Chapter 6:The Normal Distribution & Other Continuous Distributions

8/9/2019 Chapter 6:The Normal Distribution & Other Continuous Distributions

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Copyright 2012 Pearson Education, Inc. publishing as Prentice Hall Chap 6-1 Chap 6-1

Chapter 6

he !or"al #istribution $ %therContinuous #istributions

Basic Business Statistics12thEdition


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&earning %b'ecti(es

In this chapter, you learn: o co"pute probabilities )ro" the nor"al distribution

Ho* to use the nor"al distribution to sol(e business

proble"s o use the nor"al probability plot to deter"ine *hether

a set o) data is appro+i"ately nor"ally distributed

o co"pute probabilities )ro" the uni)or" distribution

o co"pute probabilities )ro" the e+ponentialdistribution


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Copyright 2012 Pearson Education, Inc. publishing as Prentice Hall Chap 6- Chap 6-

Continuous Probability #istributions

continuous rando" (ariableis a (ariable thatcan assu"e any (alue on a continuu" canassu"e an uncountable nu"ber o) (alues/

thicness o) an ite" ti"e reuired to co"plete a tas te"perature o) a solution height, in inches

hese can potentially tae on any (aluedepending only on the ability to precisely andaccurately "easure


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he !or"al #istribution

Bell Shaped Symmetrical Mean, Median and Mode

are Equal

Location is determined by themean,

Spread is determined by thestandard de!iation, "

#he random !ariable has anin$inite theoretical ran%e:3 to

Mean& Median& Mode

'

$(')

4


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he !or"al #istribution#ensity unction

2)(X

2

1

e21f(X)

=

he )or"ula )or the nor"al probability density )unction is

Where e = the mathematical constant approximated by 2.71828

= the mathematical constant approximated by .1!1"#

= the pop$lation mean

% = the pop$lation standard de&iation

X = any &al$e of the contin$o$s &ariable


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By !aryin% the parameters and ", *e obtaindi$$erent normal distributions

7any !or"al #istributions


7/66Copyright 2012 Pearson Education, Inc. publishing as Prentice Hall Chap 6-8 Chap 6-8

he !or"al #istribution9hape

'

$(')

"

Changing shi)ts thedistribution le)t or right.

Changing 4increasesor decreases thespread.


8/66Copyright 2012 Pearson Education, Inc. publishing as Prentice Hall Chap 6-: Chap 6-:

he 9tandardi;ed !or"al

nynor"al distribution *ith any "ean andstandard de(iation co"bination/ can be

trans)or"ed into the standardi;ed nor"aldistribution


9/66Copyright 2012 Pearson Education, Inc. publishing as Prentice Hall Chap 6-> Chap 6->

ranslation to the 9tandardi;ed!or"al #istribution

ranslate )ro" = to the standardi;ed nor"althe ?



he 9tandardi;ed!or"al #istribution

lso no*n as the ?


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Co"paring = and < units

+

.-

/0

./ .'

1ote that the shape o$ the distribution is the same,only the scale has chan%ed0 2e can e3press theproblem in the ori%inal units (' in dollars) or in

standardi4ed units (+)

B 100, 4 50/

B 0, 4 1/



inding !or"al Probabilities

a b '

$(') P a ' b /5

Probability is "easured by the areaunder the cur(e

5

P a ' b /

!ote that the probability

o) any indi(idual (alue is;ero/



$(')

'

Probability asrea Fnder the Cur(e

0.50.5

he total area under the cur(e is 1.0, and the cur(e issy""etric, so hal) is abo(e the "ean, hal) is belo*

1.0/=P =



he 9tandardi;ed !or"al able

he Cu"ulati(e 9tandardi;ed !or"al tablein the te+tboo ppendi+ table E.2/gi(es theprobability less thana desired (alue o) < i.e.,)ro" negati(e in)inity to



he 9tandardi;ed !or"al able

he (alue *ithin the

table gi(es theprobability )ro" < up to the desired



eneral Procedure )orinding !or"al Probabilities

#ra* the nor"al cur(e )or the proble" in ter"s o) =

ranslate =-(alues to


18/66Copyright 2012 Pearson Education, Inc. publishing as Prentice Hall Chap 6-1: Chap 6-1:


&et = represent the ti"e it taes in seconds/to do*nload an i"age )ile )ro" the internet.

