I'm currently working on a quasi-experimental research paper. I only have a sample size of 15 due to low population within the chosen area and that only 15 fit my criteria. Is 15 the minimum sample size to compute for t-test and F-test? If so, where can I get an article or book to support this small sample size?
This paper was already defended last Monday and one of the panel asked to have a supporting reference because my sample size is too low. He said it should've been at least 40 respondents.
Best Answer
There is no minimum sample size for the t test to be valid other than it be large enough to calculate the test statistic. Validity requires that the assumptions for the test statistic hold approximately. Those assumptions are in the one sample case that the data are iid normal (or approximately normal) with mean 0 under the null hypothesis and a variance that is unknown but estimated from the sample. In the two sample case it is that both samples are independent of each other and each sample consists of iid normal variables with the two samples having the same mean and a common unknown variance under the null hypothesis. A pooled estimate of variance is used for the statistic.
In the one sample case the distribution under the null hypothesis is a central t with n-1 degrees of freedom. In the two sample cases with sample sizes n and m not necessarily equal the null distribution of the test statistics is t with n+m-2 degrees of freedom. The increased variability due to low sample size is accounted for in the distribution which has heavier tails when the degrees of freedom is low which corresponds to a low sample size. So critical values can be found for the test statistic to have a given significance level for any sample size (well, at least of size 2 or larger).
The problem with low sample size is with regard to the power of the test. The reviewer may have felt that 15 per group was not a large enough sample size to have high power of detecting a meaningful difference say delta between the two means or a mean greater than delta in absolute value for a one sample problem. Needing 40 would require a specification of a certain power at a particular delta that would be achieved with n equal 40 but not lower than 40.
I should add that for the t test to be performed the sample must be large enough to estimate the variance or variances.