Final answer:
The z distribution is used in hypothesis testing of two population means when the population variances are known, which allows for the use of a z-test.
Step-by-step explanation:
When testing the difference between two population means under independent sampling, we use the z distribution if the population variances are known. Therefore, the correct answer is: a. the population variances are known. This is because the z-test is appropriate when we have a standard normal distribution, which we can only be confident in using when both the population variances are known and the populations are normally distributed. In such cases, the standard deviations of the populations can be used to calculate the standard error and subsequently, the z-score for hypothesis testing.
For example, consider a scenario where a powder diet is tested on 49 people, and a liquid diet is tested on 36 different people, where the population standard deviations are two pounds and three pounds, respectively. In this case, a z-test can be used to see whether the liquid diet yields a higher mean weight loss than the powder diet because the standard deviations are known.