 # Test statistic calculator: Estimation and Theorem

The test statistics calculator is also known as the T-Test Calculator. It is basically used with the motive to estimate the significance of the differences between the mean of two samples where the null hypothesis is present. In case there is a null hypothesis, that means there is no significant difference between the mean with the standard deviations.

There are two categories in which the test statistic calculator has been categorized and used to Calculate Test Statistic:

• One sample t-test calculator
• Two sample t-test calculator

## One-Sample T-Test Calculator:

Single sample test statistics calculator is basically used for comparing the mean that is derived with the help of a single sample of scores. It is also abbreviated as t test calculator. This calculator is used for performing very specific calculations related to mean for hypothetical population or diastolic blood pressure of a particular group of patients.

This One-Sample T-Test Calculator plays an important role in forming a complete one-sample t-test. In this, the sample mean, the sample size, the hypothesized mean, and the sample standard deviation are provided. The result that gets generated with the help of this calculator includes the statistics, and also the degrees of freedom along with the critical t values for both the directional as well as the non-directional hypothesis. Along with this, the directional and non-directional probability values are always associated with the results and the tests conducted.

## Two sample t-test calculator:

This calculator plays an important role in testing the mean difference between two samples of the data that is continuous in nature. In this, that two-sample t-test procedure is always followed. From the student distribution, this calculator uses the probabilities.

## T Stat Calculator Formula:

If you are looking for the formula behind Null Hypothesis Calculator, then take a look over the below section: For conducting a hypothesis test having null hypothesis H_0 : \mu_1 = \mu_2, the test statistics values will be determined by using the formula as:
Here in this formula:

• ¯X1 and ¯X2X¯2 are sample means,
• s1 and s2 are standard deviation samples
• n1 and n2 are sizes of samples given

## How to Calculate Test Statistic?

If you are willing to complete the calculation with the help of the T Statistic Calculator then you are suggested to take a look over the below section of instructions:

• First of all, you need to enter two samples that you have observed
• Must remember that the values are real numbers or variable
• After entering the details, you need to select and hit on the calculate tab
• Within seconds, the Hypothesis Test Calculator will provide you the exact results.

## Test Statistic Formula with steps (Theorem):

In order to let you understand the working of the T-Test Calculator, here we have stated an example with respect to Probability. Here, There are two hypotheses included in a hypothesis test. The two hypotheses include the null hypothesis and the alternative hypothesis. The alternative hypothesis is also known as research hypothesis.
To represent the null hypothesis, the symbol H_0 is used. The null hypothesis shows that the experiment will have no observed effect. It can also be considered as an equal sign. On the other hand, the alternative hypothesis represent that there is an effect on the experiment that is observed. To represent the alternative hypothesis, the symbol Ha is used.
The first step that needs to be followed in this texting is about determining the null hypothesis and the alternative hypothesis. While completing the hypothesis testing, it is important remember some important terms like the rejection region, and the non-rejection region along with the critical values. The value that leads to the rejection of the null hypothesis is considered as the rejection region. On the other hand, the value that leads to non-rejection of the null hypothesis is considered as non-rejection region. There are several values that separate the rejection and the non-rejection regions. Such values are known as critical values.
In order to compare the mean of two populations, the t-test is used. There are two approaches that are followed.

• When the two population samples are independent
• When the samples derived from the two populations are paired or dependent

In case, we are having the samples that are independent then it is a possibility that the pair of samples are likely to be equal to the pair of samples estimated. Let's take an example that X is a variable that is distributed normally in the each set of both populations. Therefore, for availing the independent samples of size N1 and n2 from the populations, the mean derived from the possible differences between the two samples will be equal to the mean that is derived from the difference between the two population means.
\mu_{\bar X_1-\bar X_2}=\mu_1-\mu_2
To determine the standard deviation between the two sample standard deviation, there is a specific formula used as:
To determine the standard deviation between the two sample standard deviation, there is a specific formula used as:
\sigma_{\bar X_1-\bar X_2}=\sqrt{\frac{\sigma^2_1}{n_1}+\frac{\sigma^2_2}{n_2}}
In this formula, ¯X1−¯X2 is distributed normally,The variable given

z=\frac{(\bar X_1-\bar X_2)-(\mu_1-\mu_2)}{\sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}}}

Also contains the standard normal distribution. Apart from this, σ1 and σ2 are the standard deviations of population.
In this distribution, there are two cases derived:
If populations standard deviations are equal, σ1−σ2 Formula used to determine the pooled sample standard deviation will be:

s_p=\sqrt{\frac{(n_1-1)s_1+(n_2-1)s_2}{n_1+n_2-2}}

Where: s_{{1}} and s_{{2}} are sample standard deviations,In case, x is a variable that is normally distributed then for the samples that are independent and having size and 1 and 2 from the different populations come the variable formula given as

t=\frac{\bar X_1-\bar X_2}{s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}

Will have the t-distribution with df=n1+n2−2. In case, the hypothesis test with a null hypothesis

H_0 : \mu_1 = \mu_2

,the variable used will be

t=\frac{\bar X_1-\bar X_2}{s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}

## Incase both the population standard deviations are different

If X is a variable that is normally distributed on both the population, the variable for the independent samples of two populations having size n_{{1}} and n_{{2}}

t=\frac{(\bar X_1-\bar X_2)-(\mu_1-\mu_2)}{\sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}}}

will have an approximate t-distribution. The formula used to determine the degree of freedom will be

\Delta=\frac{\Big[\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}\Big]^2}{\frac{\Big(\frac{\sigma_1^2}{n_1}\Big)^2}{n_1-1}+\frac{\Big(\frac{\sigma_2^2}{n_2}\Big)^2}{n_2-1}}

The variable used to determine the hypothesis test having a null hypothesis will be

t=\frac{(\bar X_1-\bar X_2)}{\sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}}}

For availing the critical values or P value, the t table will be used. The hypothesis testing procedure given above is called the non-pooled t test. There are two formats in which table is available. The formats are one tail and two tail.

## What is the Value Of Test Statistic Calculator?

The Null Hypothesis Calculator plays an important role in performing a complete 1-sample-based t-test. In order to Find Test Statistic Calculator calculation, it is necessary to have the following parameters:

• Sample Mean
• Sample Size
• hypothesized Mean
• Sample Standard Deviation

Once the calculation gets completed, the result includes the value of the T-Statistic along with the Degrees of Freedom, Two-Tailed Hypotheses (Non-Directional), One-Tailed Hypotheses (Directional), Critical T-Values, One-Tailed Probability Values, and Two-Tailed Probability Values. Hence, these are all the details about Hypothesis Testing Calculator. You can use this calculator and can easily perform the calculation.