Submit Search
Upload
Bbs11 ppt ch06
•
0 likes
•
626 views
T
Tuul Tuul
Follow
The Normal Distribution and Other Continuous Distributions
Read less
Read more
Education
Report
Share
Report
Share
1 of 57
Recommended
Bbs11 ppt ch05
Bbs11 ppt ch05
Tuul Tuul
Basic Probability
Basic Probability
Yesica Adicondro
Chap03 numerical descriptive measures
Chap03 numerical descriptive measures
Uni Azza Aunillah
Bbs11 ppt ch03
Bbs11 ppt ch03
Tuul Tuul
Some Important Discrete Probability Distributions
Some Important Discrete Probability Distributions
Yesica Adicondro
Chap06 normal distributions & continous
Chap06 normal distributions & continous
Uni Azza Aunillah
Bbs11 ppt ch10
Bbs11 ppt ch10
Tuul Tuul
Bbs11 ppt ch11
Bbs11 ppt ch11
Tuul Tuul
Recommended
Bbs11 ppt ch05
Bbs11 ppt ch05
Tuul Tuul
Basic Probability
Basic Probability
Yesica Adicondro
Chap03 numerical descriptive measures
Chap03 numerical descriptive measures
Uni Azza Aunillah
Bbs11 ppt ch03
Bbs11 ppt ch03
Tuul Tuul
Some Important Discrete Probability Distributions
Some Important Discrete Probability Distributions
Yesica Adicondro
Chap06 normal distributions & continous
Chap06 normal distributions & continous
Uni Azza Aunillah
Bbs11 ppt ch10
Bbs11 ppt ch10
Tuul Tuul
Bbs11 ppt ch11
Bbs11 ppt ch11
Tuul Tuul
Bbs11 ppt ch02
Bbs11 ppt ch02
Tuul Tuul
Bbs11 ppt ch07
Bbs11 ppt ch07
Tuul Tuul
Bbs11 ppt ch01
Bbs11 ppt ch01
Tuul Tuul
Chap04 basic probability
Chap04 basic probability
Uni Azza Aunillah
Chap05 discrete probability distributions
Chap05 discrete probability distributions
Uni Azza Aunillah
Bbs11 ppt ch08
Bbs11 ppt ch08
Tuul Tuul
Basic business statistics 2
Basic business statistics 2
Anwar Afridi
The Normal Distribution and Other Continuous Distributions
The Normal Distribution and Other Continuous Distributions
Yesica Adicondro
Chap04 discrete random variables and probability distribution
Chap04 discrete random variables and probability distribution
Judianto Nugroho
Bbs11 ppt ch04
Bbs11 ppt ch04
Tuul Tuul
Chap02 presenting data in chart & tables
Chap02 presenting data in chart & tables
Uni Azza Aunillah
Chap01 intro & data collection
Chap01 intro & data collection
Uni Azza Aunillah
Newbold_chap08.ppt
Newbold_chap08.ppt
cfisicaster
Slides for ch05
Slides for ch05
Firas Husseini
Business Statistics Chapter 6
Business Statistics Chapter 6
Lux PP
Chap02 describing data; numerical
Chap02 describing data; numerical
Judianto Nugroho
Quantitative Analysis For Management 11th Edition Render Solutions Manual
Quantitative Analysis For Management 11th Edition Render Solutions Manual
Shermanne
Chap14 multiple regression model building
Chap14 multiple regression model building
Uni Azza Aunillah
Chap07 interval estimation
Chap07 interval estimation
Uni Azza Aunillah
Chap12 simple regression
Chap12 simple regression
Uni Azza Aunillah
chapter_6_modified_slides_0_2.ppt
chapter_6_modified_slides_0_2.ppt
LORAINE FRANCISCO
chapter_6_modified_slides_4.ppt
chapter_6_modified_slides_4.ppt
alvinmadahan
More Related Content
What's hot
Bbs11 ppt ch02
Bbs11 ppt ch02
Tuul Tuul
Bbs11 ppt ch07
Bbs11 ppt ch07
Tuul Tuul
Bbs11 ppt ch01
Bbs11 ppt ch01
Tuul Tuul
Chap04 basic probability
Chap04 basic probability
Uni Azza Aunillah
Chap05 discrete probability distributions
Chap05 discrete probability distributions
Uni Azza Aunillah
Bbs11 ppt ch08
Bbs11 ppt ch08
Tuul Tuul
Basic business statistics 2
Basic business statistics 2
Anwar Afridi
The Normal Distribution and Other Continuous Distributions
The Normal Distribution and Other Continuous Distributions
Yesica Adicondro
Chap04 discrete random variables and probability distribution
Chap04 discrete random variables and probability distribution
