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 a z - test to test the difference between two population means:
• Independent samples (A) are necessary as the z - test for two - population means assumes no relationship between the two samples.
• If each population has a normal distribution with a known standard deviation (B), the z - statistic can be calculated using the formula \(z=\frac{(\bar{x}_1-\bar{x}_2)-(\mu_1 - \mu_2)}{\sqrt{\frac{\sigma_1^{2}}{n_1}+\frac{\sigma_2^{2}}{n_2}}}\).
• Randomly selected samples (C) are required to ensure that the samples are representative of the populations, which is a fundamental assumption in statistical hypothesis testing.
• Paired samples (D) are used in a paired - t test, not in a z - test for the difference between 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.