Normal Distributions Z Transformations Central Limit Theorem Standard Normal Distribution Z Distribution Table Confidence Intervals Levels of Significance Critical Values Population Parameter Estimations

Normal Distribution

Mean 

Normal Distribution Mean  Variance  2

Normal Distribution Mean  Variance  2 Standard Deviation 

Normal Distribution Mean  Variance  2 Standard Deviation  Z Transformation

Normal Distribution Mean  Variance  2 Standard Deviation  Pick any point X along the abscissa.

Normal Distribution Mean  Variance  2 Standard Deviation  x

Normal Distribution Mean  Variance  2 Standard Deviation  x Measure the distance from x to .

Normal Distribution Mean  Variance  2 Standard Deviation   x –  x Measure the distance from x to .

Normal Distribution Mean  Variance  2 Standard Deviation  Measure the distance using z as a scale; where z = the number of  ’s.  x

Normal Distribution Mean  Variance  2 Standard Deviation  Measure the distance using z as a scale; where z = the number of  ’s.  x zz

Normal Distribution Mean  Variance  2 Standard Deviation   x –  zz x Both values represent the same distance.

Normal Distribution Mean  Variance  2 Standard Deviation   x x –  = z 

Normal Distribution Mean  Variance  2 Standard Deviation   x x –  = z  z = (x –  ) / 

Z Transformation for Normal Distribution Z = ( x –  ) / 

Central Limit Theorem The distribution of all sample means of sample size n from a Normal Distribution ( ,  2 ) is a normally distributed with Mean =  Variance =  2 / n Standard Error =  / √n

Sampling Normal Distribution Sample Size n Mean  Variance  2 / n Standard Error  / √n 

Sampling Normal Distribution Sample Size n Mean  Variance  2 / n Standard Error  / √n Pick any point X along the abscissa.  x

Sampling Normal Distribution Sample Size n Mean  Variance  2 / n Standard Error  / √n z = ( x –  ) / (  / √n)  x

Z Transformation for Sampling Distribution Z = ( x –  ) / (  / √n)

Standard Normal Distribution & The Z Distribution Table What is a Standard Normal Distribution?

Standard Normal Distribution Mean  = 0

Standard Normal Distribution Mean  = 0 Variance  2 = 1

Standard Normal Distribution Mean  = 0 Variance  2 = 1 Standard Deviation  = 1

Standard Normal Distribution Mean  = 0 Variance  2 = 1 Standard Deviation  = 1 What is the Z Distribution Table?

Z Distribution Table The Z Distribution Table is a numeric tabulation of the Cumulative Probability Values of the Standard Normal Distribution.

Z Distribution Table The Z Distribution Table is a numeric tabulation of the Cumulative Probability Values of the Standard Normal Distribution. What is “Z” ?

Define Z as the number of standard deviations along the abscissa. Practically speaking, Z ranges from to  (-4.00) = and  (+4.00) =

Standard Normal Distribution Mean  = 0 Variance  2 = 1 Standard Deviation  = 1 Area under the curve = 100% z = z = +4.00

Normal Distribution Mean  Variance  2 Standard Deviation  Area under the curve = 100% z = -4.00z = And the same holds true for any Normal Distribution !

Sampling Normal Distribution Sample Size n Mean  Variance  2 / n Standard Error  / √n Area = 100% As well as Sampling Distributions ! z = z = +4.00

Confidence Intervals Levels of Significance Critical Values

Confidence Intervals Example: Select the middle 95% of the area under a normal distribution curve.

Confidence Interval 95% 95%

Confidence Interval 95% 95% 95% of all the data points are within the 95% Confidence Interval

Confidence Interval 95% 95% Level of Significance  = 100% - Confidence Interval

Confidence Interval 95% 95% Level of Significance  = 100% - Confidence Interval  = 100% - 95% = 5%

Confidence Interval 95% 95% Level of Significance  = 100% - Confidence Interval  = 100% - 95% = 5%  /2 = 2.5%

 / 2  5% Confidence Interval 95% Level of Significance  5%

 / 2  5% Confidence Interval 95% Level of Significance  5% From the Z Distribution Table For  (z) = z = And  (z) = z = +1.96

 / 2  5% Confidence Interval 95% Level of Significance  5%      

Calculating X Critical Values X critical values are the lower and upper bounds of the samples means for a given confidence interval. For the 95% Confidence Interval X lower = (  - X) Z  /2 / ( s / √n) where Z  /2 = X upper = (  - X) Z  /2 / ( s / √n) where Z  /2 = +1.96

 / 2  5% Confidence Interval 95% Level of Significance  5%       X lower X upper

Estimating Population Parameters Using Sample Data

A very robust estimate for the population variance is  2 = s 2. A Point Estimate for the population mean is  = X. Add a Margin of Error about the Mean by including a Confidence Interval about the point estimate. FromZ = ( X –  ) / (  / √n)  = X ± Z  /2 (s / √n) For 95%, Z  /2 = ±1.96