How to perform a Student's t-test?

To perform a Student's t-test, first determine the null and alternative hypotheses. Then calculate the t-statistic and compare it to the critical value.

The Student's t-test is a statistical test used to determine if there is a significant difference between the means of two groups. The null hypothesis states that there is no significant difference between the means, while the alternative hypothesis states that there is a significant difference.

To perform the t-test, first calculate the sample means and standard deviations for each group. Then calculate the pooled standard deviation using the formula:

s_p = sqrt(((n1-1)s1^2 + (n2-1)s2^2)/(n1+n2-2))

where n1 and n2 are the sample sizes, s1 and s2 are the sample standard deviations, and s_p is the pooled standard deviation.

Next, calculate the t-statistic using the formula:

t = (x1 - x2)/(s_p * sqrt(1/n1 + 1/n2))

where x1 and x2 are the sample means.

Finally, compare the t-statistic to the critical value from the t-distribution table with degrees of freedom equal to n1 + n2 - 2 and the desired level of significance. If the t-statistic is greater than the critical value, reject the null hypothesis and conclude that there is a significant difference between the means. Otherwise, fail to reject the null hypothesis.

It is important to note that the t-test assumes that the populations are normally distributed and have equal variances. If these assumptions are not met, alternative tests such as the Welch's t-test or the Mann-Whitney U test may be more appropriate.