Download Normal Distribution

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
Normal Distribution
Mean m
Normal Distribution
Mean m
Variance s2
Normal Distribution
Mean m
Variance s2
Standard Deviation s
Normal Distribution
Mean m
Variance s2
Standard Deviation s
Z Transformation
Normal Distribution
Mean m
Variance s2
Standard Deviation s
Pick any point X along the abscissa.
Normal Distribution
Mean m
Variance s2
Standard Deviation s
x
Normal Distribution
Mean m
Variance s2
Standard Deviation s
x
Measure the distance from x to m.
Normal Distribution
Mean m
Variance s2
Standard Deviation s
x–m
m
x
Measure the distance from x to m.
Normal Distribution
Mean m
Variance s2
Standard Deviation s
m
x
Measure the distance using z as a scale;
where z = the number of s’s.
Normal Distribution
Mean m
Variance s2
Standard Deviation s
zs
m
x
Measure the distance using z as a scale;
where z = the number of s’s.
Normal Distribution
Mean m
Variance s2
Standard Deviation s
x–m
zs
m
x
Both values represent the same distance.
Normal Distribution
Mean m
Variance s2
Standard Deviation s
x – m = zs
m
x
Normal Distribution
Mean m
Variance s2
Standard Deviation s
x – m = zs
z = (x – m) / s
m
x
Z Transformation for Normal Distribution
Z=(x–m)/s
Central Limit Theorem
• The distribution of all sample means of sample
size n from a Normal Distribution (m, s2) is a
normally distributed with
Mean = m
Variance = s2 / n
Standard Error = s / √n
Sampling Normal Distribution
Sample Size n
Mean m
Variance s2/ n
Standard Error s / √n
m
Sampling Normal Distribution
Sample Size n
Mean m
Variance s2 / n
Standard Error s / √n
m
x
Pick any point X along the abscissa.
Sampling Normal Distribution
Sample Size n
Mean m
Variance s2 / n
Standard Error s / √n
m
x
z = ( x – m ) / (s / √n)
Z Transformation for Sampling Distribution
Z = ( x – m ) / (s / √n)
Standard Normal Distribution
&
The Z Distribution Table
What is a Standard Normal Distribution?
Standard Normal Distribution
Mean m = 0
Standard Normal Distribution
Mean m = 0
Variance s2 = 1
Standard Normal Distribution
Mean m = 0
Variance s2 = 1
Standard Deviation s = 1
Standard Normal Distribution
Mean m = 0
Variance s2 = 1
Standard Deviation s = 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
(z)  P(Z  z) 


1
2
e
1

2
m du
2
Z Distribution Table
• The Z Distribution Table is a numeric
tabulation of the Cumulative Probability
Values of the Standard Normal Distribution.
z
(z)  P(Z  z) 


1
2
What is “Z” ?
e
1

2
m du
2
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 Distribution
Mean m = 0
Variance s2 = 1
Standard Deviation s = 1
Area under the curve = 100%
z = -4.00
z = +4.00
Normal Distribution
Mean m
Variance s2
Standard Deviation s
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 n
Mean m
Variance s2/ n
Standard Error s / √n
Area = 100%
z = -4.00
z = +4.00
As well as Sampling Distributions !
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 a = 100% - Confidence Interval
Confidence Interval 95%
95%
Level of Significance a = 100% - Confidence Interval
a = 100% - 95% = 5%
Confidence Interval 95%
95%
Level of Significance a = 100% - Confidence Interval
a= 100% - 95% = 5%
a/2 = 2.5%
Confidence Interval 95%
Level of Significance a  5%
a / 2  2.5%
a / 2  2.5%
Confidence Interval 95%
Level of Significance a  5%
a / 2  2.5%
a / 2  2.5%
From the Z Distribution Table
For (z) = 0.025 z = -1.96
And (z) = 0.975 z = +1.96
Confidence Interval 95%
Level of Significance a  5%
a / 2  2.5%
Za/2  1.96
a / 2  2.5%
+Za/2  1.96
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 = (m - X) Za/2 / ( s / √n) where Za/2 = -1.96
X upper = (m - X) Za/2 / ( s / √n) where Za/2 = +1.96
Confidence Interval 95%
Level of Significance a  5%
a / 2  2.5%
a / 2  2.5%
Za/2  1.96
+Za/2  +1.96
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 s2 = s2.
A Point Estimate for the population mean is m = X.
Add a Margin of Error about the Mean by including a
Confidence Interval about the point estimate.
From Z = ( X – m ) / (s / √n)
m = X ± Za/2 (s / √n)
For 95%, Za/2 = ±1.96