5.5 Normal Approximations to Binomial Distributions

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5.5 Normal Approximations to

Binomial Distributions

Statistics

Mrs. SpitzFall 2008



Objecties!Assi"nment

# $o% to &eci&e %'en t'e normal &istribution can

approximate t'e binomial &istribution

# $o% to (in& t'e correction (or continuit)

# $o% to use t'e normal &istribution to

approximate binomial probabilities

# Assi"nment* pp. 2+,-2+ /,-,8 all



Approximatin" a Binomial Distribution

# n Section +.21 )ou learne& 'o% to (in& binomialprobabilities. For instance1 a sur"ical proce&ure 'as an85 c'ance o( success an& a &octor per(orms t'eproce&ure on ,0 patients1 it is eas) to (in& t'e probabilit)

o( exactl) t%o success(ul sur"eries.# But %'at i( t'e &octor per(orms t'e sur"ical proce&ure on

,50 patients an& )ou %ant to (in& t'e probabilit) o( (e%ert'an ,00 success(ul sur"eries3

# 4o &o t'is usin" t'e tec'niues &escribe& in +.21 )ou

%oul& 'ae to use t'e binomial (ormula ,00 times an&(in& t'e sum o( t'e resultin" probabilities. 4'is is notpractical an& a better approac' is to use a normal&istribution to approximate t'e binomial &istribution.



Normal Approximation to a Binomial

Distribution

4o see %') t'is result is ali&1 loo6 at t'e (ollo%in" sli&e

an& binomial &istributions (or p 7 0.25 an& n 7 +1 ,01 25an& 50. Notice t'at as n increases1 t'e 'isto"ram

approac'es a normal cure.





Stu&) 4ip

# roperties o( a binomial experiment

9 n in&epen&ent trials

9 4%o possible outcomes* success or (ailure

9 robabilit) o( success is p: probabilit) o( a

(ailure is , 9 p 7

9 p is constant (or eac' trial



;x. ,* Approximatin" t'e Binomial

Distribution

# 4%o binomial experiments are liste&. Deci&e

%'et'er )ou can use t'e normal &istribution to

approximate x1 t'e number o( people %'o repl)

)es. ( so1 (in& t'e mean an& stan&ar& &eiation.( not1 explain %').

,. 4'irt)-seen percent o( Americans sa) t'e)

al%a)s (l) an American (la" on t'e Fourt' o(

<ul). =ou ran&oml) select ,5 Americans an& as6

eac' i( 'e or s'e al%a)s (lies an American (la"

on t'e Fourt' o( <ul).



;x. ,* Approximatin" t'e Binomial

Distribution - Solution

,. 4'irt)-seen percent o( Americans sa) t'e) al%a)s (l)an American (la" on t'e Fourt' o( <ul). =ou ran&oml)select ,5 Americans an& as6 eac' i( 'e or s'e al%a)s(lies an American (la" on t'e Fourt' o( <ul).

n t'is binomial experiment1 n 7 ,51 p 7 0.> an& 7

0.?1 so*

np 7 ,5@0.> 7 5.55 an& nq 7 ,5@0.? 7 .+5

Because np C 5 an& n C 51 )ou can use t'e normal&istribution %it' µ 7 5.55 an&

87.163.037.015 ≈••== npqσ

to approximate t'e &istribution o( x.



;x. ,b* Approximatin" t'e Binomial

Distribution - Solution

2. Ninet)-t'ree percent o( Americans %ant t'e nationalant'em to remain t'e same. =ou ran&oml) select ?5 Americans an& as6 eac' i( 'e or s'e %ant t'e nationalant'em to remain t'e same.

n t'is binomial experiment1 n 7 ?51 p 7 0. an& 7

0.0>1 so*

np 7 ?5@0. 7 ?0.+5 an& nq 7 [email protected]> 7 +.55

Because n ≤ 51 )ou cannot use t'e normal &istribution

to approximate t'e &istribution o( x.



