In this article, we explain the three important applications of the T-test in statistics. The T-test is a parametric test based on the T distribution. The three main applications are:

- One Sample T-test – To test if the population mean differs significantly from the

given hypothetical value. - Paired Two Sample T-test – To test the significance of the difference between the means of two paired samples.
- Unpaired Two Sample T-test – To test the significance of the difference between the means of two unpaired samples.

**1) One Sample T-Test:**

Here we want to check if the population mean is equal to a given value or not. We formulate the null and alternative hypotheses as follows,

H_{0} : µ=µ_{0} vs H_{1}: µ ≠ µ_{0}

The T statistic can be calculated using the formula,

t = (x̄- µ_{0})/(s/√n) where,

**x̄ is the sample mean, **

**µ _{0} is the assumed mean value, **

**s is the sample standard deviation and,**

**n is the sample size.**

**Example**:

Suppose that a random sample of 10 boys had the following I.Q.s:

70, 120, 110, 101, 88, 83, 95, 98, 107, 100. We want to test the hypothesis that the average IQ of the population is 100. The test is carried out as follows:

- We calculate the sample mean and sample variance. We find that to x̄ = 97.2 and s
^{2}= 203.73 - We are given that µ
_{0}= 100. - The value of the T statistic is t = (x̄- µ
_{0})/(s/√n) = 0.62 - Checking the p-value from the T table for (n-1) = 9 degrees of freedom we get that,

P Value = 0.5506

Since the P value is greater than 0.05 we accept the null hypothesis and conclude that the average IQ is indeed 100.

**2) Paired Two Sample T-Test:**

Here, we want to test the significance of the difference between the means of two paired samples. By a paired sample we mean that the two samples are related and their variances are equal. The two sample sizes are also equal.

For example, suppose that the average annual sales of a company are 10 million units. The company runs an advertising campaign and the yearly annual sales increase to 110.3 million units.

In the above example, the two samples are related to each other. We formulate the null and alternative hypotheses as follows,

*H*_{0 }: *μ*_{1}=*μ*_{2} vs H_{1}: *μ*_{1}≤*μ*_{2}

Here, *μ*_{1} denotes the average sales before the advertising campaign and *μ*_{2 }denotes the average sales after the advertising campaign. The paired two-sample T-test will tell us if the advertising campaign was effective or not.

Another example, where we can apply a paired T-test is the before and after effect of a given brand of fertilizer on the growth rate of crops.

**3) Unpaired Two Sample T-Test:**

This test is applied when we are given two samples that are not related to each other. The population variances as well as the sample sizes need not be equal.

An example of a situation where we can apply an unpaired T-test is as follows. Suppose that we are given a sample of 70 people from City A and a sample of 50 people from City B.

We want to test the hypothesis that the average weight of people in both cities is equal. In this case, since the weight distribution in the two cities is independent we apply the two-sample T-test.