Final answer:
Following a failure to reject the null hypothesis in an ANOVA, no further tests are usually required unless specific pairwise comparisons are planned, in such case, post-hoc tests like Tukey's HSD may be used.
Step-by-step explanation:
The subject of the question is hypothesis testing using ANOVA (Analysis of Variance) and related post-hoc tests in the field of statistics. If a research comparing four groups fails to reject the null hypothesis of no mean difference using an F-test from an ANOVA, deciding the next step should be based on the analysis plan and the context of the research. If there are no significant differences among the group means, typically, no further tests are required. However, in some cases, further tests may be warranted if the research has a specific interest in comparing certain groups or if there is reason to believe that a type II error might have occurred. In academic settings, the answer might be to conduct no more statistical tests, unless specific pairs of groups are of interest, in which case a post-hoc test such as a Tukey's HSD might be appropriate for pairwise comparisons. A t-test for each pair of means could also be used for specific comparisons but would require a correction for multiple comparisons to control for Type I error rate. Conducting a chi-square test would not be applicable in this scenario, as it is used for categorical data, not for comparing means.