9uppose = is nor"al *ith a "ean o) 1:.0

seconds and a standard de(iation o) 5.0seconds. ind P= G 1:.6/

-06

'-0


19/66Copyright 2012 Pearson Education, Inc. publishing as Prentice Hall Chap 6-1> Chap 6-1>

&et = represent the ti"e it taes, in seconds to do*nload an i"age )ile)ro" the internet.

9uppose = is nor"al *ith a "ean o) 1:.0 seconds and a standardde(iation o) 5.0 seconds. ind P= G 1:.6/

+0-/'-06-

B 1:4 5

B & " & -

(continued)


'.12".'

8.'118.*

%

X

=

=

=

P= G 1:.6/ P< G 0.12/



+

0-/

< .00 .01

0.0 .5000 .500 .50:0

.5>: .5:

0.2 .58> .5:2 .5:81

0. .618> .6218 .6255

9olutionA inding P< G 0.12/

0.58:0/

0- 058:

9tandardi;ed !or"al Probabilityable Portion/

0

P< G 0.12/

P= G 1:.6/



inding !or"alFpper ail Probabilities

9uppose = is nor"al *ith "ean 1:.0and standard de(iation 5.0.

!o* ind P= J 1:.6/

'

-06

-0



!o* ind P= J 1:.6/

(continued)

+

0-/

+

0-/

0.58:

1.000 1.0 - 0.58: 0.522

P= J 1:.6/ P< J 0.12/ 1.0 - P< K 0.12/

1.0 - 0.58: 0.522

inding !or"alFpper ail Probabilities



inding a !or"al ProbabilityLet*een *o Dalues

9uppose = is nor"al *ith "ean 1:.0 andstandard de(iation 5.0. ind P1: G = G 1:.6/

P1: G = G 1:.6/

P0 G < G 0.12/

+0-/'-06-

'"

8118

%

X =

=

=

'.12"

8118.*

%

X =

=

=

Calculate


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+

0-/

9olutionA inding P0 G < G 0.12/

0.08:

0

P0 G < G 0.12/

P1: G = G 1:.6/

P< G 0.12/ M P< K 0/

0.58: - 0.5000 0.08:

0.5000

< .00 .01

0.0 .5000 .500 .50:0

.5>: .5:

0.2 .58> .5:2 .5:81

0. .618> .6218 .6255

0/

0- 058:

9tandardi;ed !or"al Probabilityable Portion/


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9uppose = is nor"al *ith "ean 1:.0and standard de(iation 5.0.

!o* ind P18. G = G 1:/

'

-80;-0

Probabilities in the &o*er ail


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Probabilities in the &o*er ail

!o* ind P18. G = G 1:/

'-80; -0

P18. G = G 1:/

P-0.12 G < G 0/ P< G 0/ M P< K -0.12/

0.5000 - 0.522 0.08:

(continued)

0.08:

0.522

+


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9teps to )ind the = (alue )or a no*nprobabilityA

1. ind the


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Copyright 2012 Pearson Education, Inc. publishing as Prentice Hall Chap 6-2: Chap 6-2:

inding the = (alue )or aNno*n Probability

E+a"pleA &et = represent the ti"e it taes in seconds/ to

do*nload an i"age )ile )ro" the internet.

9uppose = is nor"al *ith "ean 1:.0 and standardde(iation 5.0

ind = such that 20O o) do*nload ti"es are less than=.

'= -0

0.2000

+=

(continued)

i d th < l )


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Copyright 2012 Pearson Education, Inc. publishing as Prentice Hall Chap 6-2> Chap 6-2>

ind the 6

0;


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2. Con(ert to = units using the )or"ulaA

inding the = (alue

8.1

'.")8!.'('.18

%X

=

+=

+=

9o 20O o) the (alues )ro" a distribution*ith "ean 1:.0 and standard de(iation5.0 are less than 1.:0


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Copyright 2012 Pearson Education, Inc. publishing as Prentice Hall Chap 6-1

Fsing E+cel ith he !or"al#istribution

Chap 6-1

>indin% 1ormal 9robabilities

>indin% ' ?i!en @ 9robability


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Fsing 7initab ith he !or"al#istribution

Chap 6-2

>indin% 9('A) *hen ' is normal

*ith a mean o$ 8 and a standard de!iation o$ /

Cumulati!e istribution >unction

!or"al *ith "ean 8 and standard de(iation 2

+ P = G + /

5 0.15:655

1

2

F i 7i it b ith h ! l


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Copyright 2012 Pearson Education, Inc. publishing as Prentice Hall Chap 6-

Fsing 7initab ith he !or"al#istribution

Chap 6-

(continued)

