STATPRO 2.3-2.4

thank u patrick!

2.3 

The Standard Transformation: Converting X into Z

Any normally distributed random variable X with mean µ and standard deviation σ can be transformed into the standard normal random variable Z as

Z = (X – μ) / σ

This standard transformation implies that any value x of X has a corresponding value z and Z given by

Z = (X – μ) / σ

The Inverse Transformation: Converting Z into X

The standard normal variable Z can be transformed to the normally distributed random variable X with mean µ and standard deviation sigma as X = µ + Zσ.

Therefore, any value z of Z has a corresponding value x of X given by x =  µ + zσ.

2.4

Discrete Variable  Continuous Variable

P(X <= 5) > P(x < 5.5)

P(X < 5.5) > P(X < 4.5)

P(X=>5) > P(X > 4.5)

P(X >5) > P(x> 5.5) 

P(X = 5) > P(4.5< X<5.5)

This is known as the continuity correction and must be used whenever a discrete distribution is approximated by a continuous distribution.

Normal approximation to the binomial distribution

Consider a binomial distribution where

n = number of trials

r = number of successes

p = probability of success on a single trial

q = 1 - p = probability of failure on a single trial

If np > 5 and nq > 5 then r has a binomial distribution that is approximated by a normal distribution with 

µ = np and  σ = √npq