Chap 006

Chapter 06 - Confidence Intervals

CHAPTER 6 CONFIDENCE INTERVALS INTERVALS 6.1.

A confidenc confidencee interval interval is an interval interval of values values believed believed to contain contain an unknown unknown population population parameter. parameter. It has a given level of our belief, or confidence, that the interval contains the unknown parameter.

6.. 6..

!nce !nce the the samp sampli ling ng has has take taken n plac place, e, X is reali"ed and, though unknown, it is either in the given interval, or not in the interval. #othing is random at this point and hence we cannot talk about probabilities. probabilities.

6.$.

If the the pre-sam pre-sampli pling ng probabil probabilit% it% that the the parame parameter ter will will be captur captured ed in in an interva intervall is 1 − α , then &1 − α '100( of the intervals that would be constructed this wa% would contain the unknown parameter. )ence the confide confidence nce level level &1 − α '100( that we attach to a given, single interval we obtain.

6.*.

An e+pens e+pensiv ivee wa% wa% of of solvi solving ng the problem problem bu% bu% more more inform informatio ation. n. As n increases, the width of interval decreases. !r, a cheap wa% to solve the problem lower %our reuired confidence level from /( to 0( or lower. As the confidence level decreases, decreases, the interval width decreases decre ases with it.

6./.

/( C.I. x

± 1.6&σ 

n ' 

2,63$.1 ± 1.6&/,/00 16 '

 2,63$.1 ± ,6/  426,32.1, ,$62.15 dollars.

Confidence Interval for Known

Population Normal? σ Population Stdev. 5500 Sample Size 16 n Sampl ample e Mean Mean 8967 89673. 3.12 12 x -ar -ar

99% 95%

6.6.

!on"iden#e $nterval 8967 89 673. 3.12 12 & 35 35'1 '1..77 8967 89 673. 3.12 12 & 26 269' 9'..95

( (

86131.3)9321'.9 86978.2)92368.1

( C.I. 2,63$.1 ± ./36&/,/00 16 '  426,1$1.1, $,1/.15 As e+pected, this interval is wider than the /( C.I. based on the same results.

6-1


6-3.

σ 

*.6

n  100

x  $

/( C.I. for µ  $ ± 1.6&*.6 100 '  $ ± .016  4$1.02, $.05 m.p.g. 6-2. 6-2.

e nee need d not not ass assum umee a norm normal al pop popul ulat atio ion. n. n  100 7 $0, so the Central 8imit 9heorem applies.

6-.

σ 

1.

n  60

x  .$

0( C.I. .$ ± &1.6*/'

1. 60

 4.0*/, .///5 percent alcohol.

&template  :stimating ;ean.+ls, sheet  sample stats' Confidence Interval for Known

Population Normal? Population St Stdev. Sample Size Sample Me Mean

σ 

*$0

n  $0

σ

n x -ar

!on"iden#e $nterval 9.3 9. 3 & 0. 0.39 3990 905 5 (  8.90095)9.69905 9.3 9. 3 & 0. 0.30 3036 36' ' (  8.99636)9.6036' 9.3 9. 3 & 0. 0.25 25'8 '82 2 (  9.0'518)9.55'82

99% 95% 90%

6-10.

1.2 60 9.3

! ! !

x  1*2


Population Normal? Population Stdev. Sample Size Sample Mean

99% 95%

'30 30 1'82

σ

n x -ar

!on"iden#e $nterval 1'82 & 202.221 1'82 & 153.87

6-

( (

1279.78)168'.22! 1328.13)1635.87!


6-3.

σ 

*.6

n  100

x  $

/( C.I. for µ  $ ± 1.6&*.6 100 '  $ ± .016  4$1.02, $.05 m.p.g. 6-2. 6-2.

e nee need d not not ass assum umee a norm normal al pop popul ulat atio ion. n. n  100 7 $0, so the Central 8imit 9heorem applies.

6-.

σ 

1.

n  60

x  .$

0( C.I. .$ ± &1.6*/'

1. 60

 4.0*/, .///5 percent alcohol.