1

2

>indin% 3 so that 9('3) & 0- *hen ' is normal

*ith a mean o$ 8 and a standard de!iation o$ /

In!erse Cumulati!e istribution >unction

!or"al *ith "ean 8 and standard de(iation 2

P =G + / +

0.1 .6>0


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E(aluating !or"ality

!ot all continuous distributions are nor"al It is i"portant to e(aluate ho* *ell the data set is

appro+i"ated by a nor"al distribution. !or"ally distributed data should appro+i"ate the

theoretical nor"al distributionA he nor"al distribution is bell shaped sy""etrical/

*here the "ean is eual to the "edian. he e"pirical rule applies to the nor"al distribution.

he interuartile range o) a nor"al distribution is 1.standard de(iations.


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Co"paring data characteristics to theoreticalproperties

Construct charts or graphs or s"all- or "oderate-si;ed data sets, construct a ste"-and-lea)

display or a bo+plot to chec )or sy""etry or large data sets, does the histogra" or polygon appear bell-

shapedQ

Co"pute descripti(e su""ary "easures #o the "ean, "edian and "ode ha(e si"ilar (aluesQ Is the interuartile range appro+i"ately 1. 4Q Is the range appro+i"ately 6 4Q

(continued)


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Co"paring data characteristics to theoreticalproperties %bser(e the distributiono) the data set

#o appro+i"ately 2R o) the obser(ations lie *ithin "ean S1standard de(iationQ

#o appro+i"ately :0O o) the obser(ations lie *ithin "eanS1.2: standard de(iationsQ

#o appro+i"ately >5O o) the obser(ations lie *ithin "ean S2

standard de(iationsQ E(aluate nor"al probability plot

Is the nor"al probability plot appro+i"ately linear i.e. a straightline/ *ith positi(e slopeQ

(continued)

Constr cting T antile T antile


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Constructing Tuantile-Tuantile!or"al Probability Plot

!or"al probability plot rrange data into ordered array

ind corresponding standardi;ed nor"al uantile(alues


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Copyright 2012 Pearson Education, Inc. publishing as Prentice Hall Chap 6-: Chap 6-:

uantile-uantile nor"alprobability plot )or data )ro" a

nor"al distribution *ill be

appro+i"ately linearA

0

60

>0

-2 -1 0 1 2 Chap 6->

Tuantile-Tuantile !or"alProbability Plot Interpretation

&e)t-9e*ed Uight-9e*ed

Uectangular

0

60

>0

-2 -1 0 1 2 0

-2 -1 0 1 2 0

-2 -1 0 1 2 1O o) the obser(ations are *ithin 1standard de(iation o) the "ean. In a nor"aldistribution this percentage is 6:.26O.

:5.O o) the obser(ations are *ithin 1.2:standard de(iations o) the "ean. In a nor"aldistribution this percentage is :0O./

E l ti ! lit


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Copyright 2012 Pearson Education, Inc. publishing as Prentice Hall Chap 6-

E(aluating !or"alityn E+a"pleA Lond unds Ueturns

Chap 6-

(continued)

#escripti(e 9tatistics >6.20O o) the returns are *ithin 2 standard

de(iations o) the "ean. In a nor"aldistribution, >5.O o) the (alues lie *ithin 2standard de(iations o) the "ean./

he se*ness statistic is 0.>0:5 and the

urtosis statistic is 2.56. In a nor"aldistribution each o) these statistics euals ;ero./

E l ti ! lit


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(continued)

Plot is not a straightline and sho*s thedistribution is se*edto the right. he

nor"al distributionappears as a straightline./

Tuantile-Tuantile !or"al Probability Plot ro" E+cel

E l ti ! lit


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(continued)

Plot is not a straightline, rises uicly inthe beginning, risesslo*ly at the end and

sho*s the distributionis se*ed to theright.

!or"al Probability Plot ro" 7initab

E l ti ! lit


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E(aluating !or"alityn E+a"pleA 7utual unds Ueturns

Conclusions he returns are right-se*ed he returns ha(e "ore (alues *ithin 1 standard

de(iation o) the "ean than e+pected he range is larger than e+pected "ostly due to the

outlier at 2/ !or"al probability plot is not a straight line %(erall, this data set greatly di))ers )ro" the

theoretical properties o) the nor"al distribution

(continued)


