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. the samples must be randomly selected. c. each population has a normal distribution with a known standard deviation. d. the samples are paired.

Explanation:

To use the z - test for the difference between two population means:

• Option A: Independent samples are necessary because if samples are dependent (paired), a paired - test (like paired t - test) is more appropriate, and the z - test for two independent means requires independence.
• Option B: Randomly selected samples are needed to ensure that the samples are representative of the populations, which is a fundamental assumption for inferential statistics including the z - test.
• Option C: For a z - test, we need to know the population standard deviations (or have large sample sizes, but here the option states each population has a normal distribution with a known standard deviation, which is a valid condition for the z - test).
• Option D: Paired samples are for paired tests (like paired t - test), not for the z - test of two independent population means, so this is not a necessary condition for the z - test of two population means.

Answer:

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