Editing Z-test (section)

==Use in location testing==

# The term "''Z''-test" is often used to refer specifically to the [[location test|one-sample location test]] comparing the mean of a set of measurements to a given constant when the sample variance is known. For example, if the observed data ''X''<sub>1</sub>, ..., ''X''<sub>n</sub> are (i) independent, (ii) have a common mean μ, and (iii) have a common variance σ<sup>2</sup>, then the sample average <span style="text-decoration: overline">''X''</span> has mean μ and variance <math>\frac{\sigma^2}{n}</math>. 
# The null hypothesis is that the mean value of X is a given number μ<sub>0</sub>.  We can use <span style="text-decoration: overline">''X''</span>&nbsp;  as a test-statistic, rejecting the null hypothesis if <span style="text-decoration: overline">''X''</span>&nbsp;− μ<sub>0</sub> is large. 
# To calculate the standardized statistic <math>Z=\frac{(\bar{X}-\mu_0)}{s}</math>, we need to either know or have an approximate value for σ<sup>2</sup>, from which we can calculate <math>s^2=\frac{\sigma^2}{n}</math> . In some applications, σ<sup>2</sup> is known, but this is uncommon. 
# If the sample size is moderate or large, we can substitute the [[Sample variance#Population variance and sample variance|sample variance]] for σ<sup>2</sup>, giving a ''plug-in'' test. The resulting test will not be an exact ''Z''-test since the uncertainty in the sample variance is not accounted for—however, it will be a good approximation unless the sample size is small. 
# A [[t-test|''t''-test]] can be used to account for the uncertainty in the sample variance when the data are exactly [[normal distribution|normal]]. 
# Difference between ''Z''-test and ''t''-test: ''Z''-test is used when sample size is large (''n'' > 50), or the population variance is known. ''t''-test is used when sample  size is small (''n'' < 50) and population variance is unknown. 
# There is no universal constant at which the sample size is generally considered large enough to justify use of the plug-in test. Typical rules of thumb: the sample size should be 50 observations or more. 
# For large sample sizes, the ''t''-test procedure gives almost identical ''p''-values as the ''Z''-test procedure. 
# Other location tests that can be performed as ''Z''-tests are the two-sample location test and the [[paired difference test]].