Final answer:
To perform a hypothesis test about the difference between two means, several conditions must be met: random sampling, small sample size relative to the population, sample independence, sufficient sample size or normal population distribution. Large samples or normally distributed populations ensure the sampling distribution of the mean is normal. for inference of two sample means, the usual qualifications that must be met include random sampling (A), independence of the samples (C), and large sample size or normality condition (D and/or E).
Step-by-step explanation:
When analyzing two sample means, we are often interested in determining whether there is a significant difference between them, or if any observed difference can be attributed to random variation. We discuss this within the context of hypothesis testing. To pursue a hypothesis test about the difference between two means, certain conditions must be met:
• A. The samples can be reasonably assumed to be random. This ensures that every member of the population had an equal chance of being selected, thereby minimizing bias.
• B. The sample sizes must be small relative to the population, specifically not more than 5% of the population. This condition minimizes the impact of the sample on the population and allows the assumption of independence between samples.
• C. The samples should be independent. This means the samples are taken from different subject groups, and the observations in one sample don't influence the observations in the other.
• D. The sample sizes must be large (both greater than or equal to 30), or the populations must be normally distributed. This is known as the Central Limit Theorem which allows us to assume that the sampling distribution of the mean will be approximately normal if the sample size is sufficiently large.
• E. The population must be normally distributed if the sample sizes are small. This condition ensures that the distribution of the sample mean will also be normal.
For matched or paired samples, conditions B and C are not directly applicable because measurements are drawn from the same set of individuals or objects, and differences are calculated from these matched or paired samples. The sample of differences is used in the hypothesis test, and we need to establish that these differences come from a population that is either normal or sufficiently large for the Central Limit Theorem to apply.
In summary, for inference of two sample means, the usual qualifications that must be met include random sampling (A), independence of the samples (C), and large sample size or normality condition (D and/or E).