Judianto Nugroho
Bbs11 ppt ch04
Bbs11 ppt ch04
Tuul Tuul
Chap02 presenting data in chart & tables
Chap02 presenting data in chart & tables
Uni Azza Aunillah
Chap01 intro & data collection
Chap01 intro & data collection
Uni Azza Aunillah
Newbold_chap08.ppt
Newbold_chap08.ppt
cfisicaster
Slides for ch05
Slides for ch05
Firas Husseini
Business Statistics Chapter 6
Business Statistics Chapter 6
Lux PP
Chap02 describing data; numerical
Chap02 describing data; numerical
Judianto Nugroho
Quantitative Analysis For Management 11th Edition Render Solutions Manual
Quantitative Analysis For Management 11th Edition Render Solutions Manual
Shermanne
Chap14 multiple regression model building
Chap14 multiple regression model building
Uni Azza Aunillah
Chap07 interval estimation
Chap07 interval estimation
Uni Azza Aunillah
Chap12 simple regression
Chap12 simple regression
Uni Azza Aunillah
What's hot
(20)
Bbs11 ppt ch02
Bbs11 ppt ch02
Bbs11 ppt ch07
Bbs11 ppt ch07
Bbs11 ppt ch01
Bbs11 ppt ch01
Chap04 basic probability
Chap04 basic probability
Chap05 discrete probability distributions
Chap05 discrete probability distributions
Bbs11 ppt ch08
Bbs11 ppt ch08
Basic business statistics 2
Basic business statistics 2
The Normal Distribution and Other Continuous Distributions
The Normal Distribution and Other Continuous Distributions
Chap04 discrete random variables and probability distribution
Chap04 discrete random variables and probability distribution
Bbs11 ppt ch04
Bbs11 ppt ch04
Chap02 presenting data in chart & tables
Chap02 presenting data in chart & tables
Chap01 intro & data collection
Chap01 intro & data collection
Newbold_chap08.ppt
Newbold_chap08.ppt
Slides for ch05
Slides for ch05
Business Statistics Chapter 6
Business Statistics Chapter 6
Chap02 describing data; numerical
Chap02 describing data; numerical
Quantitative Analysis For Management 11th Edition Render Solutions Manual
Quantitative Analysis For Management 11th Edition Render Solutions Manual
Chap14 multiple regression model building
Chap14 multiple regression model building
Chap07 interval estimation
Chap07 interval estimation
Chap12 simple regression
Chap12 simple regression
Similar to Bbs11 ppt ch06
chapter_6_modified_slides_0_2.ppt
chapter_6_modified_slides_0_2.ppt
LORAINE FRANCISCO
chapter_6_modified_slides_4.ppt
chapter_6_modified_slides_4.ppt
alvinmadahan
normal distribution_normal distribution_
normal distribution_normal distribution_
NzAh3
Chap006.ppt
Chap006.ppt
najwalyaa
Newbold_chap06.ppt
Newbold_chap06.ppt
cfisicaster
BBS10_ppt_ch07_Sampling_Distribution.ppt
BBS10_ppt_ch07_Sampling_Distribution.ppt
Hermita3
normal distribtion best lecture series.ppt
normal distribtion best lecture series.ppt
syedMinhajuddin7
The Normal Distribution.ppt
The Normal Distribution.ppt
Aravind Reddy
Normal Distribution
Normal Distribution
Shubham Mehta
chap06-1.pptx
chap06-1.pptx
SoujanyaLk1
Applied Business Statistics ,ken black , ch 6
Applied Business Statistics ,ken black , ch 6
AbdelmonsifFadl
Newbold_chap12.ppt
Newbold_chap12.ppt
cfisicaster
Chap05 continuous random variables and probability distributions
Chap05 continuous random variables and probability distributions
Judianto Nugroho
Bbs10 ppt ch03
Bbs10 ppt ch03
Anwar Afridi
Chap09
Chap09
Marouane Zouzhi
Statistik 1 6 distribusi probabilitas normal
Statistik 1 6 distribusi probabilitas normal
Selvin Hadi
Lecture 14.ppt
Lecture 14.ppt
TanvirIslam461642
Newbold_chap05.ppt
Newbold_chap05.ppt
cfisicaster
Chapter 6 2022.pdf
Chapter 6 2022.pdf
Mohamed Ali
Advanced Statistics Homework Help
Advanced Statistics Homework Help
Statistics Homework Helper
Similar to Bbs11 ppt ch06
(20)
chapter_6_modified_slides_0_2.ppt
