesis testing. It represents the probability threshold below which the null hypothesis will be rejected. Common levels are 0.05 (5%) or 0.01 (1%). The choice of α affects the Z-critical value, which is used to determine whether to reject the null hypothesis based on the computed Z-score.

The test value calculator transforms the data analysis by simplifying the hypothesis testing. Attach to the guide below to utilize the test statistics calculator. Input: Choose the point that you want to calculate. Put the values according to the chosen value. Tap on "Calculate".

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.

This test statistic is then used in statistical hypothesis testing to determine if there is enough evidence to reject the null hypothesis at a certain significance level. Limitations The calculator presumes that the sample data are representative of the population and that the sampling distribution of the mean is approximately normally ...

Scale of measurement should be interval or ratio. The two sets of scores are paired or matched in some way. Null Hypothesis. H 0: U D = U 1 - U 2 = 0, where U D equals the mean of the population of difference scores across the two measurements. Equation. A T-test calculator that compares 2 dependent population means for statistical significance.

A hypothesis test for a proportion is sometimes known as a z-test because of the use of a z-score in analyzing results. If we find the probability is below the significance level, we reject the null hypothesis. Interpret Your Results. The conclusion of a hypothesis test for a proportion is always either: Reject the null hypothesis

Inferential statistics are techniques that allow statisticians to use data from a sample to make inferences or predictions for a larger population. Central to inferential statistics is the idea of hypothesis testing where data from a subset of the population is used to provide probabilistic support for a hypothesis about the larger population.