Shapiro-Wilk Test of Normality

Is a qq GOF Test for Class of Distributions, in this case Normal Distributions.

How to perform the test

Make qq-Plot for any Normal Distribution against Standard Normal Distribution. Use simple Linear Regression to fit a diagonal line to the plot and read off the slope $S$ .

Calculate Test Statistic

T=\frac{S^2}{s^2}

where $s$ is the empirical standard deviation for the data sequence.

Given $H_{0}$ the test statistic will be close to $1$ . You can quanitfy this by simulating the test statistic and determening its distribution. You can calculate an Acceptance Domain with

[z_{\alpha / 2}, z_{1- \alpha / 2}]

where $\alpha$ is the probability for Type 1 Error.