chapter_6_modified_slides_0_2.ppt
chapter_6_modified_slides_4.ppt
chapter_6_modified_slides_4.ppt
normal distribution_normal distribution_
normal distribution_normal distribution_
Chap006.ppt
Chap006.ppt
Newbold_chap06.ppt
Newbold_chap06.ppt
BBS10_ppt_ch07_Sampling_Distribution.ppt
BBS10_ppt_ch07_Sampling_Distribution.ppt
normal distribtion best lecture series.ppt
normal distribtion best lecture series.ppt
The Normal Distribution.ppt
The Normal Distribution.ppt
Normal Distribution
Normal Distribution
chap06-1.pptx
chap06-1.pptx
Applied Business Statistics ,ken black , ch 6
Applied Business Statistics ,ken black , ch 6
Newbold_chap12.ppt
Newbold_chap12.ppt
Chap05 continuous random variables and probability distributions
Chap05 continuous random variables and probability distributions
Bbs10 ppt ch03
Bbs10 ppt ch03
Chap09
Chap09
Statistik 1 6 distribusi probabilitas normal
Statistik 1 6 distribusi probabilitas normal
Lecture 14.ppt
Lecture 14.ppt
Newbold_chap05.ppt
Newbold_chap05.ppt
Chapter 6 2022.pdf
Chapter 6 2022.pdf
Advanced Statistics Homework Help
Advanced Statistics Homework Help
More from Tuul Tuul
2019 12etses
2019 12etses
Tuul Tuul
2018 12etses
2018 12etses
Tuul Tuul
2017 12etses
2017 12etses
Tuul Tuul
2016 12etses
2016 12etses
Tuul Tuul
Jishee
Jishee
Tuul Tuul
Bbs11 ppt ch19
Bbs11 ppt ch19
Tuul Tuul
Bbs11 ppt ch17
Bbs11 ppt ch17
Tuul Tuul
Bbs11 ppt ch16
Bbs11 ppt ch16
Tuul Tuul
Bbs11 ppt ch14
Bbs11 ppt ch14
Tuul Tuul
Bbs11 ppt ch13
Bbs11 ppt ch13
Tuul Tuul
Bbs11 ppt ch12
Bbs11 ppt ch12
Tuul Tuul
The Theory of Consumer Choice
The Theory of Consumer Choice
Tuul Tuul
Income Inequality and Poverty
Income Inequality and Poverty
Tuul Tuul
Earnings and Discrimination
Earnings and Discrimination
Tuul Tuul
The Economics of Labor Markets
The Economics of Labor Markets
Tuul Tuul
Monopolistic Competition
Monopolistic Competition
Tuul Tuul
Oligopoly
Oligopoly
Tuul Tuul
Monopoly
Monopoly
Tuul Tuul
Firms in Competitive Markets
Firms in Competitive Markets
Tuul Tuul
The Costs of Production
The Costs of Production
Tuul Tuul
More from Tuul Tuul
(20)
2019 12etses
2019 12etses
2018 12etses
2018 12etses
2017 12etses
2017 12etses
2016 12etses
2016 12etses
Jishee
Jishee
Bbs11 ppt ch19
Bbs11 ppt ch19
Bbs11 ppt ch17
Bbs11 ppt ch17
Bbs11 ppt ch16
Bbs11 ppt ch16
Bbs11 ppt ch14
Bbs11 ppt ch14
Bbs11 ppt ch13
Bbs11 ppt ch13
Bbs11 ppt ch12
Bbs11 ppt ch12
The Theory of Consumer Choice
The Theory of Consumer Choice
Income Inequality and Poverty
Income Inequality and Poverty
Earnings and Discrimination
Earnings and Discrimination
The Economics of Labor Markets
The Economics of Labor Markets
Monopolistic Competition
Monopolistic Competition
Oligopoly
Oligopoly
Monopoly
Monopoly
Firms in Competitive Markets
Firms in Competitive Markets
The Costs of Production
The Costs of Production
Recently uploaded
Introduction to AI in Higher Education_draft.pptx
Introduction to AI in Higher Education_draft.pptx
pboyjonauth
MARGINALIZATION (Different learners in Marginalized Group
MARGINALIZATION (Different learners in Marginalized Group
JonathanParaisoCruz
भारत-रोम व्यापार.pptx, Indo-Roman Trade,
भारत-रोम व्यापार.pptx, Indo-Roman Trade,
Virag Sontakke
Proudly South Africa powerpoint Thorisha.pptx
Proudly South Africa powerpoint Thorisha.pptx
thorishapillay1
ECONOMIC CONTEXT - LONG FORM TV DRAMA - PPT
ECONOMIC CONTEXT - LONG FORM TV DRAMA - PPT
iammrhaywood
Software Engineering Methodologies (overview)
Software Engineering Methodologies (overview)
eniolaolutunde
How to Make a Pirate ship Primary Education.pptx
How to Make a Pirate ship Primary Education.pptx
manuelaromero2013
OS-operating systems- ch04 (Threads) ...
