Basic Concepts of Statistics and Probability (2 of 3)

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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Numerical



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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Numerical



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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Numerical




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NumericalandGraphicalExamples



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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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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


22 2 / )(

2

1)(

xe x f


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Normal Distribution


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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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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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


),(~ 2 N x

n,N~

2

x


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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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Samplingfrom

Distributions



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

Distributions



Chi-square ( 2) Distribution for various degrees of freedom.



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Samplingfrom

Distributions



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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Samplingfrom

Distributions



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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Samplingfrom

Distributions



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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Samplingfrom

Distributions


Industrial Engineering Basic Concepts of Statistics & Probability

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Basic Concepts of Statistics and Probability (2 of 3)