Final answer:
The distribution of scores in the populations might be identical under specific statistical hypotheses like the null hypothesis in ANOVA, but it is not universally true as distributions can differ based on factors such as sample sizes and shapes of distributions.
Step-by-step explanation:
The statement 'The distribution of scores in the populations underlying each group are identical' can be true or false based on the context of statistical analysis being discussed. For instance, if we are discussing the assumption under the null hypothesis in ANOVA (Analysis of Variance), then the statement might be considered true because the null hypothesis assumes that all group population means are equal, which suggests that the populations have identical normal distributions, given equal variances are also assumed.
However, two populations can share the same range and number of individuals but have different distribution patterns, which might not make their distributions identical. Moreover, when comparing the standard normal distribution to the Student's t-distribution, you will find that they are centered at zero but are not identical as Student's t-distribution has heavier tails, especially with smaller sample sizes.
Thus, whether the distribution of scores in the populations underlying each group are identical is dependent on the specific statistical assumptions or conditions being discussed.