OS-operating systems- ch04 (Threads) ...
Dr. Mazin Mohamed alkathiri
9953330565 Low Rate Call Girls In Rohini Delhi NCR
9953330565 Low Rate Call Girls In Rohini Delhi NCR
9953056974 Low Rate Call Girls In Saket, Delhi NCR
ESSENTIAL of (CS/IT/IS) class 06 (database)
ESSENTIAL of (CS/IT/IS) class 06 (database)
Dr. Mazin Mohamed alkathiri
KSHARA STURA .pptx---KSHARA KARMA THERAPY (CAUSTIC THERAPY)————IMP.OF KSHARA ...
KSHARA STURA .pptx---KSHARA KARMA THERAPY (CAUSTIC THERAPY)————IMP.OF KSHARA ...
M56BOOKSTORE PRODUCT/SERVICE
Hierarchy of management that covers different levels of management
Hierarchy of management that covers different levels of management
mkooblal
POINT- BIOCHEMISTRY SEM 2 ENZYMES UNIT 5.pptx
POINT- BIOCHEMISTRY SEM 2 ENZYMES UNIT 5.pptx
Sayali Powar
CARE OF CHILD IN INCUBATOR..........pptx
CARE OF CHILD IN INCUBATOR..........pptx
GaneshChakor2
ECONOMIC CONTEXT - PAPER 1 Q3: NEWSPAPERS.pptx
ECONOMIC CONTEXT - PAPER 1 Q3: NEWSPAPERS.pptx
iammrhaywood
Types of Journalistic Writing Grade 8.pptx
Types of Journalistic Writing Grade 8.pptx
Eyham Joco
Meghan Sutherland In Media Res Media Component
Meghan Sutherland In Media Res Media Component
InMediaRes1
Biting mechanism of poisonous snakes.pdf
Biting mechanism of poisonous snakes.pdf
adityarao40181
CELL CYCLE Division Science 8 quarter IV.pptx
CELL CYCLE Division Science 8 quarter IV.pptx
JiesonDelaCerna
call girls in Kamla Market (DELHI) 🔝 >༒9953330565🔝 genuine Escort Service 🔝✔️✔️
call girls in Kamla Market (DELHI) 🔝 >༒9953330565🔝 genuine Escort Service 🔝✔️✔️
9953056974 Low Rate Call Girls In Saket, Delhi NCR
Recently uploaded
(20)
Introduction to AI in Higher Education_draft.pptx
Introduction to AI in Higher Education_draft.pptx
MARGINALIZATION (Different learners in Marginalized Group
MARGINALIZATION (Different learners in Marginalized Group
भारत-रोम व्यापार.pptx, Indo-Roman Trade,
भारत-रोम व्यापार.pptx, Indo-Roman Trade,
Proudly South Africa powerpoint Thorisha.pptx
Proudly South Africa powerpoint Thorisha.pptx
ECONOMIC CONTEXT - LONG FORM TV DRAMA - PPT
ECONOMIC CONTEXT - LONG FORM TV DRAMA - PPT
Software Engineering Methodologies (overview)
Software Engineering Methodologies (overview)
How to Make a Pirate ship Primary Education.pptx
How to Make a Pirate ship Primary Education.pptx
OS-operating systems- ch04 (Threads) ...
OS-operating systems- ch04 (Threads) ...
9953330565 Low Rate Call Girls In Rohini Delhi NCR
9953330565 Low Rate Call Girls In Rohini Delhi NCR
ESSENTIAL of (CS/IT/IS) class 06 (database)
ESSENTIAL of (CS/IT/IS) class 06 (database)
KSHARA STURA .pptx---KSHARA KARMA THERAPY (CAUSTIC THERAPY)————IMP.OF KSHARA ...
