The minimum sample size for a t-test is two (one observation in each group). However, with such a small sample size, the power of the test to detect a difference between the groups would be very low. In general, the larger the sample size, the greater the power of the test to detect a difference between the groups.

The appropriate sample size for a t-test depends on several factors, including the desired level of statistical power, the significance level, and the expected effect size. A power analysis can be used to determine the minimum sample size needed to achieve a desired level of power for a given significance level and effect size.

The Student's t-test is a parametric test, i.e. it assumes that the quantitative variable under study follows a normal distribution. For its use, the normality of the distribution of the quantitative variable must be evaluated. There are multiple tests for this, such as the Kolmorogov Smirnov test or the Saphiro Wilk test. The homogeneity of variances should also be checked by means of a Levene's test.

The nonparametric alternative to the Student's t-test is the Mann Whitney U test.

Ramkrishna Mohanta Ramkrishna, This is a quote from the US National Institute of Health. "In most studies, a sample size of at least 40 can guarantee that the sample mean is approximately normally distributed, and the one-sample t-test can then be safely applied. It is used to know whether the unknown means of two populations are different from each other based on independent samples from each population." For more details use this link to read, "The bread and butter of statistical analysis “t-test”: Uses and misuses". Article The bread and butter of statistical analysis “t-test”: Uses ...

For a sample size of n, the t distribution has n-1 degrees of freedom. A t-distribution with 1 degree of freedom has a two-sided 95% level of 12.7, with 2 degrees of freedom 4.3, etc. The standard normal two-sided 95% level is 1.96. The idea of the t-distribution is to adjust confidence levels for the small sample. Thus with 2 degrees of freedom the confidence interval is

(xbar - 4.3 se, xbar+4.3se) - the broader interval accounting for the larger sample. (Of course, there is an assumption that the parent distribution is normal - otherwise, the exact t-distribution is not valid regardless of the size of the sample)

William Sealy Gosset introduced the t-distribution to cater for small samples One can think of it as adjusting for the inferior estimates of the variance. In summary, it can, in theory, be used for any sample size from 2 up. (I would not recommend it for a sample size of 2)

The minimum sample size for a t-test depends on several factors, including the desired level of statistical power, the effect size you want to detect, and the desired level of significance (alpha).

In general, a larger sample size is preferred as it increases the statistical power and improves the reliability of the results. However, there is no fixed minimum sample size for a t-test that applies universally to all scenarios.

As a rule of thumb, some researchers suggest a minimum sample size of around 30 to 40 observations per group for a t-test to provide reasonably reliable results. This is based on assumptions of normality and other underlying conditions of the t-test. However, it's important to note that this is a general guideline and the specific requirements may vary based on the nature of the data and the research context.

To determine the appropriate sample size for your specific study, it is recommended to conduct a power analysis. A power analysis takes into account factors such as the desired level of power, effect size, alpha level, and variability in the data. It can help estimate the minimum sample size needed to detect a statistically significant effect.

Consulting with a statistician or using statistical software that incorporates power analysis can assist you in determining the minimum sample size required for your t-test based on the specific parameters of your study.