Use the t-test to compare the means of two different samples (the formula for the t-test will be provided, as shown in the Mathematical requirements)

To perform a t-test to compare the means of two samples, we follow these steps:

Formula for the T-test

Given two samples with means X1ˉbar{X_1}[latex]X1​ˉ​[/latex] and X2ˉbar{X_2}[latex]X2​ˉ​[/latex], sample sizes n1n_1[latex]n1​[/latex] and n2n_2[latex]n2​[/latex], and variances s12s_1^2[latex]s12​[/latex] and s22s_2^2[latex]s22​[/latex], the formula for the t-test is:

where:

• X1ˉbar{X_1}[latex]X1​ˉ​[/latex] and X2ˉbar{X_2}[latex]X2​ˉ​[/latex] are the means of the two samples,
• s12s_1^2[latex]s12​[/latex] and s22s_2^2[latex]s22​[/latex] are the variances of the two samples,
• n1n_1[latex]n1​[/latex] and n2n_2[latex]n2​[/latex] are the sample sizes.

Steps for Performing the T-Test

1. Calculate the Means bar{X_1} and (bar{X_2} of each sample.
2. Calculate the Variances s_1^2 and (s_2^2 of each sample.
3. Plug values into the formula to calculate the t-value.
4. Compare the t-value to a critical value from a t-distribution table (based on the chosen significance level, e.g., 0.05) and the degrees of freedom.

The degrees of freedom (df) can be approximated as:

If you have specific data for two samples, I can calculate the t-test result for you. Just provide the values, and I’ll walk you through the process.