Normal Distributions Z Transformations Central Limit Theorem Standard Normal Distribution Z...

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

Normal Distribution

Normal DistributionMean


Variance 2


Variance 2

Standard Deviation


Variance 2

Standard Deviation

Z Transformation


Variance 2

Standard Deviation

Pick any point X along the abscissa.


Variance 2

Standard Deviation

x


Variance 2

Standard Deviation

x

Measure the distance from x to .


Variance 2

Standard Deviation

x –

x

Measure the distance from x to .


Variance 2

Standard Deviation

Measure the distance using z as a scale;

where z = the number of ’s.

x


Variance 2

Standard Deviation

Measure the distance using z as a scale;

where z = the number of ’s.

x

z


Variance 2

Standard Deviation

x – z

x

Both values represent the same distance.


Variance 2

Standard Deviation

x

x – = z


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 DistributionSample Size nMean Variance 2/ nStandard Error / √n

Sampling Normal DistributionSample Size nMean Variance 2 / n

Standard Error / √n

Pick any point X along the abscissa.

x

Sampling Normal DistributionSample Size nMean 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 DistributionMean = 0


Variance 2 = 1


Variance 2 = 1Standard 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.

2z 1

21

(z) P(Z z) du2 e

Z Distribution Table

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

2z 1

21

(z) P(Z z) du2 e

What is “Z” ?

What is “Z” ?

Define Z as the number of standard deviations along the abscissa.

Practically speaking,Z ranges from -4.00 to +4.00

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



Area under the curve = 100%

z = -4.00 z = +4.00


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 DistributionSample Size nMean Variance 2/ nStandard Error / √n

Area = 100%

As well as Sampling Distributions !

z = -4.00 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%


95%

95% of all the data points are within the

95% Confidence Interval


95%

Level of Significance = 100% - Confidence Interval


95%


= 100% - 95% = 5%


95%


= 100% - 95% = 5%

/2 = 2.5%

/ 25% / 25%

Confidence Interval 95%Level of Significance 5%

/ 25% / 25%


From the Z Distribution Table

For (z) = 0.025 z = -1.96

And (z) = 0.975 z = +1.96

/ 25% / 25%


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 = -1.96

X upper = ( - X) Z/2 / ( s / √n) where Z/2 = +1.96

/ 25% / 25%


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.From Z = ( X – ) / ( / √n)

= X ± Z/2 (s / √n) For 95%, Z/2 = ±1.96

Normal Distributions Z Transformations Central Limit Theorem Standard Normal Distribution Z...

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