Note (or 4 users

# 4'e 4s cannot calculate t'e cumulatie

binomial probabilit) (or n 7 ,010001 p 7 0.+

an& x 7 000. 4'e probabilit) can be

calculate& usin" a normal approximation.

4'ere are issues %it' memor) limitation

(or )our calculator (or t'e binomial

&istribution.



Eorrection (or Eontinuit)

# 4'e binomial &istribution is &iscrete an& can berepresente& b) a probabilit) 'isto"ram. 4o calculate anexact binomial probabilit)1 )ou can use t'e binomial(ormula (or eac' alue o( x an& a&& t'e results.

eometricall)1 t'is correspon&s to a&&in" t'e areas o(bars in t'e probabilit) 'isto"ram. G'en )ou &o t'is1remember t'at eac' bar 'as a %i&t' o( one unit an& x ist'e mi&point o( t'e interal.

# G'en )ou use a continuous normal &istribution to

approximate a binomial probabilit)1 )ou nee& to moe 0.5units to t'e le(t an& ri"'t o( t'e mi&point to inclu&e allpossible x-alues in t'e interal. G'en )ou &o t'is1 )ouare ma6in" a correction (or continuit).



Eorrection (or Eontinuit)



;x. 2* Hsin" a Eorrection (or Eontinuit)

Hse t'e correction (or continuit) to conert eac' o( t'e (ollo%in"

binomial interals to a normal &istribution interal.

,. 4'e probabilit) o( "ettin" bet%een 2>0 an& ,0 successes1 inclusie

2. 4'e probabilit) o( "ettin" more t'an ,5> an& less t'an +20

successes

. 4'e probabilit) o( "ettin" less t'an ? successes.

SOIH4ON*

,. 4'e probabilit) o( "ettin" bet%een 2>0 an& ,0 successes1 inclusie

4'e mi&point alues are 2>01 2>,1 . . . ,0. 4'e boun&aries (or t'e

normal &istribution are 2?.5 J x J ,0.5







successes

SOIH4ON*


successes

4'e mi&point alues are ,581 ,51 . . . +,. 4'e boun&aries (or t'e

normal &istribution are ,5>.5 J x J +,.5 @less t'an means but not

eual to







SOIH4ON*


4'e mi&point alues are . . . ?01 ?,1 ?2. 4'e boun&ar) (or t'e normal&istribution is x J ?2.5 @less t'an means but not eual to





;x. * Approximatin" a Binomial robabilit)

4'irt)-seen percent o( Americans sa) t'e) al%a)s (l) an

American (la" on t'e Fourt' o( <ul). =ou ran&oml) select,5 Americans an& as6 eac' i( 'e or s'e (lies an American (la" on t'e Fourt' o( <ul). G'at is t'eprobabilit) t'at (e%er t'an ei"'t o( t'em repl) )es3

SOIH4ON* From ;xample ,1 )ou 6no% t'at )ou can

use a normal &istribution %it' µ 7 5.55 an& σ K,.8> to

approximate t'e binomial &istribution. B) appl)in" t'e

continuit) correction1 )ou can re%rite t'e &iscrete

probabilit) @x J 8 as @x J >.5. 4'e "rap' on t'enext sli&e s'o%s a normal cure %it' µ 7 5.55 an& σ

K,.8> an& a s'a&e& area to t'e le(t o( >.5. 4'e z-score

t'at correspon&s to x 7 >.5 is



Continued . . .

