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Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 6-1
Chapter 6
he !or"al #istribution $ %therContinuous #istributions
Statistics for Managers usingMicrosoft Excel 6 th Edition
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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"so use the nor"al probability plot to deter"ine *hether
a set o) data is appro+i"ately nor"ally distributed
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Continuous Probability #istributions
continuous rando" (ariable is a (ariable thatcan assu"e any (alue on a continuu" canassu"e an uncountable nu"ber o) (alues/
thic ness o) an ite"ti"e re uired to co"plete a taste"perature o) a solutionheight, in inches
hese can potentially ta e 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 EqualLocation 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
'
$(')
4
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he !or"al #istribution#ensity unction
he )or"ula )or the nor"al probability density )unction is
Where e = the mathematical constant approximated by 2.71828π = the mathematical constant approximated by 3.14159μ = the pop lation mean! = the pop lation standard de"iation# = any "al e o$ the contin o s "ariable
2μ%
e2π1$&#%
−
−= σ
σ
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By !aryin% the parameters and " , *e obtaindi$$erent normal distributions
7any !or"al #istributions
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he !or"al #istribution9hape
'
$(')
"
Changing shi)ts thedistribution le)t or right .
Changing 4 increasesor decreases thespread.
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he 9tandardi;ed !or"al
ny nor"al distribution *ith any "ean andstandard de(iation co"bination/ can betrans)or"ed into the standardi;ed nor"al distribution </
!eed to trans)or" = units into < units
he standardi;ed nor"al distribution </ has a"ean o) 0 and a standard de(iation o) 1
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ranslation to the 9tandardi;ed!or"al #istribution
ranslate )ro" = to the standardi;ed nor"althe ?<@ distribution/ bysubtracting the "ean
o) = and di(iding by its standard de(iation A
he < distribution al*ays has "ean B 0 and
standard de(iation B 1
4
C=< −=
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he 9tandardi;ed !or"alProbability #ensity unction
he )or"ula )or the standardi;ed nor"alprobability density )unction is
Dhere e B the "athe"atical constant appro+i"ated by 2.81:2: B the "athe"atical constant appro+i"ated by .1 15>
< B any (alue o) the standardi;ed nor"al distribution
21F2/<e2E1) </ −=
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he 9tandardi;ed !or"al #istribution
lso no*n as the ?<@ distribution7ean is 0
9tandard #e(iation is 1
+
$(+)
-
Galues abo(e the "ean ha(e positi(e <-(alues,(alues belo* the "ean ha(e negati(e <-(alues
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E+a"ple
I) = is distributed nor"ally *ith "ean o) 100 and standard de(iation o) 50 , the < (alue)or = B 200 is
his says that = B 200 is t*o standardde(iations 2 incre"ents o) 50 units/ abo(ethe "ean o) 100.
2.'(5'
1''((2''!
μ#) =
−=
−=
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Co"paring = and < units
+.-
/0./ .'
1ote that the shape o$ 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 B 50/
B 0, 4 B 1/
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inding !or"al Probabilities
a b '
$(') P a ' b /5
Probability is "easured by the areaunder the cur(e
5
P a ' b /B!ote that the probability
o) any indi(idual (alue is;ero/
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$(')
'
Probability as rea nder 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 =∞<<−∞
0.5/=P C =∞<<0.5C/=P =<<−∞
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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 than a desired (alue o) < i.e.,)ro" negati(e in)inity to </
+/0
0788/E+a"pleA
P < J 2.00/ B 0.>882
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he 9tandardi;ed !or"al able
he (alue *ithin the
table gi(es theprobability )ro" < B ∞ up to the desired <(alue
0788/
2.09(+ /0 ) & 0788/
he ro* sho*sthe (alue o) <to the )irst
deci"al point
he colu"n gi(es the (alue o)< to the second deci"al point
/0
..
.
(continued)
< 0.00 0.01 0.02 K
0.0
0.1
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Leneral Procedure )orinding !or"al Probabilities
#ra* the nor"al cur(e )or the proble" in ter"s o) =
ranslate =-(alues to <-(alues
se the 9tandardi;ed !or"al able
o )ind P a J = J b/ *hen = isdistributed nor"allyA
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inding !or"al Probabilities
&et = represent the ti"e it ta es in seconds/to do*nload an i"age )ile )ro" the internet.9uppose = is nor"al *ith a "ean o)1:.0seconds and a standard de(iation o) 5.0seconds. ind P = J 1:.6/
- 06
'- 0
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&et = represent the ti"e it ta es, 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 = J 1:.6/
+0-/'- 06-
B 1: 4 B 5
& " & -
(continued)inding !or"al Probabilities
P = J 1:.6/ P < J 0.12/
'.125.'8.'118.*
!μ#
) =
−=
−=
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+
0-/
< .00 .01
0.0 .5000 .50 0 .50:0
.5 >: .5 :
0.2 .58> .5: 2 .5:81
0. .618> .6218 .6255
9olutionA inding P < J 0.12/
0.5 8:0 /
0- 05 8:
9tandardi;ed !or"al Probabilityable Portion/
0
B P < J 0.12/P = J 1:.6/
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inding !or"alpper ail Probabilities
9uppose = is nor"al *ith "ean 1:.0and standard de(iation 5.0.
