Chapter 6 Notes

©2008 by S. Gramlich

Normal Distribution

! = Important Note

! These Notes are not meant to replace Reading.  Read Chapter first.

 

6-2

Normal (Gaussian) Distribution Properties:

            1) Bell-Shaped & Symmetric (Frequency Polygon with zero skewness)

            2) Total Area Under the curve is 100%  or 1 (so each half is 50% or .5)

            3) Continuous

            4) Asymptotic to the horizontal axis (runs infinitesimally close to but never touches)

 

Standard Normal (or Z) Distribution when mean (μ) = 0, standard deviation (σ) = 1

Steps to find Area Under Curve (as a decimal) or Probability of selecting 1 item @ random from the population,

            1) Sketch Normal curve and shade area representing claim.

2) Use 1 of the 2 Table A-2's and look up the given z score.

! Tables can be found in Chapter Contents (in left margin) - Formulas and Tables Fold-Out Card

! Body of table gives Area Under Curve to 4 decimal places.

! Normal curve is Symmetric so you can use the "NEGATIVE z Scores" table for positive z scores

   and the "POSITIVE z Scores" table for negative z scores

! You can also use the extra Table (A-2.3) I've posted on the Web.

! Table must correspond to your sketch.

! Triola teaches subtracting from 1 to get sketched region.

! You may also have to add or subtract 2 found Areas to get the sketched region.

 

To find z score when given the percentile or probability:

                        ! This is the reverse of the above procedure.

            1) Find the CLOSEST prob value in the body of 1 of the Table A-2's (to 4 decimal places)

            2) the corresponding z score tenth spot is in the first column, hundredth spot in top row

                        ! If the z score is to the left of the mean 0, then it will be negative.

6-3

To find Area Under Curve (as a decimal) or Probability of selecting 1 item @ random from the population

when μ ≠ 0 or σ ≠ 1, or both, convert Original Score (X) to a z score using z = (X-μ)/σ

then follow steps above to find the area.

 

To find Original Score (X) when given the percentile or probability, follow same steps above

then use X = μ ± z*σ to convert the z score into the original x score.

 

6-5

To find Area Under Curve (as a decimal) or Probability of selecting N items @ random from the population

when μ ≠ 0 or σ ≠ 1, or both, convert Original Mean Score (xbar) to a z score using z = (xbar-μ) / (σ/√n)

then follow steps above (6-2) to find the area.

 

(σ/√n) = σxbar is called the Standard Error.

 

Central Limit Theorem (CLT) = for sample sizes greater than 30, the sampling distribution can be approximated by a Normal Distribution even if the original population is not normally distributed.

 

TECHNOLOGY

using StatCrunch:

! Don't need to enter any info in StatCrunch spreadsheet for this procedure

to find Prob, z, or original X score:   Stat - Calculators - Normal - (enter data) - Compute

 

EXCEL commands:

! enter the given value inside the parentheses

! Excel Probability commands will only give area to the Left of z or x

Find                                         Excel Command

Standard Probability              =NormSdist(z score)

z score                                     =NormSinv(prob)

any Probability                        =NormDist(x, μ, σ, true)

original X score                      =NormInv(prob, μ, σ)