KSHARA STURA .pptx---KSHARA KARMA THERAPY (CAUSTIC THERAPY)————IMP.OF KSHARA ...
Hierarchy of management that covers different levels of management
Hierarchy of management that covers different levels of management
POINT- BIOCHEMISTRY SEM 2 ENZYMES UNIT 5.pptx
POINT- BIOCHEMISTRY SEM 2 ENZYMES UNIT 5.pptx
CARE OF CHILD IN INCUBATOR..........pptx
CARE OF CHILD IN INCUBATOR..........pptx
ECONOMIC CONTEXT - PAPER 1 Q3: NEWSPAPERS.pptx
ECONOMIC CONTEXT - PAPER 1 Q3: NEWSPAPERS.pptx
Types of Journalistic Writing Grade 8.pptx
Types of Journalistic Writing Grade 8.pptx
Meghan Sutherland In Media Res Media Component
Meghan Sutherland In Media Res Media Component
Biting mechanism of poisonous snakes.pdf
Biting mechanism of poisonous snakes.pdf
CELL CYCLE Division Science 8 quarter IV.pptx
CELL CYCLE Division Science 8 quarter IV.pptx
call girls in Kamla Market (DELHI) 🔝 >༒9953330565🔝 genuine Escort Service 🔝✔️✔️
call girls in Kamla Market (DELHI) 🔝 >༒9953330565🔝 genuine Escort Service 🔝✔️✔️
Bbs11 ppt ch06
1.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc. Chap 6-1 Chapter 6 The Normal Distribution and Other Continuous Distributions Basic Business Statistics 11th Edition
2.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-2 Learning Objectives In this chapter, you learn: To compute probabilities from the normal distribution To use the normal probability plot to determine whether a set of data is approximately normally distributed To compute probabilities from the uniform distribution To compute probabilities from the exponential distribution To compute probabilities from the normal distribution to approximate probabilities from the binomial distribution
3.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-3 Continuous Probability Distributions A continuous random variable is a variable that can assume any value on a continuum (can assume an uncountable number of values) thickness of an item time required to complete a task temperature of a solution height, in inches These can potentially take on any value depending only on the ability to precisely and accurately measure
4.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-4 The Normal Distribution ‘Bell Shaped’ Symmetrical Mean, Median and Mode are Equal Location is determined by the mean, μ Spread is determined by the standard deviation, σ The random variable has an infinite theoretical range: + to Mean = Median = Mode X f(X) μ σ
5.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-5 The Normal Distribution Density Function 2 μ)(X 2 1 e 2π 1 f(X) The formula for the normal probability density function is Where e = the mathematical constant approximated by 2.71828 π = the mathematical constant approximated by 3.14159 μ = the population mean σ = the population standard deviation X = any value of the continuous variable
6.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-6 By varying the parameters μ and σ, we obtain different normal distributions Many Normal Distributions
7.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-7 The Normal Distribution Shape X f(X) μ σ Changing μ shifts the distribution left or right. Changing σ increases or decreases the spread.
