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DistribusiNormal

CHAPTER TOPICS

The Normal Distribution

The Standardized Normal Distribution

Evaluating the Normality Assumption

The Uniform Distribution

The Exponential Distribution

CONTINUOUS PROBABILITYDISTRIBUTIONS

Continuous Random Variable Values from interval of numbers Absence of gaps

Continuous Probability Distribution Distribution of continuous random variable

Most Important Continuous ProbabilityDistribution The normal distribution

THE NORMAL DISTRIBUTION

“Bell Shaped” Symmetrical Mean, Median and

Mode are Equal Interquartile Range

Equals 1.33 σ Random Variable

Has Infinite Range

MeanMedianMode

THE MATHEMATICAL MODEL

2(1/ 2) /1

: density of random variable

3.14159; 2.71828

: population mean

: population standard deviation

: value of random variable

Xf X e

MANY NORMALDISTRIBUTIONS

Varying the Parameters and , We ObtainDifferent Normal Distributions

There are an Infinite Number of Normal Distributions

THE STANDARDIZEDNORMAL DISTRIBUTION

When X is normally distributed with a mean and a

standard deviation , follows a standardized

(normalized) normal distribution with a mean 0 and a

standard deviation 1.

FINDING PROBABILITIES

Probability isthe area underthe curve!

?P c X d

WHICH TABLE TO USE?

Infinitely Many Normal DistributionsMeans Infinitely Many Tables to Look Up!

SOLUTION: THE CUMULATIVESTANDARDIZED NORMAL DISTRIBUTION

Z .00 .01

0.0 .5000 .5040 .5080

.5398 .5438

0.2 .5793 .5832 .5871

0.3 .6179 .6217 .6255

.5478.02

0.1 .5478

Cumulative Standardized NormalDistribution Table (Portion)

Probabilities

Only One Table is Needed

0 1Z Z

Z = 0.120

STANDARDIZING EXAMPLE6.2 5

0.1210

Normal Distribution StandardizedNormal Distribution

5 6.2 X Z

0Z 0.12

EXAMPLE:

Normal Distribution StandardizedNormal Distribution10 1Z

5 7 .1 X Z0Z

2 .9 5 7 .1 5.21 .21

X XZ Z

2 .9 0 .21

2.9 7 .1 .1664P X

EXAMPLE:

Z .00 .010.0 .5000 .5040 .5080

.5398 .54380.2 .5793 .5832 .5871

0.3 .6179 .6217 .6255

.5832.02

0.1 .5478

Cumulative Standardized NormalDistribution Table (Portion) 0 1Z Z

Z = 0.21

2.9 7.1 .1664P X (continued)

EXAMPLE:

Z .00 .01

-0.3 .3821 .3783 .3745

.4207 .4168

-0.1 .4602 .4562 .4522

0.0 .5000 .4960 .4920

.4168.02

-0.2 .4129

Z = -0.21

2.9 7.1 .1664P X (continued)

NORMAL DISTRIBUTIONIN PHSTAT

PHStat | Probability & Prob. Distributions |Normal …

Example in Excel Spreadsheet

Microsoft ExcelWorksheet

EXAMPLE:

8 .3821P X

8 X Z0Z 0.30

8 5.30

EXAMPLE: (continued)

8 .3821P X

Z .00 .01

0.0 .5000 .5040 .5080

.5398 .5438

0.2 .5793 .5832 .5871

0.3 .6179 .6217 .6255

.6179.02

0.1 .5478

Z = 0.300

FINDING Z VALUES FORKNOWN PROBABILITIES

Z .00 0.2

0.0 .5000 .5040 .5080

0.1 .5398 .5438 .5478

0.2 .5793 .5832 .5871

.6179 .6255

Cumulative StandardizedNormal Distribution Table

(Portion)What is Z GivenProbability = 0.6217 ?

0 1Z Z

.31Z 0

RECOVERING X VALUES FORKNOWN PROBABILITIES

5 .30 10 8X Z

5 ? X Z0Z 0.30

.3821.6179

MORE EXAMPLES OF NORMAL DISTRIBUTIONUSING PHSTAT

A set of final exam grades was found to be normallydistributed with a mean of 73 and a standard deviation of 8.

What is the probability of getting a grade no higher than 91on this exam?

