T Distribution Formula

T Distribution Formula (Table of Contents)

• T Distribution Formula
• Examples of T Distribution Formula (With Excel Template)
• T Distribution Calculator

T Distribution Formula

T Distribution is a statistical method which is used in the probability distribution formula and it has been widely recommended and use in the past and by various statisticians. The method is appropriate and it is used to estimate the population parameters when the sample size is small and or when the population variance is unknown.

In this article, we will try and understand the logic behind the formula and the relevance and the example of the same.

We often use this formula as the Sample size can also be often small and population variance and the standard deviation of the population is unknown.

So the formula of T Distribution can be defined as :

• t = T – Distribution
• x = Sample Mean
• μ = Population Mean
• S = Standard Deviation
• n = Sample Size

Examples of T Distribution Formula (With Excel Template)

Let’s take an example to understand the calculation of T Distribution in a better manner.

Example #1

The engineer of a water company which manufactures water pumps claims that the average water pumps can last 200 days. A researcher while searching finds out that an average water pump can last only 180 days with a standard deviation of 40 when selected from a random sample of 15 pumps for the purpose of testing. If the engineer claims were true we need to find out the probability that a randomly selected 15 pumps would have an average life of more than 200 days?

First, to solve this problem we need to identify the variables in the problem

Here the variables are

T Distribution is calculated using the formula given below

t = (x – μ) / (S / √n)

• T Distribution = (200 – 180) / (40 /√15)
• T Distribution = 20 / 10.32
• T Distribution = 1.94

Example #2

The CEO of a startup is evaluating the license terms of its contracts and finds out that the average contract lasts 300 days. An audit while auditing the license terms of the contracts finds out that an average contract lasts 260 days with a standard deviation of 35 when selected from a random sample of 12 contracts for the purpose of testing. If the auditor claims were true we need to find out the probability that a randomly selected 12 contracts would have an average life of more than 300 days?

First, to solve this problem we need to identify the variables in the problem

Here the variables are

T Distribution is calculated using the formula given below

t = (x – μ) / (S / √n)

• T Distribution = (300 – 260) / (35 / √12)
• T Distribution = 40 / 10.10
• T Distribution = 3.96

Example #3

Solve the given variables to find the T- distribution

T Distribution is calculated using the formula given below

t = (x – μ) / (S / √n)

• T Distribution = (360 – 200) / (40 / √17)
• T Distribution = 160 / 9.70
• T Distribution = 16.49

Explanation

Going to the formula in detail a random variable which is being standardized is being subtracted from the mean of the distribution and is then divided by the Standard deviation of the distribution. Once these terms are determined we can simply compute the T distribution.

We have assumed here that the sampling distribution will follow a normal distribution as long as the sample size is small.

The T-Distribution formula is used to compare the expected data with the sample data to be obtained from a specific hypothesis. T- Distribution is also one of the most useful formula available to a behavioral scientist. T-Distribution is used when the population parameters (mean and standard deviation) are not known

Relevance and Use of T Distribution

The formula is used in the following fields-

• Sampling and probability distribution.
• Used in a specific hypothesis.
• The test of significance level hypothesis the formula is being used.

T Distribution Calculator

You can use the following T Distribution Calculator

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