Welcome back to the AI Bayeslab Statistics series. I understand that data is abundant in your work. Using the AI Agent online tool, you’ll discover that nearly 90% of your data analysis tasks can be handled without needing a dedicated data analyst- you can resolve them independently.

If you are comparing two means from two different groups, what analysis are you conducting? We have to choose the right Statistical Test.

Let’s proceed with discussing hypothesis testing for the means of two populations. Previously, we explored an example where sigma was known. Today, we will wrap up this section with an example involving an unknown sigma.

The t-distribution and its use in Hypothesis Testing

When σ₁² equals σ₂², both populations exhibit a normal distribution. However, this applies even when the populations are not normally distributed, as long as sampling involves a large sample size-defined as having more than 30 individuals per sample group.

So when the data sample accords with the condition as mentioned above, we can use a hypothesis with a t-statistic; the formula is as follows:

I discussed some typical statistical distributions before. You can refer to the post “That means the data follows the T-distribution. We discussed some typical statistical distributions before. You can refer to the post “Parameter Estimation with a Real-Life Example: A Guide for Beginners in 2025.”

What is df used for in statistics?

Therefore, when our data sample groups conform to the t-distribution, we must consider another important idea: “degrees of freedom.” We introduced the definition before, but let’s recall it here.

Degrees of Freedom: The number of values that can vary independently in the parameter estimates of a population.

This indicates the count of individuals in each group. For instance, if Group A had 10 patients, the degrees of freedom would be 10.

Unlike the normal distribution, the shape of the t-distribution is a family of distributions with varying degrees of freedom (n).

• When n is less than 30, the dispersion of the t-distribution is significantly wider than that of a normal distribution, and its density function is more stable.

• As n increases, the dispersion of the t-distribution increasingly overlaps with that of the normal distribution.

• When n is more than 30, the dispersion of the t-distribution almost completely overlaps with the normal distribution.

In other words, normal distribution is a limiting form of the t-distribution. Please refer to the image below.

Example: Two Population Means with Unknown Standard Deviations (σ₁², σ₂² unknown)

Data sample description:

• Group1: male-student.csv — Sample size: 100, Mean 81.6, Standard Deviation 7.57

• Group2: female-student.csv — Sample size: 80, Mean 76.98, Standard Deviation 9.20

Question: Is there a significance between the two different sexual populations?

This is a case of two populations with Unknown Standard Deviations, so let’s analyze it step by step as we always do.

Step1. State the Hypothesis

• Null Hypothesis (H₀):

• Typically represents no significant difference or effect. For example,

• H₀: μ = 100 (the population mean equals 100).

• Alternative Hypothesis (H₁):

• Indicates a significant difference or effect.

• For example, H₁: μ ≠ 100 (the population mean does not equal 100).

In this scenario, we use a two-tailed test. Null Hypothesis (H₀) vs Alternative Hypothesis (H₁)

H₀: μ₁ = μ₂ or μ₁-μ₂= 0

H₁: μ₁ ≠ μ₂ or μ₁-μ₂≠ 0

Step2. Choose the Significance Level and Appropriate Test

The population variances are both unknown and adhere to a t-distribution curve.

To refresh our memory, we established four criteria that guide the selection of an appropriate hypothesis test.

We created a summary sheet to evaluate both sample and population, which helped us select the test statistic and the corresponding formula.

In this scenario, we can utilize the t-distribution and the test statistic detailed below:

Step3. Calculate the Test Statistic

We have calculated the means of the two samples above:

▶︎ Female: X̅₁ = 76.98

▶︎ Male: X̅₂ =81.6

▶︎ 𝑛₁ =100

▶︎ 𝑛₂ = 80

So the degrees of freedom = 𝑛₁ + 𝑛₂-1 = 178

We can calculate the corresponding t value by substituting these into the formula.

(𝑥̅₁ − 𝑥̅₂)

𝑡 = ____________________

√(𝑠ₚ² (1/𝑛₁ + 1/𝑛₂))

Where:

(𝑛₁ − 1)𝑠₁² + (𝑛₂ − 1)𝑠₂²

𝑠ₚ² = ____________________

(𝑛₁ + 𝑛₂ − 2)

However, at this point, we will not perform the calculation manually.

We will integrate it into one step and let the AI Agent perform the calculation based on the selected formula and the null hypothesis.

Step4. Make a Decision

AI prompts are powerful tools that enable us to accomplish various tasks and obtain desired results efficiently. By leveraging technology in this way, I believe we have an incredible opportunity to reintegrate statistics into people’s daily routines, rekindling interest in data and analytical thinking.

This revival is essential because statistics can provide valuable insights into various aspects of life, helping individuals make more informed decisions.

By making statistics accessible and relevant, we enhance understanding and empower people to engage with data meaningfully, ultimately enriching their everyday experiences.

Stay tuned, subscribe to Bayeslab, and help everyone gain mastery in the wisdom of statistics affordably.