The z test is used when
The z-test is used in statistics when you want to determine whether there is a significant difference between the means of two groups or between a sample mean and a population mean. Here are some specific conditions under which a z-test is appropriate:
Sample Size: The z-test is typically used when the sample size is large (usually n > 30). For smaller sample sizes, a t-test is often more appropriate unless the population standard deviation is known.
Population Standard Deviation: The z-test is used when the population standard deviation is known. If the population standard deviation is unknown and the sample size is small, a t-test should be used instead.
Normality: The z-test assumes that the data follows a normal distribution. If the sample size is large, the Central Limit Theorem suggests that the sampling distribution of the sample mean will be approximately normal, even if the data itself is not normally distributed.
Independent Samples: When comparing two means, the z-test assumes that the samples are independent of each other.
Hypothesis Testing: The z-test is used to test hypotheses about population parameters, such as testing whether a sample mean is significantly different from a known population mean.
In summary, the z-test is appropriate when you have a large sample size, know the population standard deviation, and want to test hypotheses about means.