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Copyright 2012 Pearson Education, Inc. publishing as Prentice Hall Chap 6-: Chap 6-:

he Fni)or" #istribution

he uni)or" distributionis a probability

distribution that has eual probabilities

)or all possible outco"es o) the rando"(ariable

lso called a rectangular distribution


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Properties o) the


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Properties o) theFni)or" #istribution

he "ean o) a uni)or" distribution is

he standard de(iation is

2

baB

+=

12

a/-b4

2

=


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Fni)or" #istribution E+a"ple

E+a"pleAFni)or" probability distribution o(er the range 2 K = K 6A

2 6

0.25

)=/ 0.25 )or 2 K = K 66 - 21

=

)=/

2

62

2

baB =

+=

+=

158.112

2/-6

12

a/-b4

22

===


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Fni)or" #istribution E+a"ple

E+a"pleAFsing the uni)or" probabilitydistribution to )ind P K = K 5/A

2 6

0.25

P K = K 5/ Lase/Height/ 2/0.25/ 0.5

=

)=/

(continued)

5


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he E+ponential #istribution

%)ten used to "odel the length o) ti"ebet*een t*o occurrenceso) an e(ent theti"e bet*een arri(als/

E+a"plesA i"e bet*een trucs arri(ing at an unloading doc i"e bet*een transactions at an 7 7achine i"e bet*een phone calls to the "ain operator


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he E+ponential #istribution

=Ve1=/ti"eParri(al =unction

E+ponential *ith "ean 0.05

+ P = G + /

0.1 0.:6665

Calculatin% the probability that an e3ponential distribution *ith amean o$ -/ is less than 0-


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Copyright 2012 Pearson Education, Inc. publishing as Prentice Hall Chap 6-5: Chap 6-5:

Chapter 9u""ary

Presented ey continuous distributions nor"al, uni)or", e+ponential

ound probabilities using )or"ulas and tables

Uecogni;ed *hen to apply di))erent distributions

pplied distributions to decision proble"s


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Copyright 2012 Pearson Education, Inc. publishing as Prentice Hall Chap 6-5>

Dn Line #opic

he !or"al ppro+i"ation o heLino"ial

Basic Business Statistics12thEdition


60/66


&earning %b'ecti(es

In this topic, you learn: hy using a continuity ad'ust"ent yields a "ore

accurate appro+i"ation

o appro+i"ate bino"ial probabilities using the nor"aldistribution

Fsing !or"al #istribution o


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Fsing !or"al #istribution oppro+i"ate Lino"ial Probability

bino"ial distribution is a discrete distribution*hich can only tae on the (alues o) 0, 1, 2, . . ,n.

hen n gets large the calculations associated*ith the bino"ial distribution beco"e tedious.

In these situations can use a nor"al distribution

*ith the sa"e "ean and standard de(iation asthe bino"ial to appro+i"ate the bino"ialprobability


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or a bino"ial rando" (ariable =, P= c/ is non;ero)or c 0, 1, 2, . . . n.

or a nor"al rando" (ariable , P c/ )or any (alue

c is ;ero. 9o to appro+i"ate a bino"ial probability using the

nor"al distribution ha(e to use a continuity ad'ust"ent. I) = is bino"ial and is nor"al *e appro+i"ate P=c/

by Pc M 0.5 G G c 3 0.5/ *here has the sa"e"ean and standard de(iation as =. dding and subtracting the 0.5 is the continuity

ad'ust"ent

he !eed or Continuity d'ust"ent

hen Can he !or"al


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hen Can he !or"alppro+i"ation Le Fsed

he nor"al appro+i"ation can be used as long asA nW X 5 and n1 M W/ X 5

Uecall 7ean o) a bino"ial is B nW

9tandard de(iation o) a bino"ial is 49TUnW1 M W//


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n E+a"ple

You select a rando" sa"ple o) n 1600 tires)ro" a production process *ith a de)ect rate o):O. You *ant to calculate the probability that

150 or )e*er tires *ill be de)ecti(e. Here B 1600Z0.0: 12: and 4

9TU1600Z0.0:Z0.>2/ 10.:5.

&et = be a nor"al rando" (ariable *ith this"ean and standard de(iation then the desiredprobability is P= G 150.5/


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E+a"ple Con[t/

his is P< G150.5 M 12:/R10.:5/ 0.>:0:

9o *e appro+i"ate the probability o) )inding150 or )e*er de)ects as 0.>:0:.


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opic 9u""ary

In this topic, you learned: hy using a continuity ad'ust"ent yields a "ore

accurate appro+i"ation

o appro+i"ate bino"ial probabilities using the nor"aldistribution

Documents

Chapter 6:The Normal Distribution & Other Continuous Distributions