Standard Normal Distribution — Hypothesis Testing

The normal distribution X∼N(0,1), μ = 0 and σ = 1, is called the standard normal distribution, and its distribution function is:

$$f(x)=\frac{e^{-x^2/2}}{\sqrt{2π}}$$

A value from any normal distribution can be transformed into its corresponding value on a standard normal distribution using the following formula:

$$Z =\frac{X - μ}{σ}$$

where Z is the value on the standard normal distribution, X is the value on the original distribution, μ is the mean of the original distribution, and σ is the standard deviation of the original distribution.

$$E(Z)= E\biggl(\frac{X - μ}{σ}\biggr)=\frac{E(X) - μ}{σ}=0$$ $$Var(Z)=Var\biggl(\frac{X - μ}{σ}\biggr)=\frac{Var(X)}{σ^2}=1$$

To computing probabilities for normal distributions, convert to standard value and lookup the distribution function tables for the standard normal distribution:

$$P(X≤b)→P\biggl(Z≤\frac{b - μ}{σ}\biggr)$$

Note: The probabilities may also be obtained using the NORM.DIST function in Excel. The Excel NORM.INV function returns the variable value, given the parameters and the probability.