normal distributions z transformations central limit theorem standard normal distribution z...
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Normal DistributionsZ TransformationsCentral Limit TheoremStandard Normal DistributionZ Distribution TableConfidence IntervalsLevels of SignificanceCritical ValuesPopulation Parameter Estimations
Normal Distribution
Normal DistributionMean
Normal DistributionMean
Variance 2
Normal DistributionMean
Variance 2
Standard Deviation
Normal DistributionMean
Variance 2
Standard Deviation
Z Transformation
Normal DistributionMean
Variance 2
Standard Deviation
Pick any point X along the abscissa.
Normal DistributionMean
Variance 2
Standard Deviation
x
Normal DistributionMean
Variance 2
Standard Deviation
x
Measure the distance from x to .
Normal DistributionMean
Variance 2
Standard Deviation
x –
x
Measure the distance from x to .
Normal DistributionMean
Variance 2
Standard Deviation
Measure the distance using z as a scale;
where z = the number of ’s.
x
Normal DistributionMean
Variance 2
Standard Deviation
Measure the distance using z as a scale;
where z = the number of ’s.
x
z
Normal DistributionMean
Variance 2
Standard Deviation
x – z
x
Both values represent the same distance.
Normal DistributionMean
Variance 2
Standard Deviation
x
x – = z
Normal DistributionMean
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
Standard Normal DistributionMean = 0
Variance 2 = 1
Standard Normal DistributionMean = 0
Variance 2 = 1Standard Deviation = 1
Standard Normal DistributionMean = 0
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
Standard Normal DistributionMean = 0
Variance 2 = 1Standard Deviation = 1
Area under the curve = 100%
z = -4.00 z = +4.00
Normal DistributionMean
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%
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%
/ 25% / 25%
Confidence Interval 95%Level of Significance 5%
/ 25% / 25%
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
/ 25% / 25%
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 = -1.96
X upper = ( - X) Z/2 / ( s / √n) where Z/2 = +1.96
/ 25% / 25%
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.From Z = ( X – ) / ( / √n)
= X ± Z/2 (s / √n) For 95%, Z/2 = ±1.96