47. What is one of the three assumptions of independent t-tests?
Answer: A
Individuals in the sample were selected randomly.
One critical assumption of independent t-tests is that individuals in the sample were selected randomly. This ensures that the samples are representative of the populations from which they are drawn, allowing for valid comparisons.
A) Individuals in the sample were selected randomly.
This option is correct because random selection of individuals helps eliminate bias and ensures that the samples reflect the true characteristics of the populations. This randomization is essential for the validity of the statistical conclusions drawn from the t-test.
B) The size of each sample from the population is greater than 30.
This option is incorrect as it misrepresents the assumption. While larger sample sizes can lead to more reliable results, the assumption for independent t-tests does not specify that each sample must exceed 30. The central limit theorem allows for the use of t-tests with smaller sample sizes, provided the data meets other assumptions.
C) Individuals in the two samples can be meaningfully paired.
This option is incorrect because it describes a characteristic of paired t-tests rather than independent t-tests. Independent t-tests are used when the samples are not paired, meaning the observations in one sample do not influence the observations in the other.
D) The population samples have the same mean and standard deviation.
This option is also incorrect. While the assumption of equal variances (homogeneity of variance) is relevant to independent t-tests, it does not require the samples to have the same mean. In fact, the purpose of the t-test is to determine if there are significant differences in means between the two samples.
Conclusion
In summary, the correct answer is that individuals in the sample were selected randomly, as this is fundamental to the integrity of the independent t-test. Options B, C, and D either misinterpret the assumptions or describe conditions relevant to other statistical tests, thereby failing to align with the core principles underpinning independent t-tests.