Final answer:
An example explaining the distribution for sigma known vs. sigma unknown typically involves using a Z-test when the population standard deviation is known and a t-test when it is unknown. The random variable for a Z-test is Z, and for a t-test, it is t, which accounts for the uncertainty of using a sample standard deviation in place of an unknown population standard deviation.
Step-by-step explanation:
When dealing with the hypothesis testing of a single population mean, two different statistical tests may be applied based on whether the population standard deviation (σ) is known or unknown. If σ is known and certain conditions are met (such as the population being normally distributed or the sample size being large), a Z-test is typically used. Conversely, if σ is unknown, a t-test is utilized, especially when the sample size is large and the population is assumed to be approximately normally distributed or the sample itself is a simple random sample from the population.
The random variable for the Z-test is often denoted as Z, which measures the number of standard deviations a sample mean is from the population mean. For the t-test, the random variable is denoted as t, which considers the sample standard deviation as a substitute for the unknown population standard deviation. The t-test accounts for the additional uncertainty in the estimate of the standard deviation when σ is unknown.