8.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-8 The Standardized Normal Any normal distribution (with any mean and standard deviation combination) can be transformed into the standardized normal distribution (Z) Need to transform X units into Z units The standardized normal distribution (Z) has a mean of 0 and a standard deviation of 1
9.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-9 Translation to the Standardized Normal Distribution Translate from X to the standardized normal (the “Z” distribution) by subtracting the mean of X and dividing by its standard deviation: σ μX Z The Z distribution always has mean = 0 and standard deviation = 1
10.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-10 The Standardized Normal Probability Density Function The formula for the standardized normal probability density function is Where e = the mathematical constant approximated by 2.71828 π = the mathematical constant approximated by 3.14159 Z = any value of the standardized normal distribution 2 (1/2)Z e 2π 1 f(Z)
11.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-11 The Standardized Normal Distribution Also known as the “Z” distribution Mean is 0 Standard Deviation is 1 Z f(Z) 0 1 Values above the mean have positive Z-values, values below the mean have negative Z-values
12.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-12 Example If X is distributed normally with mean of 100 and standard deviation of 50, the Z value for X = 200 is This says that X = 200 is two standard deviations (2 increments of 50 units) above the mean of 100. 2.0 50 100200 σ μX Z
13.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-13 Comparing X and Z units Z 100 2.00 200 X Note that the shape of the distribution is the same, only the scale has changed. We can express the problem in original units (X) or in standardized units (Z) (μ = 100, σ = 50) (μ = 0, σ = 1)
14.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-14 Finding Normal Probabilities a b X f(X) P a X b( )≤ Probability is measured by the area under the curve ≤ P a X b( )<<= (Note that the probability of any individual value is zero)
15.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-15 f(X) Xμ Probability as Area Under the Curve 0.50.5 The total area under the curve is 1.0, and the curve is symmetric, so half is above the mean, half is below 1.0)XP( 0.5)XP(μ 0.5μ)XP(
16.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-16 The Standardized Normal Table The Cumulative Standardized Normal table in the textbook (Appendix table E.2) gives the probability less than a desired value of Z (i.e., from negative infinity to Z) Z0 2.00 0.9772 Example: P(Z < 2.00) = 0.9772
17.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-17 The Standardized Normal Table The value within the table gives the probability from Z = up to the desired Z value .9772 2.0P(Z < 2.00) = 0.9772 The row shows the value of Z to the first decimal point The column gives the value of Z to the second decimal point 2.0 . . . (continued) Z 0.00 0.01 0.02 … 0.0 0.1
18.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-18 General Procedure for Finding Normal Probabilities Draw the normal curve for the problem in terms of X Translate X-values to Z-values Use the Standardized Normal Table To find P(a < X < b) when X is distributed normally:
19.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-19 Finding Normal Probabilities Let X represent the time it takes to download an image file from the internet. Suppose X is normal with mean 8.0 and standard deviation 5.0. Find P(X < 8.6) X 8.6 8.0
20.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-20 Let X represent the time it takes to download an image file from the internet. Suppose X is normal with mean 8.0 and standard deviation 5.0. Find P(X < 8.6) Z0.120X8.68 μ = 8 σ = 10 μ = 0 σ = 1 (continued) Finding Normal Probabilities 0.12 5.0 8.08.6 σ μX Z P(X < 8.6) P(Z < 0.12)
21.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-21 Z 0.12 Z .00 .01 0.0 .5000 .5040 .5080 .5398 .5438 0.2 .5793 .5832 .5871 0.3 .6179 .6217 .6255 Solution: Finding P(Z < 0.12) .5478.02 0.1 .5478 Standardized Normal Probability Table (Portion) 0.00 = P(Z < 0.12) P(X < 8.6)
22.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-22 Finding Normal Upper Tail Probabilities Suppose X is normal with mean 8.0 and standard deviation 5.0. Now Find P(X > 8.6) X 8.6 8.0
23.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-23 Now Find P(X > 8.6)… (continued) Z 0.12 0 Z 0.12 0.5478 0 1.000 1.0 - 0.5478 = 0.4522 P(X > 8.6) = P(Z > 0.12) = 1.0 - P(Z ≤ 0.12) = 1.0 - 0.5478 = 0.4522 Finding Normal Upper Tail Probabilities
24.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-24 Finding a Normal Probability Between Two Values Suppose X is normal with mean 8.0 and standard deviation 5.0. Find P(8 < X < 8.6) P(8 < X < 8.6) = P(0 < Z < 0.12) Z0.120 X8.68 0 5 88 σ μX Z 0.12 5 88.6 σ μX Z Calculate Z-values:
25.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-25 Z 0.12 Solution: Finding P(0 < Z < 0.12) 0.0478 0.00 = P(0 < Z < 0.12) P(8 < X < 8.6) = P(Z < 0.12) – P(Z ≤ 0) = 0.5478 - .5000 = 0.0478 0.5000 Z .00 .01 0.0 .5000 .5040 .5080 .5398 .5438 0.2 .5793 .5832 .5871 0.3 .6179 .6217 .6255 .02 0.1 .5478 Standardized Normal Probability Table (Portion)
26.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-26 Suppose X is normal with mean 8.0 and standard deviation 5.0. Now Find P(7.4 < X < 8) X 7.4 8.0 Probabilities in the Lower Tail
