Final answer:
One-way ANOVA requires populations to be normally distributed with equal variances and samples to be selected randomly and independently. The mentioned scenario with different standard deviations among samples suggests a violation of these assumptions, necessitating further testing before proceeding with ANOVA.
Step-by-step explanation:
Understanding One-Way ANOVA
One-way ANOVA (Analysis of Variance) is a statistical method used to test if there are any statistically significant differences between the means of three or more independent (unrelated) groups. The core assumptions for conducting a one-way ANOVA include normal distribution of populations, equal variances (or standard deviations) among populations, and randomly and independently selected samples. If these assumptions are not met, the results of the ANOVA could be misleading.
In the scenario provided, even though the samples were independent and randomly selected, the differing standard deviations suggest a violation of one of the ANOVA assumptions. Differing standard deviations imply the underlying populations might have different variances, which contradicts the assumption of homogeneity of variance necessary for a one-way ANOVA. Therefore, further statistical tests, such as the Levene's test for equality of variances, should be conducted to verify if it is still appropriate to proceed with one-way ANOVA.
When the key assumptions are not fully met, other statistical techniques or adjustments might be necessary to analyze the data appropriately. For example, if the assumption of normal distribution or equal variance is not satisfied, you could consider transformations of the data or non-parametric tests as alternatives to the traditional one-way ANOVA.