mate 14 - lecture 3 normal pr

7/30/2019 MatE 14 - Lecture 3 Normal Pr

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NORMAL DISTRIBUTION anPROBABILITY

lecture 3

Mat E 14 Design and Analysis of Experiments in Materials Engineering


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Probability likelihood of the occurrence of an event rfrom statistical experiments

ranges from 0 to 1

notation: P(A) = The Number Of Ways Event A Can OccThe total number Of Possible Outcomes


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Example:Choose a number at random from 1 to 5.

What is the probability of each outcome?What is the probability that the number chosen i

even?What is the probability that the number chosen i

odd?


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Normal Probability Distribution As we make more and more flips, the graph of the probability of an eventbecomes smoother and approaches the bell curve, or normal di


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CENTRAL LIMIT THEOREM

If all samples of a particular size are selected from any population, sampling distribution of the sample means is approximately a normadistribution. This approximation improves as the sample size increa

Implication: For large random samples, the shape of the samplingdistribution of the sample means is close to a normal probabilitydistribution.

Application: Allows the use of the normal probability distribution toconfidence intervals for the population mean.

(sampling and sampling distributions and confidence intervals)


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Normal Probability DistributionCharacteristics:

single peak at the center of distribution mean, median, and mode are equal symmetrical about its mean Falls off smoothly in either direction from the central value, asym Has long tapering tails that extend indefinitely in both direction total area of the curve is equal to 1 Density function is given as:

to avoid the difficult task of solving integrals of normal density functions, tables havmade for standard normal distributions.


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As the value of becomes bigger, the curve becomflatter or becomes more platykurtic (distribution is mdispersed)

Two parameters that define the normal distribution ar and . = indicates the position of the normal curve along the horizontal axis

= d etermines the peakedness of the curve at its center

Normal Probability Distribution


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Standard Normal Probability Distribution

There is an infinite number of possible normaldistributions but the most important member of thisfamily is the one which has a mean of 0 andstandard deviation of 1 . This is the so-calledstandard normal distribution .


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Standard Normal Probability Distribution

most important type of distribution because areas under the curve of a standard normalprobability distribution can be found instatistical tables

any normal probability distribution may betransformed into a standard normal distributby performing the Z transformation


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Introducing the Z-ScoreYou are asked to evaluate a customer complain

about the time it took to be serviced on atelephone customer help line. The session to9 minutes.

Question: Is the event typical or atypical?


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Introducing the Z-Score Approach:Step 1: How different is this from the typical time?

Assume the process mean = 4Calculate the distance from the mean: (x- )

Step 2: Is 5 minutes a typical difference? Assume the process standard deviation = 4.5Calculate the ratio of the distance with the stddeviation:

(x- ) / = 5 / 4.5 = 1.11 standard deviations


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Introducing the Z-ScoreZ Score

The number of standard deviations between a valueand the meanZ = (x- ) /

How far the value is from the average?

How typical is the distance

1 std dev distance = standard

2 std dev distances = atypical

3 std dev distances = rare


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Areas Under the Normal Curve

The area under the curve boundtwo ordinates a and b equals theprobability that the random variableassumes a value between a and b


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About 68% of the area under the normal curve is within plus oand minus one standard deviation of the mean. This can be writteas 1 . About 95% of the area is within plus and minus two standards othe mean, written 2 . Practically all of the area under the normal curve is within threestandard deviations of the mean, written 3 . Showndiagrammatically:

Areas Under the Normal Curve


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Z TransformationExample:

If the z-value is computed to be 1.9what is the area under the normalcurve between the mean and x?

resulting area = 0.47193


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Z TransformationExample:

If the z-value is computed to be 1.9what is the area under the normalcurve between the mean and x?

resulting area = 0.47193


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Z TransformationSome Z table gives the proportion of populatioLESS THAN or EQUAL a specific Z

P (z < 0.10) = 0.53983P (z < 0.13) = 0.55172

P (z < 1.56) = 0.94062


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Z TransformationSince all probabilities add up to 1.0:

P (z > 0.10) = 1- P (z < 0.10)= 1- 0.53983= 0.46017

46 .017%53.983%


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Z TransformationSince the Standard Normal Distribution is symmetrical:

P (z < -0.10) = P (z > 0.10)= 1- 0.53983= 0.46017

P (z < -1.00) = P (z > 1.00)= 1- 0.84134= 0.15866


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Z-Tfrom

to Z


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Z TransformationExamp le 1 : Given a standard normal distribution, find the

area under the curve that lies to the right of z = 1.84, and between z = -1.97 and z = 0.86

Examp le 2 : Given a standard normal distribution, find the value ofsuch that

P (z >k ) = 0.3015 P (k < z < -0.18) = 0.4197


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Z TransformationExam ple 3 : Given a normal distribution with = 50 and =10,

probability that X assumes a value between 45 and 62.

Exam ple 4 : Given a normal distribution with = 300 and = 50probability that X assumes a value greater than 362.

Exam ple 5 : A certain type of storage batter lasts, on average, 3.0 yearwith a standard deviation of 0.5 year. Assuming that the battery livesarea normally distributed, find the probability that a given battery wilast less than 2.3 years.

mate 14 - lecture 3 normal pr

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