Population Normal? Population St Stdev. Sample Size Sample Me Mean

σ 

*$0

n  $0

σ

n x -ar

!on"iden#e $nterval 9.3 9. 3 & 0. 0.39 3990 905 5 (  8.90095)9.69905 9.3 9. 3 & 0. 0.30 3036 36' ' (  8.99636)9.6036' 9.3 9. 3 & 0. 0.25 25'8 '82 2 (  9.0'518)9.55'82

99% 95% 90%

6-10.

1.2 60 9.3

! ! !

x  1*2



99% 95%

'30 30 1'82

σ

n x -ar

!on"iden#e $nterval 1'82 & 202.221 1'82 & 153.87

6-

( (

1279.78)168'.22! 1328.13)1635.87!


6.11.

σ 

200 n  /0

x  1300



99% 95% 90%

800 50 1700

σ

n x -ar -ar

!on"iden#e $nterval 1700 & 291.'22 1700 & 221.7'' 1700 & 186.09'

( ( (

1'08.58)1991.'2 1'78.26)1921.7' 1513.91)1886.09

9he 0( C.I. 1300 ±126.0* 41/1$.1, 1226.05 b. 9he 0( C.I. contains contains the value value of 1200. 9he airline airline should should fl% the route. 6-1.

σ 

//,000

n  20

x  16/,000

20( C.I. for µ  16/000 ±3220./, 41/310, 132205 &template  :stimating ;ean.+ls, sheet  sample stats'

Confidence Interval for Known

Population Normal? Populati ation Stdev. ev. Sample Size Sample ple Mean

99% 95% 90% 80%

55000 80 16500 5000

σ

n x -ar -ar

!on"iden#e $nterval 165 650 000 & 15 1583 839. 9.3 3 165 650 000 & 12 1205 052. 2.2 2 165 650 000 & 10 101 11'. '.5 5 165000 & 7880.5

6-$

( ( ( (

 1'9161)180839  1529'8)177052  15'885)175115  157120)172880


6-1$.

n  /0

x  1*6.3/

σ 

$/.

/( C.I. 1*6.3/ ± 1.6&$/.

/0

'  41$6., 1/6./15

0( C.I. 1*6.3/ ± 1.6*/&$/.

/0

'  41$2./6, 1/*.*5

( C.I. 1*6.3/ ± ./36&$/.

/0

'  41$$.$, 1/./35



n  100 σ  $0

σ

n x -ar

!on"iden#e $nterval 1'6.75 & 12.8226 1'6.75 & 9.75675 1'6.75 & 8.18813

99% 95% 90%

6-1*.

35.2 50 1'6.75

X  1/0

0( C.I. 1/0 ± 1.6*/&$0 100 '  41*/.06, 1/*.*5 &template  :stimating ;ean.+ls, sheet  sample stats' Confidence Interval for Known


99% 95% 90%

6-1/.

n  $6

30 100 150

σ

n x -ar

!on"iden#e $nterval 150 & 7.7275 (  1'2.272 ) 157.728 ! 150 & 5.87988 (  1''.12 ) 155.88 ! 150 & '.93'56 (  1'5.065 ) 15'.935 !

σ 

2(

X  0(

/( C.I. 0 ± 1.6&2

$6 '  413.*(, .6(5

6-*

( ( (

133.927)159.573! 136.993)156.507! 138.562)15'.938!


&template  :stimating ;ean.+ls, sheet  sample stats' Known


8 36 20

n x -ar

!on"iden#e $nterval 20 & 3.'3''5 20 & 2.61328

99% 95%

6.16.

σ

( (

 

16.5656 ) 23.'3'' ! 17.3867 ) 22.6133 !

Interval width   z   &σ  n ' α

Consider m  *n. 9he width of the interval is now *n ' =  z α   &σ  

 z α   &σ 

n'

 1 of the old width.

9hus the answer is * times the sample si"e. 6.13.

&1.6 σ  1.6& σ  σ



n

n n

'  10

'/

 /1.6  .//1

z α  

 1.6*/

idth of 0( C.I.  &1.6*/&.//1''  2.$$ units. :asier wa% new width  &old width'1.6*/1.6  10&1.6*/'1.6  2.$$ 6-12.

n  /0

x  1*./ s  /.6

/( C.I. 1*./

± 1.6 &/.6

0( C.I. 1*./

± 1.6*/&/.6

/0 '  41./, 16.0/5



/0 '  41$.0, 1/.205



&template  :stimating ;ean.+ls, sheet  sample stats' "n#nown

Population Normal? Sample Size Sample Mean Sample Stdev.

n  16/

n x -ar s

!on"iden#e $nterval 1'.5 & 2.122'1 (  12.3776 ) 16.622' ! 1'.5 & 1.5915 (  12.9085 ) 16.0915 ! 1'.5 & 1.32776 (  13.1722 ) 15.8278 !

