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Normal DistributionsZ TransformationsCentral Limit TheoremStandard Normal DistributionZ Distribution TableConfidence IntervalsLevels of SignificanceCritical ValuesPopulation Parameter Estimations

Normal Distribution

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 

Normal Distribution

Mean 

Variance 2

Standard Deviation 

Measure the distance from x to .

Normal Distribution

Mean 

Variance 2

Standard Deviation 



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.



Normal Distribution

Mean 

Variance 2

Standard Deviation 

Measure the distance using z as a scale;

where z = the number of ’s.



z

Normal Distribution

Mean 

Variance 2

Standard Deviation 



x – 

z

Both values represent the same distance.

Normal Distribution

Mean 

Variance 2

Standard Deviation 



x –  = z

Normal Distribution

Mean 

Variance 2

Standard Deviation 



x –  = z

z = (x –) /

Z Transformation for Normal Distribution

Z = ( x – ) / 

Central Limit Theorem

•The distribution of all sample means of samplesize n from a Normal Distribution (, 2) is anormally distributed with Mean =  Variance = 2 / n Standard Error =  / √n

Sampling Normal Distribution

Sample Size nMean 

Variance 2/ n

Standard Error  / √n



Sampling Normal Distribution

Sample Size nMean 

Variance 2 / n

Standard Error / √n

Pick any point X along the abscissa.



Sampling Normal Distribution

Sample Size nMean 

Variance 2 / n

Standard Error / √n

z = ( x – ) / ( / √n)



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 numerictabulation of the Cumulative ProbabilityValues of the Standard Normal Distribution.

Z Distribution Table

•The Z Distribution Table is a numerictabulation of the Cumulative ProbabilityValues of the Standard Normal Distribution.

What is “Z” ?

What is “Z” ?

Define Z as the number of standard deviationsalong the abscissa.

Practically speaking,Z ranges from -4.00 to +4.00

(-4.00) = 0.00003 and (+4.00) = 0.99997

Standard Normal Distribution

Mean  = 0

Variance 2 = 1

Standard Deviation  = 1

Area under the curve = 100%

z = -4.00

z = +4.00

Normal Distribution

Mean 

Variance 2

Standard Deviation 

Area under the curve = 100%

z = -4.00

z = +4.00

And the same holds true for any Normal Distribution !

Sampling Normal Distribution

Sample Size nMean 

Variance 2/ n

Standard Error  / √n

Area = 100%

As well as Sampling Distributions !

z = -4.00

z = +4.00

Confidence Intervals Levels of SignificanceCritical Values

Confidence Intervals

•Example: Select the middle 95% of thearea 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) = 0.025 z = -1.96

And (z) = 0.975 z = +1.96

 / 25%

Confidence Interval 95%

Level of Significance 5%





Calculating X Critical Values

X critical values are the lower and upperbounds of the samples means for a givenconfidence interval.

For the 95% Confidence Interval

X lower = ( - X) Z/2 / ( s / √n) where Z/2 = -1.96

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

Estimating Population Parameters Using Sample Data

A very robust estimate for the population variance is 2 = s2.

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