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ENGR 610
Applied Statistics
Fall 2007 - Week 4
Marshall University
CITE
Jack Smith
Overview for Today
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Review of Ch 5
Homework problems for Ch 5
Estimation Procedures (Ch 8)
Homework assignment
About the 1st exam
Chapter 5 Review
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Continuous probability distributions
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Uniform
Normal
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Standard Normal Distribution (Z scores)
Approximation to Binomial, Poisson distributions
Normal probability plot
LogNormal
Exponential
Sampling of the mean, proportion
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Central Limit Theorem
Continuous Probability Distributions
P(a  X  b) 
P(X  b) 

  E(X) 
b

b

f (x)dx
a
f (x)dx


 xf (x)dx
(Mean, expected value)


2 

2
(x


)
f (x)dx



(Variance)
Uniform Distribution
 1
b  a

f (x)  
 0


b

 x
a
a x b
elsewhere
1
(b  a)
dx 
(b  a)
2
2
(
b

a
)
1
(
b

a
)


 2   x 
dx 

2  (b  a)
12
a 
b
2
Normal Distribution
f ( x) 
1
2  x
e
Gaussian with
peak at µ and
inflection points at +/- σ
FWHM = 2(2ln(2))1/2 σ
(1/ 2 )[( x   x ) /  x ]2
Standard Normal Distribution
1 (1/ 2)Z 2
f (x) 
e
2
where
Z
X  x
x
Is the standard normal score (“Z-score”)

With and effective mean of zero
and a standard deviation of 1
68, 95, 99.7%
Normal Approximation to
Binomial Distribution

For binomial distribution
x  np
 x  np(1 p)
and so
X  x
Z



x
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X  np
np(1 p)
Variance, 2, should be at least 10
Normal Approximation to
Poisson Distribution
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For Poisson distribution
x  
x  
and so


Z


X  x
x

X 

Variance, , should be at least 5
Normal Probability Plot
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Use normal probability graph paper
to plot ordered cumulative percentages,
Pi = (i - 0.5)/n * 100%, as Z-scores
- or Use Quantile-Quantile plot (see directions in
text)
- or Use software (PHStat)!
Lognormal Distribution
1
f (x) 
(X )  e



2 ln(x )
e
(1/ 2)[(ln(X )  ln(x ) )/ ln(x ) ]2
2
 ln(X )  ln(X
) /2
X  e
2
2  ln(X )  ln(X
)
2
 ln(X
)
(e
1)
Exponential Distribution
f (x)  e x
   1/ 

Poisson, with continuous rate of change, 
Only memoryless random distribution
 X
P(x  X) 1 e



Sampling Distribution of the Mean,
Proportion
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Central Limit Theorem
x  x
x  x / n
p  
p 
Continuous data
(proportion)
 (1  )
n
Attribute data
Homework Problems (Ch 5)
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5.66
5.67
5.68
5.69
Estimation Procedures
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Estimating population mean ()
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Estimating population variance (2)
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from sample mean (X-bar) and population variance (2)
using Standard Normal Z distribution
from sample mean (X-bar) and sample variance (s2)
using Student’s t distribution
from sample variance (s2)
using 2 distribution
Estimating population proportion ()
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from sample proportion (p) and binomial variance (npq)
using Standard Normal Z distribution
Estimation Procedures, cont’d
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Predicting future individual values (X)
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from sample mean (X-bar) and sample variance (s2)
using Student’s t distribution
Tolerance Intervals
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One- and two-sided
Using k-statistics
Parameter Estimation
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Statistical inference
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Conclusions about population parameters from
sample statistics (mean, variation, proportion,…)
Makes use of CLT, various sampling distributions,
and degrees of freedom
Interval estimate
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With specified level of confidence that population
parameter is contained within
When population parameters are known and
distribution is Normal,
E(X)  Z X  X  E(X)  Z X
Point Estimator Properties
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Unbiased
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Efficient
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Average (expectation) value over all possible
samples (of size n) equals population parameter
Arithmetic mean most stable and precise measure
of central tendency
Consistent
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Improves with sample size n
Estimating population mean ()

From sample mean (X-bar) and known population
variance (2)
X Z
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

n
 X Z

n
Using Standard Normal distribution (and CLT!)
Where Z, the critical value, corresponds to area of
(1-)/2 for a confidence level of (1-)100%

For example, from Table A.2, Z = 1.96 corresponds to area =
0.95/2 = 0.475 for 95% confidence interval, where  = 0.05
is the sum of the upper and lower tail portions
Estimating population mean ()

From sample mean (X-bar) and sample variance (s2)
X  tn1
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
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s
s
   X  t n1
n
n
Using Student’s t distribution with
n-1 degrees of freedom
Where tn-1, the critical value, corresponds to area of
(1-)/2 for a confidence level of (1-)100%
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For example, from Table A.4, t = 2.0639 corresponds to area
= 0.95/2 = 0.475 for 95% confidence interval, where
/2 = 0.025 is the area of the upper tail portion, and 24 is the
number of degrees of freedom for a sample size of 25
Estimating population variance (2)
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From sample variance (s2)
(n 1)
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s

2
2
U
   (n 1)
2
s
2

2
L
Using 2 distribution with
n-1 degrees of freedom
Where U and L, the upper and lower critical values,
corresponds to areas of /2 and 1-/2 for a
confidence level of (1-)100%
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For example, from Table A.6, U = 39.364 and L = 12.401
correspond to the areas of 0.975 and 0.025 for 95%
confidence interval and 24 degrees of freedom
Predicting future individual values (X)
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
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From sample mean (X-bar) and sample
variation (s2)
1
1
X  tn1s 1  X  X  t n1s 1
n
n
Using Student’s t distribution
Prediction interval
Analogous to   Z  X    Z
Tolerance intervals
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An interval that includes at least a certain proportion
of measurements with a stated confidence based on
sample mean (X-bar) and sample variance (s2)
X  K 2s
X  K1s
X  K1s

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
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Two-sided
Lower Bound
Upper Bound
Using k-statistics (Tables A.5a, A.5.b)
Where K1 and K2 corresponds to a confidence level of
(1-)100% for p100% of measurements and a
sample size of n
Estimating population proportion ()
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From binomial mean (np) and variation (npq)
from sample (size n, and proportion p)
p(1 p)
p(1 p)
pZ
  p Z
n
n
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
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Using Standard Normal Z distribution as
approximation to binomial distribution
Analogous to   Z  X    Z
where p = X/n
Homework
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Ch 8
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Appendix 8.1
Problems: 8.43-44
Exam #1 (Ch 1-5,8)
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Take home
Given out (electronically) after in-class review
Open book, notes
No collaboration - honor system
Use Excel w/ PHStat where appropriate, but
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Explain, explain, explain!
Due by beginning of class, Sept 27