. The null hypothesis is that the mean value of X is a given number μ0. We can use X as a test-statistic, rejecting the null hypothesis if X − μ0 is large. To calculate the standardized statistic , we need to either know or have an approximate value for σ2, from which we can calculate . In some applications, σ2 is known, but this is uncommon. If the sample size is moderate or large, we can substitute the sample variance for σ2, 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 can be used to account for the uncertainty in the sample variance when the data are exactly 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. Conditions For the Z-test to be applicable, certain conditions must be met. Nuisance parameters should be known, or estimated with high accuracy (an example of a nuisance parameter would be the standard deviation in a one-sample location test). Z-tests focus on a single parameter, and treat all other unknown parameters as being fixed at their true values. In practice, due to Slutsky's theorem, "plugging in" consistent estimates of nuisance parameters can be justified. However if the sample size is not large enough for these estimates to be reasonably accurate, the Z-test may not perform well. The test statistic should follow a normal distribution. Generally, one appeals to the central limit theorem to justify assuming that a test statistic varies normally. There is a great deal of statistical research on the question of when a test statistic varies approximately normally. If the variation of the test statistic is strongly non-normal, a Z-test should not be used. If estimates of