Statistics question?

Question

List the steps in determining the probabitlity of an event occurring given the "Z" value in a normal probability distribution?

Answer 1

For any normal random variable X with mean μ and standard deviation  ÃÂƒ , X ~ Normal( μ ,  ÃÂƒ ), (note that in most textbooks and literature the notation is with the variance, i.e., X ~ Normal( μ ,  ÃÂƒÃ‚² ).  Most software denotes the normal with just the standard deviation.)

You can translate into standard normal units by:

Z = ( X - μ ) /  ÃÂƒ

Moving from the standard normal back to the original distribuiton using:

X = μ + Z * σ

Where Z ~ Normal( μ  = 0,  ÃÂƒ  = 1).  You can then use the standard normal cdf tables to get probabilities.

If you are looking at the mean of a sample, then remember that for any sample with a large enough sample size the mean will be normally distributed.  This is called the Central Limit Theorem.

If a sample of size is is drawn from a population with mean μ and standard deviation σ then the sample average xBar is normally distributed

with mean μ and standard deviation  ÃÂƒ /√(n)

An applet for finding the values

http://www-stat.stanford.edu/~naras/jsm/...

calculator

http://stattrek.com/Tables/normal.aspx

how to read the tables

http://rlbroderson.tripod.com/statistics...

Example:

X ~ Normal( μx = 3.23 , σx² = 4.41 )

X ~ Normal( μx = 3.23 , σx = 2.1 )

Find P( X < 5.23 )

P( ( X - μ ) / σ < ( 5.23 - 3.23 ) / 2.1 )

= P( Z < 0.952381 )

= 0.8295481

Find P( X > 5.23 )

P( ( X - μ ) / σ > ( 5.23 - 3.23 ) / 2.1 )

= P( Z > 0.952381 )

= P( Z < -0.952381 )

= 0.1704519