© 2003 Prentice-Hall, Inc.Chap 6-1 The Normal Distribution and Other Continuous Distributions IE 440 PROCESS IMPROVEMENT THROUGH PLANNED EXPERIMENTATION

© 2003 Prentice-Hall, Inc. Chap 6-1

The Normal Distribution and Other Continuous

Distributions

IE 440

PROCESS IMPROVEMENT THROUGH PLANNED EXPERIMENTATION

Dr. Xueping LiUniversity of Tennessee


Chapter Topics

The Normal Distribution The Standardized Normal Distribution Evaluating the Normality Assumption The Uniform Distribution The Exponential Distribution


Continuous Probability Distributions

Continuous Random Variable Values from interval of numbers Absence of gaps

Continuous Probability Distribution Distribution of continuous random variable

Most Important Continuous Probability Distribution The normal distribution


The Normal Distribution

“Bell Shaped” Symmetrical Mean, Median and

Mode are Equal Interquartile Range

Equals 1.33 Random Variable

Has Infinite Range

Mean Median Mode

X

f(X)


The Mathematical Model

2(1/ 2) /1

2

: density of random variable

3.14159; 2.71828

: population mean

: population standard deviation

: value of random variable

Xf X e

f X X

e

X X


Many Normal Distributions

Varying the Parameters and , We Obtain Different Normal Distributions

There are an Infinite Number of Normal Distributions


The Standardized Normal 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.

XZ

X

f(X)

Z

0Z

1Z

f(Z)


Finding Probabilities

Probability is the area under the curve!

c dX

f(X)

?P c X d


Which Table to Use?

Infinitely Many Normal Distributions Means Infinitely Many Tables to Look

Up!


Solution: The Cumulative Standardized 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 Normal Distribution Table (Portion)

Probabilities

Only One Table is Needed

0 1Z Z

Z = 0.12

0


Standardizing Example

6.2 50.12

10

XZ

Normal Distribution

Standardized Normal

Distribution10 1Z

5 6.2 X Z

0Z 0.12


Example:

Normal Distribution

Standardized Normal

Distribution10 1Z

5 7.1 X Z0Z

0.21

2.9 5 7.1 5.21 .21

10 10

X XZ Z

2.9 0.21

.0832

2.9 7.1 .1664P X

.0832


Z .00 .01

0.0 .5000 .5040 .5080

.5398 .5438

0.2 .5793 .5832 .5871

0.3 .6179 .6217 .6255

.5832.02

0.1 .5478


0 1Z Z

Z = 0.21

Example: 2.9 7.1 .1664P X

(continued)

0


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


0 1Z Z

Z = -0.21

Example: 2.9 7.1 .1664P X

(continued)

0


Normal Distribution in PHStat

PHStat | Probability & Prob. Distributions | Normal …

Example in Excel Spreadsheet

Microsoft Excel Worksheet


Example: 8 .3821P X

Normal Distribution

Standardized Normal

Distribution10 1Z

5 8 X Z0Z

0.30

8 5.30

10

XZ

.3821


Example: 8 .3821P X

(continued)

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


0 1Z Z

Z = 0.30

0


.6217

Finding Z Values for Known Probabilities

Z .00 0.2

0.0 .5000 .5040 .5080

0.1 .5398 .5438 .5478

0.2 .5793 .5832 .5871

.6179 .6255

.01

0.3

Cumulative Standardized Normal Distribution Table

(Portion)

What is Z Given Probability = 0.6217 ?

.6217

0 1Z Z

.31Z 0


Recovering X Values for Known Probabilities

5 .30 10 8X Z

Normal Distribution

Standardized Normal

Distribution10 1Z

5 ? X Z0Z 0.30

.3821.6179


More Examples of Normal Distribution Using PHStat

A set of final exam grades was found to be normally distributed with a mean of 73 and a standard deviation of 8.What is the probability of getting a grade no higher than 91 on 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 <=

2.250

X

Z91

8

73


What percentage of students scored between 65 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

20

X

Z8965

-1

73


(continued)


73

Only 5% of the students taking the test scored 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.

1.6450

X

Z? =86.16

(continued)



The middle 50% of the students scored between what two scores?

273,8X N

Cumulative Percentage 75.00%Z Value 0.67449X Value 78.39592


Cumulative Percentage 25.00%Z Value -0.67449X Value 67.60408


0.670

X

Z78.467.6

-0.67

.50P a X b

.25.25

73


(continued)


Assessing Normality

Not All Continuous Random Variables are Normally Distributed

It is Important to Evaluate How Well the Data Set Seems to Be Adequately Approximated by a 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

? Is the range approximately 6 ?

(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 Standardized Normal Quantile Values on the Horizontal Axis

Evaluate the Plot for Evidence of Linearity

(continued)


Assessing Normality

Normal Probability Plot for Normal Distribution

Look for Straight Line!

30

60

90

-2 -1 0 1 2

Z

X

(continued)


Normal Probability Plot

Left-Skewed Right-Skewed

Rectangular U-Shaped

30

60

90

-2 -1 0 1 2

Z

X

30

60

90

-2 -1 0 1 2

Z

X

30

60

90

-2 -1 0 1 2

Z

X

30

60

90

-2 -1 0 1 2

Z

X


Obtaining Normal ProbabilityPlot in PHStat

PHStat | Probability & Prob. Distributions | Normal Probability Plot

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


The Uniform Distribution

Properties: The probability of occurrence of a value is

equally likely to occur anywhere in the range between the smallest value a and the largest value b

Also called the rectangular distribution

2

a b

2

2

12

b a


The Uniform Distribution 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 the probability that a mark on the wheel will point to somewhere between the North and the East?

(continued)

1

if f X a X bb a

900 90 0.25

360P X


Exponential Distributions

arrival time 1

: any value of continuous random variable

: the population average number of

arrivals per unit of time

1/ : average time between arrivals

2.71828

XP X e

X

e

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


Exponential Distributions

Describes Time or Distance between Events Used for queues

Density Function

Parameters

(continued)

f(X)

X

= 0.5

= 2.0

1 x

f x e


Example

E.g., Customers arrive at the checkout line of a supermarket at the rate of 30 per hour. What is the probability that the arrival time between consecutive customers will be greater than 5 minutes?

30 5/ 60

30 5 / 60 hours

arrival time > 1 arrival time

1 1

.0821

X

P X P X

e


Exponential Distributionin PHStat

PHStat | Probability & Prob. Distributions | Exponential

Example in Excel Spreadsheet

Microsoft Excel Worksheet


Chapter Summary

Discussed the Normal Distribution

Described the Standard Normal

Distribution

Evaluated the Normality Assumption

Defined the Uniform Distribution

Described the Exponential Distribution

Documents

© 2003 Prentice-Hall, Inc.Chap 6-1 The Normal Distribution and Other Continuous Distributions IE 440 PROCESS IMPROVEMENT THROUGH PLANNED EXPERIMENTATION