!o* ind P = M 1:.6/
'
- 06
- 0
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!o* ind P = M 1:.6/K(continued)
+
0-/
+
0-/
0.5 8:
1.000 1.0 - 0.5 8:B 0. 522
P = M 1:.6/ B P < M 0.12/ B 1.0 - P < N 0.12/
B 1.0 - 0.5 8: B 0. 522
inding !or"alpper ail Probabilities
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inding a !or"al ProbabilityOet*een *o Galues
9uppose = is nor"al *ith "ean 1:.0 andstandard de(iation 5.0. ind P 1: J = J 1:.6/
P 1: J = J 1:.6/
B P 0 J < J 0.12/
+0-/'- 06-
Calculate <-(aluesA
'5
8118!
μ#) =−=−=
'.125
8118.*!
μ#) =−=−=
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+
0-/
9olutionA inding P 0 J < J 0.12/
0.0 8:
0
B P 0 J < J 0.12/P 1: J = J 1:.6/
B P < J 0.12/ P < N 0/
B 0.5 8: - 0.5000 B 0.0 8:
0.5000
< .00 .01
0.0 .5000 .50 0 .50:0
.5 >: .5 :
0.2 .58> .5: 2 .5:81
0. .618> .6218 .6255
0 /
0- 05 8:
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 P 18. J = J 1:/
'
-80;- 0
Probabilities in the &o*er ail
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Probabilities in the &o*er ail
!o* ind P 18. J = J 1:/K
'-80; - 0
P 18. J = J 1:/
B P -0.12 J < J 0/B P < J 0/ P < N -0.12/
B 0.5000 - 0. 522 B 0.0 8:
(continued)
0.0 8:
0. 522
+< 0-/
he !or"al distribution issy""etric, so this probabilityis the sa"e as P 0 J < J 0.12/
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E"pirical Qules
= -" encloses about6 0/6> o$ '’s
$(')
'?-"<-"
Dhat can *e say about the distribution o) (aluesaround the "eanR or any nor"al distributionA
""
6 0/6>
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he E"pirical Qule
= /" co!ers about 7@> o$ '’s
= A" co!ers about 7708> o$ '’s
3
/" /"
3
A" A"
7@0;;> 7708A>
(continued)
( l b b l
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9teps to )ind the = (alue )or a no*nprobabilityA
1. ind the < (alue )or the no*n probability2. Con(ert to = units using the )or"ulaA
Li(en a !or"al Probabilityind the = Galue
<4C= +=
i di h ( l )
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inding the = (alue )or aSno*n Probability
E+a"pleA&et = represent the ti"e it ta es in seconds/ todo*nload an i"age )ile )ro" the internet.
9uppose = is nor"al *ith "ean 1:.0 and standardde(iation 5.0ind = such that 20T o) do*nload ti"es are less than
=.
'- 0
0.2000
+
(continued)
i d h ( l )
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ind the < (alue )or20T in the &o*er ail
20T area in the lo*ertail is consistent *ith a< (alue o) -0.:< .0
-0.> .1862 .18 6
.20-0.8 .2 28 .22>6
0 ;
< 0 02005
9tandardi;ed !or"al Probabilityable Portion/
.05
.1811
.1>88
.2266
K
K
KK
'- 0
0.2000
+< 0 ;
1. ind the < (alue )or the no*n probability
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2. Con(ert to = units using the )or"ulaA
inding the = (alue
9o 20T o) the (alues )ro" a distribution*ith "ean 1:.0 and standard de(iation5.0 are less than 1 .:0
8.13
'.5%84.'&'.18
)!μ#
=−+=
+=
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E(aluating !or"ality
!ot all continuous distributions are nor"alIt is i"portant to e(aluate ho* *ell the data set isappro+i"ated by a nor"al distribution.!or"ally distributed data should appro+i"ate thetheoretical nor"al distributionA
he nor"al distribution is bell shaped sy""etrical/*here the "ean is e ual to the "edian.
he e"pirical rule applies to the nor"al distribution.
he inter uartile range o) a nor"al distribution is 1.standard de(iations.
