Final answer:
A small p-value in the ANOVA test indicates evidence that the population means for the three groups are different, leading to the rejection of the null hypothesis in favor of the alternative hypothesis that at least two of the group means are not equal.
Step-by-step explanation:
In ANOVA (Analysis of Variance), the scenario that would indicate discernible evidence that the population means for the three groups were different is a) Small p-value in the ANOVA test. This is because the p-value indicates the probability of obtaining the observed results if the null hypothesis were true. A small p-value (typically less than 0.05) suggests that there is a statistically significant difference between the group means, leading us to reject the null hypothesis in favor of the alternative hypothesis (Ha: At least two of the group means μ₁, μ₂, μ₃ are not equal). Contrarily, a large p-value indicates a greater likelihood that the observed data is consistent with the null hypothesis, and thus there is no significant evidence to suggest the means are different.
It’s crucial to remember that the purpose of a one-way ANOVA test is to determine whether there are any statistically significant differences among the group means. Key assumptions for this test include all populations being normally distributed, having equal standard deviations, samples being randomly and independently selected, and the response being numerical. The test statistic is the F ratio, and a high F statistic typically corresponds to a low p-value, indicating that the null hypothesis is likely to be rejected.