Final answer:
Using ANOVA, we test whether there are significant differences between the means of four populations with a pooled sample variance of 14. The F-statistic calculated from the means and variance will help determine if we reject the hypothesis of equal means.
Step-by-step explanation:
The question involves testing for differences between all pairs of means from four populations, which requires the use of analysis of variance (ANOVA). This statistical method is used to compare the means of three or more samples to see if at least one sample mean is significantly different from the others. Since the pooled sample variance is given as 14, and the means of the populations are 27, 26, 23, and 30, the hypothesis test will involve these values along with the assumption of equal variances across populations.
To test the hypothesis that all population means are equal (null hypothesis H0: μ1 = μ2 = μ3 = μ4) against the alternative hypothesis that at least two of the means are different (Ha), an F-statistic is calculated. If this F-value is significantly high, it suggests that the variances between groups are larger than expected under the null hypothesis, and thus, the null hypothesis is likely to be rejected.
The given data provides a starting point for such an analysis, which would require further calculations and the use of an F-distribution table to determine the p-value and make a final decision on the hypotheses. Therefore, assuming an ANOVA test, a pooled sample variance, and controlling for factors such as equal population variances, normal distribution of samples, and independent sampling, one can determine whether there are statistically significant differences between the group means.