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 [latex]X1ˉ[/latex][latex]bar{X_1}[/latex][latex]X1​ˉ​[/latex] and [latex]X2ˉ[/latex][latex]bar{X_2}[/latex][latex]X2​ˉ​[/latex], sample sizes [latex]n1[/latex][latex]n_1[/latex][latex]n1​[/latex] and [latex]n2[/latex][latex]n_2[/latex][latex]n2​[/latex], and variances [latex]s12[/latex][latex]s_1^2[/latex][latex]s12​[/latex] and [latex]s22[/latex][latex]s_2^2[/latex][latex]s22​[/latex], the formula for the t-test is:

[latex display=”true”] t = frac{bar{X_1} – bar{X_2}}{sqrt{frac{s_1^2}{n_1} + frac{s_2^2}{n_2}}} [/latex]

where:

• [latex]X1ˉ[/latex][latex]bar{X_1}[/latex][latex]X1​ˉ​[/latex] and [latex]X2ˉ[/latex][latex]bar{X_2}[/latex][latex]X2​ˉ​[/latex] are the means of the two samples,
• [latex]s12[/latex][latex]s_1^2[/latex][latex]s12​[/latex] and [latex]s22[/latex][latex]s_2^2[/latex][latex]s22​[/latex] are the variances of the two samples,
• [latex]n1[/latex][latex]n_1[/latex][latex]n1​[/latex] and [latex]n2[/latex][latex]n_2[/latex][latex]n2​[/latex] are the sample sizes.

Steps for Performing the T-Test

1. Calculate the Means [latex]bar{X_1}[/latex] and ([latex]bar{X_2}[/latex] of each sample.
2. Calculate the Variances [latex]s_1^2[/latex] and ([latex]s_2^2[/latex] 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:

[latex display=”true”] df = min(n_1 – 1, n_2 – 1) [/latex]

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.