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8/6/2019 Basic Concepts of Statistics and Probability (2 of 3)
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Basic Concepts of Statistics & Probability
Review ofStatisticalConcepts
SamplingfromDistributions
Numerical
andGraphicalExamples
Industrial Engineering
Define the following.
Probability Population
Sample Mean
Median Mode
Standard Deviation Variance
Range Box-plot
Histogram Descriptive Statistics
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Basic Concepts of Statistics & Probability
Review ofStatisticalConcepts
SamplingfromDistributions
Numerical
andGraphicalExamples
Industrial Engineering
Box plot
A graph (also known as a box and whisker plot) and summarizes the following
statistical measures:
MedianUpper and lower quartiles
Minimum and maximum data values
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SamplingfromDistributions
Numerical
andGraphicalExamples
Industrial Engineering
Example
Data on hole diameters for aircraft leading edge hole120.5, 120.9, 120.3, 121.3,
120.4, 120.2, 120.1, 120.5,
120.7, 121.1, 120.9, 120.8
1st quartile 120.35 3rd quartile 120.9
Median 120.6
Minimum 120.1 Maximum 121.3
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Basic Concepts of Statistics & Probability
Review ofStatisticalConcepts
SamplingfromDistributions
Numerical
andGraphicalExamples
Industrial Engineering
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Review ofStatisticalConcepts
SamplingfromDistributions
NumericalandGraphicalExamples
Industrial Engineering
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Review ofStatisticalConcepts
SamplingfromDistributions
NumericalandGraphicalExamples
Industrial Engineering
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Basic Concepts of Statistics & Probability
Review ofStatisticalConcepts
SamplingfromDistributions
NumericalandGraphicalExamples
Industrial Engineering
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Review ofStatisticalConcepts
SamplingfromDistributions
NumericalandGraphicalExamples
Industrial Engineering
Types of Distributions
Continuous DistributionsNormal Distribution
Chi-square ( 2) Distribution
t-Distribution
F-DistributionExponential Distribution
Weibull Distribution
Discrete DistributionsBinomial Distribution
Poisson Distribution
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SamplingfromDistributions
NumericalandGraphicalExamples
Industrial Engineering
Normal Distribution
The probability of the normal random variableProbabilities for the normal random variable are given by areas under thecurve.
Where for Standard Normal Distribution
= 0
= 1
= 3.14159
e = 2.71828
Basic Concepts of Statistics & Probability
22 2 / )(
2
1)(
xe x f
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Review ofStatisticalConcepts
SamplingfromDistributions
NumericalandGraphicalExamples
Industrial Engineering
Normal Distribution
Basic Concepts of Statistics & Probability
43210-1-2-3-4
x
For a population that isnormally distributed:
approx. 68% of the data will lie within +1standard deviation of the mean;
approx. 95% of the data will lie within +2
standard deviations of the mean, and approx. 99.7% of the data will lie within +3
standard deviations of the mean .
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Review ofStatisticalConcepts
SamplingfromDistributions
NumericalandGraphicalExamples
Industrial Engineering
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Review ofStatisticalConcepts
SamplingfromDistributions
NumericalandGraphicalExamples
Industrial Engineering
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Review ofStatisticalConcepts
SamplingfromDistributions
NumericalandGraphicalExamples
Industrial Engineering
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Review ofStatisticalConcepts
SamplingfromDistributions
NumericalandGraphicalExamples
Industrial Engineering
Example
The tensile strength of paper is modelled by a normal distribution with amean of 35 lbs/in2 and a standard deviation of 2 lbs/in2.
What is the probability that the tensile strength of a selected item isless than 40 lbs/in2?
If the specifications require the tensile strength to exceed 30 lbs/in2,what is the probability that a selected item is scrapped? The normal
distribution is an important continuous distribution.
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Review ofStatisticalConcepts
SamplingfromDistributions
NumericalandGraphicalExamples
Industrial Engineering
Example
The tensile strength of paper is modelled by a normal distribution with amean of 35 lbs/in2 and a standard deviation of 2 lbs/in2.
What is the probability that the tensile strength of a selected item is lessthan 40 lbs/in2?
If the specifications require the tensile strength to exceed 30 lbs/in2, whatis the probability that a selected item is scrapped? The normal distribution
is an important continuous distribution.Determine P(x30) = 1 - .99379= .00621
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Let X represent measurements taken from a normal distribution.
X
Select a sample of size n, at random, and calculate the sample mean,
Then
And
Basic Concepts of Statistics & Probability
),(~ 2 N x
n,N~
2
x
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SamplingfromDistributions
NumericalandGraphicalExamples
Industrial Engineering
Probability example
The life of an automotive battery is normally distributed with mean 900days and standard deviation 35 days. What is the probability that arandom sample of 25 batteries will have an average life of more than 910days?
Let X represent measurements taken from a normal distribution. X
Select a sample of size n , at random, and calculate the sample mean,
Then
Z = (910-900)/[(35/SQRT(25)] = 1.429
P(Xbar > 910) = 1 - .9235 = .0765
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Review ofStatisticalConcepts
Samplingfrom
Distributions
NumericalandGraphicalExamples
Industrial Engineering
Chi-square ( 2) Distribution
If x1, x2, , x n are normally and independently distributed randomvariables with mean zero and variance one, then the random variable
is distributed as chi-square with n degrees of freedom.
Furthermore, the sampling distribution of
is chi-square with n 1 degrees of freedom when sampling from a normalpopulation
222
21 ... n x x x y
2
2
21
2
)1()(
Sn x x
y
n
ii
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Samplingfrom
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NumericalandGraphicalExamples
Industrial Engineering
Chi-square ( 2) Distribution for various degrees of freedom.
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NumericalandGraphicalExamples
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t-distribution
If x is a standard normal random variable and if y is a chi-square randomvariable with k degrees of freedom, then
is distributed as t with k degrees of freedom.k y
xt
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Distributions
NumericalandGraphicalExamples
Industrial Engineering
F-distribution
If w and y are two independent chi-square random variables with u and v
degrees of freedom, respectively, then
is distributed as F with u numerator degrees of freedom and v denominator
degrees of freedom.
v yuw
F / /
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Review ofStatisticalConcepts
Samplingfrom
Distributions
NumericalandGraphicalExamples
Industrial Engineering
Point Estimation of Process ParametersParameters are values representing the population such as populationmean and variance
Parameters in reality are often unknown and must be estimated.
Statistics are estimates of parameters. (Ex.)
are the sample mean and sample variance, respectively.
Two properties of good point estimatorsThe point estimator should be unbiased. E( ) =
The point estimator should have minimum variance.
2,
2, S X
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NumericalandGraphicalExamples
Industrial Engineering Basic Concepts of Statistics & Probability
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