04.187.1

55.55.7≈

−=

−=

σ

µ x z

Using the StandardNormal Table,

@zJ,.0+ 7 0.8508

So, the probability that fewer than eight people

respond yes is 0.8508



;x. +* Approximatin" a Binomial robabilit)

4%ent)-nine percent o( Americans sa) t'e) are con(i&ent

t'at passen"er trips to t'e moon %ill occur &urin" t'eirli(etime. =ou ran&oml) select 200 Americans an& as6 i('e or s'e t'in6s passen"er trips to t'e moon %ill occur in'is or 'er li(etime. G'at is t'e probabilit) t'at at least 50%ill sa) )es3

SOIH4ON* Because np 7 200 L 0.2 7 58 an& n 7

200 L 0.>, 7 ,+21 t'e binomial ariable x is

approximatel) normall) &istribute& %it'

58== np µ

42.671.029.0200 ≈••== npqσ

an&



Ex. 4 ontin!ed

Hsin" t'e correction (or continuit)1 )ou can re%rite

t'e &iscrete probabilit) @x C 50 as t'econtinuous probabilit) @ x C +.5. 4'e "rap'

s'o%s a normal cure %it' µ 7 58 an& σ 7 ?.+21

an& a s'a&e& area to t'e ri"'t o( +.5.



Ex. 4 ontin!ed

4'e z-score t'at correspon&s to +.5 is

So1 t'e probabilit) t'at at least 50 %ill sa) )es is*

@x C +.5 7 , 9 @z ≤ -,.2

7 , 9 0.0+

7 0.0??

32.142.6

585.49−≈

−=

−=

σ

µ x z



St!dy Tip

# n a &iscrete &istribution1 t'ere is a

&i((erence bet%een @x C c an& @ x c.

4'is is true because t'e probabilit) t'at x

is exactl) c is not zero. N a continuous&istribution1 'o%eer1 t'ere is no

&i((erence bet%een @x C c an& @x c

because t'e probabilit) t'at x is exactl) cis zero.



;x. 5* Approximatin" a Binomial robabilit)

A sure) reports t'at +8 o( nternet users use

Netscape as t'eir bro%ser. =ou ran&oml) select,25 nternet users an& as6 eac' %'et'er 'e ors'e uses Netscape as 'is or 'er bro%ser. G'atis t'e probabilit) t'at exactl) ? %ill sa) )es3

SOIH4ON* Because np 7 ,25 L 0.+8 7 ?0 an& n 7

,25 L 0.52 7 ?51 t'e binomial ariable x is approximatel)

normall) &istribute& %it'

60== np µ

59.552.048.0125 ≈••== npqσ

an&



Ex. 5 ontin!ed

Hsin" t'e correction (or continuit)1 )ou can re%rite

t'e &iscrete probabilit) @x C ? as t'econtinuous probabilit) @ ?2.5 J x J ?.5. 4'e"rap' s'o%s a normal cure %it' µ 7 ?0 an& σ 75.51 an& a s'a&e& area bet%een ?2.5 an& ?.5.



Ex. 5 ontin!ed

4'e z-scores t'at correspon&s to ?2.5 an& ?.5are*

45.059.5

605.62

≈

−

=

−

= σ

µ x

z

63.059.5

605.63≈

−=

−=

σ

µ x z



Ex. 5 ontin!ed

So1 t'e probabilit) t'at at least 50 %ill sa) )es is*

@?2.5 J x J ?.5 7 @0.+5 J z J 0.?

7 @z J 0.? 9 @z J 0.+5 7 0.>5> 9 0.?>?

7 0.0?2,

4'ere is a probabilit) o( about 0.0? t'at exactl)

? o( t'e nternet users %ill sa) t'e) use

Netscape.



"eminders for the next #o!ple of wee$s

Fri&a) an& Mon&a)9 Gor6 in Elass on 5.5 an&eie% E'apter 5.

4ues&a) 9 E'apter 5 4est!Bin&er ra&e

Ge&nes&a) 9 eie% (or Semester ;xam 9 ol&

exams4'urs&a)9 Semester ;xam

Fri&a) 9 ?.,Eon(i&ence nterals (or t'e Mean@Iar"e Samples

4ues&a) 9?.2 Eon(i&ence nterals (or t'e Mean@Small Samples

4'urs&a) 9 ?. Eon(i&ence nterals (oropulation roportions

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5.5 Normal Approximations to Binomial Distributions