# One-Sample T-Test

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## What is a T-Test?

A t-test is an inferential statistical test that is used to determine how significant the differences are between groups of data. It does so by finding the means of these datasets and comparing them to one another. T-tests are used in research as a tool for hypothesis testing, allowing researchers to test assumptions made on the significant difference between different populations.

There are three main types of t-tests, namely:

1. One-Sample T-Test: We use the one-sample t-test when we want to test the mean of a single group against a known or hypothesized mean
2. Independent Samples T-Test: We use the independent samples t-test when we want to compare the means of two independent groups
3. Paired Sample T-Test: We use the paired sample t-test when we want to compare the means of the same group at different periods of times

Within this article, we will specifically explore the one-sample t-test to understand what it is and how it’s used.

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## What is the One-Sample T-Test

The one-sample t-test, also known as the single sample t-test, is a statistical test that is used to determine the significant difference between the mean of a single data set with a known or hypothesized mean. This mean value is often taken from previous data.

The one-sample t-test is suited to situations wherein the population standard deviation is unknown or the sample size is small. Like other forms of hypothesis testing, this test is also used to determine whether there is sufficient evidence to reject the null hypothesis (H0) and accept the alternative hypothesis (H1).

It is important to note that the one-sample t-test can be used to determine the significant difference in only one direction (one-tailed t-test) from the standard value or the significant difference in either direction from the standard value (two-tailed t-test).

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## Steps to Conduct a One-Sample T-Test

In order to conduct a one-sample t-test, you must first have the following information:

• Sample Mean (x̄)
• Population Mean ()
• Sample Standard Deviation (s)
• Number of Observations (n)

Once you have the aforementioned values, you can use the following steps to conduct the test:

• Step 1: Write your Null Hypothesis

In a one-sample t-test, the null hypothesis will generally take the following form; “the population mean equals the specified mean value”.

• Step 2: Write your Alternative Hypothesis

The alternative hypothesis will generally take the following form; “the population mean does not equal the specified mean value”.

• Step 3: Input the known Information into the T Score Formula

The t-score formula is;

t=x̄-sn

In this step, you must input your sample mean (x̄), population mean  (), sample standard deviation (s), and the number of observations (n) into the formula. Once you’ve done the calculations, you will arrive at your t-score.

• Step 4: Identify the T-Table Value

To find this value, you must have the following numbers;

1. Alpha level: Generally taken as 0.05
2. Degree of Freedom: The number of items in the sample (n) subtracted by 1 (n-1)
• Step 5: Accept or Reject the Null Hypothesis

In the final step, you must accept or reject your null hypothesis. If the t-score calculated in step 3 does not fall into the range calculated in step 4, the null hypothesis is rejected and the alternative hypothesis is accepted. On the other hand, if the t-score calculated in step 3 does fall into the range calculated in step 4, the null hypothesis is accepted and the alternative hypothesis is rejected.

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## FAQs on One-Sample T-Test

A t-test is an inferential statistical test that is used to test whether or not the means of the different unknown populations are equal.

A one-sample t-test is a type of inferential statistical test that is used to examine the extent to which the mean of a population differs statistically from a known/hypothesized value.

The three main types of t-test are; the one-sample t-test, the independent samples t-test (also known as the unpaired samples t-test), and the paired sample t-test.

A one-sample t-test should be used when the standard deviation of the population is unknown or when the sample size is small.

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