Final answer:
A z-test is most appropriate when the population standard deviation is known, as this parameter is necessary to standardize the sample mean. Large sample sizes or a normal population distribution can also validate the use of a z-test, and the population mean is typically unknown and being tested for.
Step-by-step explanation:
A z-test regarding a mean is most appropriate when the population standard deviation is known. This is because the z-test is built on the premise that the population standard deviation, which is used to standardize the sample mean, is known and fixed. In the scenarios where we use a z-test, we're typically dealing with sample sizes that are either large enough that the Central Limit Theorem ensures a normal distribution of the sample means, or the underlying population itself is normally distributed.
The options presented in the question highlight different conditions for different types of tests. For a z-test specifically:
- The population standard deviation must be known (hence option a is correct).
- A small sample size would generally require a different test, such as a t-test, as the standard deviation would likely be estimated from the sample rather than known a priori.
- A large sample size can justify the use of a z-test even if the population distribution is not normal, thanks to the Central Limit Theorem.
- Not knowing the population mean is typical for hypothesis testing, as that is usually what is being tested.
Therefore, it is essential to ensure that these conditions are met before conducting a z-test, as the validity of the test results depends heavily on them.