27.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-27 Probabilities in the Lower Tail Now Find P(7.4 < X < 8)… X7.4 8.0 P(7.4 < X < 8) = P(-0.12 < Z < 0) = P(Z < 0) – P(Z ≤ -0.12) = 0.5000 - 0.4522 = 0.0478 (continued) 0.0478 0.4522 Z-0.12 0 The Normal distribution is symmetric, so this probability is the same as P(0 < Z < 0.12)
28.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-28 Empirical Rules μ ± 1σ encloses about 68.26% of X’s f(X) X μ μ+1σμ-1σ What can we say about the distribution of values around the mean? For any normal distribution: σσ 68.26%
29.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-29 The Empirical Rule μ ± 2σ covers about 95% of X’s μ ± 3σ covers about 99.7% of X’s xμ 2σ 2σ xμ 3σ 3σ 95.44% 99.73% (continued)
30.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-30 Steps to find the X value for a known probability: 1. Find the Z value for the known probability 2. Convert to X units using the formula: Given a Normal Probability Find the X Value ZσμX
31.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-31 Finding the X value for a Known Probability Example: Let X represent the time it takes (in seconds) to download an image file from the internet. Suppose X is normal with mean 8.0 and standard deviation 5.0 Find X such that 20% of download times are less than X. X? 8.0 0.2000 Z? 0 (continued)
32.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-32 Find the Z value for 20% in the Lower Tail 20% area in the lower tail is consistent with a Z value of -0.84Z .03 -0.9 .1762 .1736 .2033 -0.7 .2327 .2296 .04 -0.8 .2005 Standardized Normal Probability Table (Portion) .05 .1711 .1977 .2266 … … … … X? 8.0 0.2000 Z-0.84 0 1. Find the Z value for the known probability
33.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-33 2. Convert to X units using the formula: Finding the X value 80.3 0.5)84.0(0.8 ZσμX So 20% of the values from a distribution with mean 8.0 and standard deviation 5.0 are less than 3.80
34.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-34 Evaluating Normality Not all continuous distributions are normal It is important to evaluate how well the data set is approximated by a normal distribution. Normally distributed data should approximate the theoretical normal distribution: The normal distribution is bell shaped (symmetrical) where the mean is equal to the median. The empirical rule applies to the normal distribution. The interquartile range of a normal distribution is 1.33 standard deviations.
35.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-35 Evaluating Normality Comparing data characteristics to theoretical properties Construct charts or graphs For small- or moderate-sized data sets, construct a stem-and-leaf display or a boxplot to check for symmetry 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)
36.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-36 Evaluating Normality Comparing data characteristics to theoretical properties Observe the distribution of the data set Do approximately 2/3 of the observations lie within mean ±1 standard deviation? Do approximately 80% of the observations lie within mean ±1.28 standard deviations? Do approximately 95% of the observations lie within mean ±2 standard deviations? Evaluate normal probability plot Is the normal probability plot approximately linear (i.e. a straight line) with positive slope? (continued)
37.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-37 Constructing A Normal Probability Plot Normal probability plot Arrange data into ordered array Find corresponding standardized normal quantile values (Z) Plot the pairs of points with observed data values (X) on the vertical axis and the standardized normal quantile values (Z) on the horizontal axis Evaluate the plot for evidence of linearity
38.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-38 A normal probability plot for data from a normal distribution will be approximately linear: 30 60 90 -2 -1 0 1 2 Z X The Normal Probability Plot Interpretation
39.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-39 Normal Probability Plot Interpretation Left-Skewed Right-Skewed Rectangular 30 60 90 -2 -1 0 1 2 Z X (continued) 30 60 90 -2 -1 0 1 2 Z X 30 60 90 -2 -1 0 1 2 Z X Nonlinear plots indicate a deviation from normality
40.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-40 Evaluating Normality An Example: Mutual Funds Returns 403020100-10 Return 2006 Boxplot of 2006 Returns The boxplot appears reasonably symmetric, with three lower outliers at -9.0, -8.0, and -8.0 and one upper outlier at 35.0. (The normal distribution is symmetric.)
41.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-41 Evaluating Normality An Example: Mutual Funds Returns Descriptive Statistics (continued) • The mean (12.5142) is slightly less than the median (13.1). (In a normal distribution the mean and median are equal.) • The interquartile range of 9.2 is approximately 1.46 standard deviations. (In a normal distribution the interquartile range is 1.33 standard deviations.) • The range of 44 is equal to 6.99 standard deviations. (In a normal distribution the range is 6 standard deviations.) • 72.2% of the observations are within 1 standard deviation of the mean. (In a normal distribution this percentage is 68.26%. • 87% of the observations are within 1.28 standard deviations of the mean. (In a normal distribution percentage is 80%.)
