Question

what conditions are necessary in order to use the z - test to test the difference between two population means? select all the necessary conditions below. a. the samples must be independent. b. each population has a normal distribution with a known standard deviation. c. the samples must be randomly selected. d. the samples are paired.

Explanation:

• For the z - test to test the difference between two population means:
• Samples must be independent (A) because the z - test for two - population means assumes no relationship between the two samples.
• Each population should have a normal distribution with a known standard deviation (B) as the formula for the z - statistic for two population means \(\ z=\frac{(\bar{x}_{1}-\bar{x}_{2})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}}\) requires known \(\sigma_{1}\) and \(\sigma_{2}\) and normal populations (or large sample sizes which, by Central Limit Theorem, approximate normality).
• Samples must be randomly selected (C) to ensure that the samples are representative of the populations. Paired samples (D) are used in a different type of test (paired - t test, for example), not the z - test for two independent population means.

Answer:

A. The samples must be independent, B. Each population has a normal distribution with a known standard deviation, C. The samples must be randomly selected.