Final answer:
For a two-sample test of means, it is assumed that the populations are normally distributed, samples are independent, populations have equal but unknown standard deviations, and the test statistic follows the t-distribution.
Step-by-step explanation:
When conducting a two-sample test of means, there are several assumptions that must be met:
- The populations from which the samples are taken are normally distributed. This assumption ensures that the sampling distribution of the means will also be approximately normal.
- The two samples are independent of each other, meaning the selection of one sample has no influence on the selection of the other.
- When the population standard deviations are unknown and the sample sizes are small, it's crucial to assume that the populations have equal but unknown standard deviations. This is required for the appropriate estimation of the pooled standard deviation.
- The test statistic follows the t-distribution when the population standard deviations are unknown. This distribution adjusts for the fact that we are estimating the standard deviation from the sample data, which adds variability.
- If both population standard deviations are known, which is often not the case, a z-test rather than a t-test can be used. However, this is a rare situation in real-world scenarios.
Therefore, the appropriate assumptions for a two-sample t-test when the population standard deviations are unknown are that the populations are normally distributed, the samples are independent, the populations have equal but unknown standard deviations, and the test statistic follows the t-distribution. If the population standard deviations are known or the sample sizes are large, other considerations apply for the test statistic.