42.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-42 Evaluating Normality An Example: Mutual Funds Returns 403020100-10 99.99 99 95 80 50 20 5 1 0.01 Return 2006 Percent Probability Plot of Return 2006 Normal (continued) Plot is approximately a straight line except for a few outliers at the low end and the high end.
43.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-43 Evaluating Normality An Example: Mutual Funds Returns Conclusions The returns are slightly left-skewed The returns have more values concentrated around the mean than expected The range is larger than expected (caused by one outlier at 35.0) Normal probability plot is reasonably straight line Overall, this data set does not greatly differ from the theoretical properties of the normal distribution (continued)
44.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-44 The Uniform Distribution The uniform distribution is a probability distribution that has equal probabilities for all possible outcomes of the random variable Also called a rectangular distribution
45.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-45 The Continuous Uniform Distribution: otherwise0 bXaif ab 1 where f(X) = value of the density function at any X value a = minimum value of X b = maximum value of X The Uniform Distribution (continued) f(X) =
46.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-46 Properties of the Uniform Distribution The mean of a uniform distribution is The standard deviation is 2 ba μ 12 a)-(b σ 2
47.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-47 Uniform Distribution Example Example: Uniform probability distribution over the range 2 ≤ X ≤ 6: 2 6 0.25 f(X) = = 0.25 for 2 ≤ X ≤ 66 - 2 1 X f(X) 4 2 62 2 ba μ 1547.1 12 2)-(6 12 a)-(b σ 22
48.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-48 Uniform Distribution Example Example: Using the uniform probability distribution to find P(3 ≤ X ≤ 5): 2 6 0.25 P(3 ≤ X ≤ 5) = (Base)(Height) = (2)(0.25) = 0.5 X f(X) (continued) 3 54
49.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-49 The Exponential Distribution Often used to model the length of time between two occurrences of an event (the time between arrivals) Examples: Time between trucks arriving at an unloading dock Time between transactions at an ATM Machine Time between phone calls to the main operator
50.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-50 The Exponential Distribution Xλ e1X)timeP(arrival Defined by a single parameter, its mean λ (lambda) The probability that an arrival time is less than some specified time X is where e = mathematical constant approximated by 2.71828 λ = the population mean number of arrivals per unit X = any value of the continuous variable where 0 < X <
51.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-51 Exponential Distribution Example Example: Customers arrive at the service counter at the rate of 15 per hour. What is the probability that the arrival time between consecutive customers is less than three minutes? The mean number of arrivals per hour is 15, so λ = 15 Three minutes is 0.05 hours P(arrival time < .05) = 1 – e-λX = 1 – e-(15)(0.05) = 0.5276 So there is a 52.76% probability that the arrival time between successive customers is less than three minutes
52.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-52 The Exponential Distribution In Excel Calculating the probability that an exponential distribution with an mean of 20 is less than 0.1
53.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-53 Normal Approximation to the Binomial Distribution The binomial distribution is a discrete distribution, but the normal is continuous To use the normal to approximate the binomial, accuracy is improved if you use a correction for continuity adjustment Example: X is discrete in a binomial distribution, so P(X = 4) can be approximated with a continuous normal distribution by finding P(3.5 < X < 4.5)
54.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-54 Normal Approximation to the Binomial Distribution The closer π is to 0.5, the better the normal approximation to the binomial The larger the sample size n, the better the normal approximation to the binomial General rule: The normal distribution can be used to approximate the binomial distribution if nπ ≥ 5 and n(1 – π) ≥ 5 (continued)
55.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-55 Normal Approximation to the Binomial Distribution The mean and standard deviation of the binomial distribution are μ = nπ Transform binomial to normal using the formula: (continued) )(1n nX σ μX Z ππ π )(1nσ ππ
56.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-56 Using the Normal Approximation to the Binomial Distribution If n = 1000 and π = 0.2, what is P(X ≤ 180)? Approximate P(X ≤ 180) using a continuity correction adjustment: P(X ≤ 180.5) Transform to standardized normal: So P(Z ≤ -1.54) = 0.0618 1.54 0.2))(1(1000)(0.2 )(1000)(0.2180.5 )(1n nX Z ππ π X180.5 200 -1.54 0 Z
57.
Basic Business Statistics,
11e © 2009 Prentice-Hall, Inc.. Chap 6-57 Chapter Summary Presented key continuous distributions normal, uniform, exponential Found probabilities using formulas and tables Recognized when to apply different distributions Applied distributions to decision problems