To determine the z-critical value, the distribution of hypothesis testing must be the standard normal distribution. The standard normal distribution is the normal distribution with mean 0 and standard deviation 1. Z-critical values are the points on the distribution which have the same probability as our test statistic, equal to the significance level α.
If the test is one-sided, then there is only one critical value, if it is two-sided, then there are two critical values, one to the left and the other to the right of the median value of the distribution.
The formulae for the z-critical values involve the quantile function, Φ-1
(z), which is the inverse of the cumulative distribution function of the standard normal distribution. The cumulative distribution function of the standard normal distribution is denoted by the formula
`Φ(z) = P (Z ≤ z) = `
` 1 `
∫x−∞ e-u2/2 du
Therefore, we need to find the inverse of the function Φ(z). The notation is to find zα
that α (which is between 0 and 1) is the probability that Z > zα
P (Z > zα) = 1 − Φ (zα) = α
In other words, zα
is the value of z after which the area under the standard normal distribution is α. Thus, area before zα
is 1−α. For finding z-critical values by hand, we need to use the table of Φ(z)
values. Some known values are:
z0 = +∞, z0.5 = 0, z1 = −∞
For example, let us find z0.25
is the z that the area before that is 0.75. So, in the table of Φ(z)
have to find the z related to 0.75.
We have the following options for selecting the test:
- In a left-tailed test, the area under the density curve from the critical value to the left is equal to α. In this case, the z-critical value can be calculated as Φ-1(α);
- In a right-tailed test, the area under the density curve from the critical value to the right is equal to α. In this case, the z-critical value can be calculated as Φ-1(1 − α);
- In a two-tailed test, the area under the density curve from the left critical value to the left is α/2 and the area under the curve from the right critical value to the right is α/2. In this case, the z-critical value can be calculated as ±Φ-1(1 - α/2)
Z Critical Value Calculator is important for conducting statistical research.