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E(aluating !or"ality
Co"paring data characteristics to theoreticalproperties
Construct charts or graphsor s"all- or "oderate-si;ed data sets, construct a ste"-and-lea)
display or a bo+plot to chec )or sy""etryor large data sets, does the histogra" or polygon appear bell-
shapedR
Co"pute descripti(e su""ary "easures#o the "ean, "edian and "ode ha(e si"ilar (aluesRIs the inter uartile range appro+i"ately 1. 4RIs the range appro+i"ately 6 4R
(continued)
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E(aluating !or"ality
Co"paring data characteristics to theoreticalproperties %bser(e the distribution o) the data set
#o appro+i"ately 2F o) the obser(ations lie *ithin "ean U1standard de(iationR#o appro+i"ately :0T o) the obser(ations lie *ithin "eanU1.2: standard de(iationsR#o appro+i"ately >5T o) the obser(ations lie *ithin "ean U2standard de(iationsR
E(aluate nor"al probability plotIs the nor"al probability plot appro+i"ately linear i.e. a straightline/ *ith positi(e slopeR
(continued)
C i
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Constructing !or"al Probability Plot
!or"al probability plot rrange data into ordered array
ind corresponding standardi;ed nor"al uantile(alues </
Plot the pairs o) points *ith obser(ed data (alues =/on the (ertical a+is and the standardi;ed nor"al
uantile (alues </ on the hori;ontal a+isE(aluate the plot )or e(idence o) linearity
h ! " l P b bilit Pl t
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nor"al probability plot )or data)ro" a nor"al distribution *ill be
appro+i"ately linear A
0
60
>0
-2 -1 0 1 2 <
=
he !or"al Probability PlotInterpretation
! " l P b bilit Pl t
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!or"al Probability PlotInterpretation
&e)t-9 e*ed Qight-9 e*ed
Qectangular
0
60
>0
-2 -1 0 1 2 <
=
(continued)
0
60
>0
-2 -1 0 1 2 <
=
0
60
>0
-2 -1 0 1 2 <
= !onlinear plots indicatea de(iation )ro"nor"ality
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E(aluating !or"ality n E+a"pleA Oond unds Qeturns
he bo+plot is s e*ed tothe le)t. he nor"aldistribution is sy""etric./
- 0 - 0 -20 -10 0 10
Bond unds / Deturns
Bond unds / Deturns
i!e<number Summary
Minimum - 1.>irst uartile -2.5
Median 2.6
#hird uartile 8.1Ma3imum 15.0
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E(aluating !or"ality n E+a"pleA Oond unds Qeturns
#escripti(e 9tatistics
(continued)
• he "ean 1. 18/ is less than the "edian 2.6/.In a nor"al distribution the "ean and "edian
are e ual./• he inter uartile range o) >.6 is appro+i"ately
1.25 standard de(iations. In a nor"al
distribution the inter uartile range is 1.standard de(iations./
• he range o) 6.> is e ual to 6.1 standardde(iations. In a nor"al distribution the range is6 standard de(iations./
• 86.1T o) the obser(ations are *ithin 1 standardde(iation o) the "ean. In a nor"al distributionthis percentage is 6:.26T.
• ::. T o) the obser(ations are *ithin 1.2:standard de(iations o) the "ean. In a nor"aldistribution this percentage is :0T./
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E(aluating !or"ality n E+a"pleA Oond unds Qeturns
(continued)
Plot is not a straightline and sho*s the
distribution is s e*edto the le)t. henor"al distributionappears as a straightline./
( l l
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E(aluating !or"ality n E+a"pleA 7utual unds Qeturns
Conclusionshe returns are le)t-s e*edhe returns ha(e "ore (alues concentrated around
the "ean than e+pectedhe range is larger than e+pected
!or"al probability plot is not a straight line%(erall, this data set greatly di))ers )ro" thetheoretical properties o) the nor"al distribution
(continued)
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Chapter 9u""ary
Presented ey continuous distributions nor"al, uni)or", e+ponential
ound probabilities using )or"ulas and tables
Qecogni;ed *hen to apply di))erent distributions
pplied distributions to decision proble"s
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ll rights reser(ed. !o part o) this publication "ay be reproduced, stored in a retrie(alsyste", or trans"itted, in any )or" or by any "eans, electronic, "echanical, photocopying,
recording, or other*ise, *ithout the prior *ritten per"ission o) the publisher.Printed in the nited 9tates o) "erica.
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