273,8X N 91 ?P X Mean 73Standard Deviation 8

X Value 91Z Value 2.25P(X<=91) 0.9877756

Probability for X <=

What percentage of students scored between65 and 89?

From X Value 65To X Value 89Z Value for 65 -1Z Value for 89 2P(X<=65) 0.1587P(X<=89) 0.9772P(65<=X<=89) 0.8186

Probability for a Range

273,8X N 65 89 ?P X

MORE EXAMPLES OF NORMAL DISTRIBUTIONUSING PHSTAT

(continued)

221.645

Only 5% of the students taking the testscored higher than what grade? 273,8X N ? .05P X

Cumulative Percentage 95.00%Z Value 1.644853X Value 86.15882

Find X and Z Given Cum. Pctage.

Z? =86.16

(continued)

MORE EXAMPLES OF NORMALDISTRIBUTION USING PHSTAT

ASSESSING NORMALITY

Not All Continuous Random Variables areNormally Distributed

It is Important to Evaluate How Well the DataSet Seems to Be Adequately Approximated bya Normal Distribution

ASSESSING NORMALITY

Construct Charts For small- or moderate-sized data sets, do the stem-and-

leaf display and box-and-whisker plot look symmetric? For large data sets, does the histogram or polygon appear

bell-shaped?

Compute Descriptive Summary Measures Do the mean, median and mode have similar values? Is the interquartile range approximately 1.33 σ? Is the range approximately 6 σ?

(continued)

ASSESSING NORMALITY

Observe the Distribution of the Data Set Do approximately 2/3 of the observations lie

between mean 1 standard deviation? Do approximately 4/5 of the observations lie

between mean 1.28 standard deviations? Do approximately 19/20 of the observations lie

between mean 2 standard deviations? Evaluate Normal Probability Plot Do the points lie on or close to a straight line

with positive slope?

(continued)

ASSESSING NORMALITY

Normal Probability Plot Arrange Data into Ordered Array Find Corresponding Standardized Normal

Quantile Values Plot the Pairs of Points with Observed Data

Values on the Vertical Axis and the StandardizedNormal Quantile Values on the Horizontal Axis

Evaluate the Plot for Evidence of Linearity

(continued)

ASSESSING NORMALITY

Normal Probability Plot for NormalDistribution

Look for Straight Line!

306090

-2 -1 0 1 2Z

(continued)

NORMAL PROBABILITY PLOT

Left-Skewed Right-Skewed

Rectangular U-Shaped

306090

-2 -1 0 1 2Z

X306090

-2 -1 0 1 2Z

306090

-2 -1 0 1 2Z

X306090

-2 -1 0 1 2Z

OBTAINING NORMAL PROBABILITYPLOT IN PHSTAT

PHStat | Probability & Prob. Distributions |Normal Probability Plot

Enter the range of the cells that contain thedata in the Variable Cell Range window

THE UNIFORMDISTRIBUTION

Properties: The probability of occurrence of a value is

equally likely to occur anywhere in the rangebetween the smallest value a and the largestvalue b

Also called the rectangular distribution

THE UNIFORMDISTRIBUTION

The Probability Density Function

Application: Selection of random numbers E.g., A wooden wheel is spun on a horizontal

surface and allowed to come to rest. What is theprobability that a mark on the wheel will point tosomewhere between the North and the East?

(continued)

iff X a X bb a

900 90 0.25

360P X

EXPONENTIALDISTRIBUTIONS

arrival tim e 1

: any value o f con tinuous random variab le

: the popu lation average num ber o f

arrivals per un it o f tim e

1 / : average tim e betw een arrivals

2 .71828

XP X e

E.g., Drivers arriving at a toll bridge;customers arriving at an ATM machine

EXPONENTIALDISTRIBUTIONS

Describes Time or Distance between Events Used for queues

Density Function

Parameters

(continued)

EXAMPLE

E.g., Customers arrive at the checkout line of asupermarket at the rate of 30 per hour. What isthe probability that the arrival time betweenconsecutive customers will be greater than 5minutes?

3 0 5 / 6 0

3 0 5 / 6 0 h o u rs

arriva l tim e > 1 arriva l tim e

.0 8 2 1

P X P X

EXPONENTIAL DISTRIBUTIONIN PHSTAT

PHStat | Probability & Prob. Distributions |Exponential

Example in Excel Spreadsheet

Microsoft ExcelWorksheet

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