99% 95% 90%

6-1.

*e+ 50 1'.5 5.6

x  16,/$0

s  /,/*

/( C.I. 16,/$0

± 1.6&/,/*

( C.I. 16,/$0

±



./36&/,/* 

16/ '  41/,62*.$3, 13,$3/.6$5 16/ '  41/,*12.6, 13,6*1.*5

6-/


6-0.

n  1

x  $*. s  /.

/( C.I. x

±

t .0/&11' & s 

n'

 $*.

±

.01&/. 

1 '  4$0.*/, $3./5

Assume a normal population. 6-1.

n'

x  $0./6/

 $0./6/

±

s  /.33/

./36&/.33/ 

$ '

 43.$, $$.15 thousands of miles. 6-.

n  100

s  .1 X  $.

/( C.I. $. ±0.//1/*, 4.6*/*6, $.3/1/*5 &template  :stimating ;ean.+ls, sheet  sample stats' Confidence Interval for "n#nown


6-*.

± t

α 

n  *0

 & s



x  21.*0

s  1/.**3

α

 0.01, we use

n'

 21.*0 ± .33&1/.**3/'  43./, 2.2215

x  *,/$

t α  

 .33, so

s  11,60

0( C.I. *,/$ ± 1.6*/&11,60 6./.

n x -ar s

!on"iden#e $nterval 3.2 & 0.5515' (  2.6'8'6 ) 3.7515' ! 3.2 & 0.'1669 (  2.78331 ) 3.61669 !

99% 95%

6-$.

*e+ 100 3.2 2.1

*0 '  4$,*2.*6, */,/3./*5 dollars.

=se the :stimating ;ean.+ls template, sheet 4>ample ?ata5. 0( C.I. 2.*$3/ ± $./3003, 4*.$62, 1.06$25 "n#nown

Population Normal? *e+ Sample Size 16 n Sample Mean 8.'9375 x -ar Sample Stdev. 8.1'5958 s

99% 95% 90%

!on"iden#e $nterval 8.'9375 & 6.00098 8.'9375 & '.3'068 8.'9375 & 3.57007

6-6

( ( (

 2.'92773 ) 1'.'9'73 !  '.153073 ) 12.83''3 !  '.92368 ) 12.06382 !


6-6.

n  *00

x  1

s  $2

/( C.I. 1

± 1.6 &$2

2( C.I. 1

±



*00 '

 402.2, 1/.35 calories.

*00 ' 

.$6&$2 

403./2, 16.*5 calories.

6.3.e need to assume normalit% of the population. ( C.I. .6 6-2.

n  /6

±

t .00/&1 ' &0.* 

x  /2

/( C.I. /2

0 '  .6

±

.261&.* 

0 '  4.$**, .2/65 da%s.

s  2/

± 1.6&2/

/6 '  4$/.3*, 20.65 dollars.



6..Assume a normal population. n  00

x  6*3,000

/( C.I. 6*3000

±

s  1*0,000

1/1.*, 463*32.6, 666/1.*5

Confidence Interval for &template  :stimating ;ean.+ls, sheet  sample stats' "n#nown

Population Normal? *e+ Sample Size 200 n Sample Mean 6'7000 x -ar Sample Stdev. 1'0000 s !on"iden#e $nterval 6'7000 & 257'6.2 (  621253.8 ) 6727'6.2 ! 6'7000 & 19521.' (  627'78.6 ) 666521.' !

99% 95%

6-$0.

n  /

x  /.6 s  /

/( C.I. /.6 6-$1.

n  *6

 4/.21, 66.$5 dollars.



/ '



*6 '  41/.26, 13.1*5 dollars.

x  16./ s  .

/( C.I. 16./ 6-$.

± 1.6&/

± 1.6 & .

Assume a normal population. /( C.I. x

±

t .0/&1' & s 

n  0 n ' 

x  /,1$

/,1$

±

s  6*0

.0$&6*0 

 4*,2$.*3, /,*$2./5 dollars.

