The Normal distribution

Example:
Prob(|Z|>=1.96) <= 0.04999565

Z =

Characteristics:
This is not a test, but a distribution. The Normal distribution, or Gaussian distribution, is the statistical distribution. It's importance flows from the fact that :

1. Any sum of Normal distributed variables is itself a Normal distributed variable
2. Sums of variables that, individually, are not Normal distributed tend to become Normal distributed (asymptotically)
The latter feature means that the distribution of the test-parameter of any statistical test can be approximated by a normal distribution if enough observations are available.
The Normal distribution has a characteristic Bell-shaped probability density curve and is symmetric. Only two parameters describe this curve: the mean and the standard deviation. The Standard Normal distribution, used in this example is the normalized variant:

Z = (x-mean)/(standard deviation)

and has a mean value of zero and a standard deviation of one.

H0:
The distribution of Z has mean 0.

Assumptions:
Z has a Standard Normal distribution with mean = 0 and standard deviation = 1.
This "test" can only be used when there is no reason to doubt this assumption.

Scale:
Interval

Procedure:
If the mean and standard deviation are not 0 and 1 respectively, calculate
Z = (x-mean)/(standard deviation).
If x is the mean of a sample of size N, use (population standard deviation)/sqrt(N) instead of (population standard deviation).
Be aware that both the mean and the standard deviation must be known a priori and should not be estimated from the sample itself (use a Student t-distribution if you do not know the mean and standard deviation).

Level of significance:
Use a table to look up the level of significance associated with Z.

Approximation:
None

Remarks:
Most used "test" because it is the approximation of all other tests.
The example used here approximates the Standard Normal distribution with an error < 1.5 * 10(-7) ( Abramowitz and Stegun 1965 (1970), p932, 26.2.17)