6-3

0 '




*e+ 20 5139 6'0

!on"iden#e $nterval 5139 & '09.'25 (  '729.58 ) 55'8.'2 ! 5139 & 299.529 (  '839.'7 ) 5'38.53 !

99% 95%

6.$$.

Assume a normal population. n  2

x  6.3

s  .*

( C.I. x ± t .00/ & 3 ' & s  6-$*.

n x -ar s

n  3/

x  2.

/( C.I. 2.

n'

 6.3

±

.331& .* 

2 '  4/.**, 3.65 %ears.

s  0./

± 1.6 &0./

3/ '  42.3, .015 units.



6.$/.Assume a normal distribution. n  1

x  61.63 s  11.

0( C.I. x ± t .0/&11' & s 

n'

 61.63

± 1.36 &11 .



1 '  4//.2/, 63.*5 containers.



n x -ar s

!on"iden#e $nterval 61.67 & 10.0595 (  51.6105 ) 71.7295 ! 61.67 & 7.12885 (  5'.5'11 ) 68.7989 ! 61.67 & 5.81675 (  55.8532 ) 67.'868 !

99% 95% 90%

6-$6.

*e+ 12 61.67 11.22

n  /, s  100 X  600

/( C.I. 600

±

*1.32, 4//2.3, 6*1.325

&template  :stimating ;ean.+ls, sheet  sample stats' Confidence Interval for

6-2


"n#nown


n x -ar s

!on"iden#e $nterval 600 & 55.939 600 & '1.278

99% 95%

6-$3.

*e+ 25 600 100

(  ( 

5''.061 ) 655.939 558.722 ) 6'1.278

0( CI 4.36$6*, 10.$20$65 &template  :stimating ;ean.+ls, sheet  sample stats' "n#nown

Population Normal? *e+ n Sample Size 25 Sample Mean 10.072 x -ar Sample Stdev. 0.901166 s

6-$2.

n  $36

99% 95% 90%

s  1

!on"iden#e $nterval 10.072 & 0.50'1 10.072 & 0.37198

( (

$%&%'( ) %&*%+*6

,

X  *1.1

0( C.I. 4*0.06, *.1*0*5 &template  :stimating ;ean.+ls, sheet  sample stats' Confidence Interval for "n#nown


*e+ 376 '1.12 12

n x -ar s

!on"iden#e $nterval 6-

  

9.567897 ) 10.5761 9.700017 ) 10.''398 -&'6*6.( / $%&*+%*6!

! !


99% 95% 90%

'1.12 & 1.60221 (  39.5178 ) '2.7222 ! '1.12 & 1.21686 (  39.9031 ) '2.3369 ! '1.12 & 1.020'' (  '0.0996 ) '2.1'0' !

6-10


6-$. &template  :stimating ;ean.+ls, sheet  sample stats' 90% C.I. 410.36, 1.165 Confidence Interval for "n#nown


n  10

n x -ar s

!on"iden#e $nterval 11.'6 & 1.09'85 (  10.3652 ) 12.55'8 ! 11.'6 & 0.83152 (  10.6285 ) 12.2915 ! 11.'6 & 0.6973 (  10.7627 ) 12.1573 !

99% 95% 90%

6-*0.

*e+ 376 11.'6 8.2

x  2

/( C.I.

@ ± 1.6 p

@q @ p



n



2     ± 1.6      10  10   10  2

10

 40.1/33, 0.$005 6-*1.

n  62 x  *

( C.I.

@ p

±

./36

@q @n p



*   6  ± ./36      62  62   62  *

62

 40.*6/2, 0.36/5 6.*.

/( CI 40./1//, 0./2*/5 &template  :stimating roportion.+ls'

Confidence Interval for Po01lation Pro0ortion

Sample +ize 800 n Sample proportion 0.55 p-,at

$

99% 95%

6.*$.

!on"iden#e $nterval 0.55 & 0.0'53 0.55 & 0.03'5

(  ( 

0( CI 40.$*$0, 0.*/305 &template  :stimating roportion.+ls'

6-11

0.50'7/ 0.5953 0.5155/ 0.58'5

! !



Sample +ize 200 n Sample proportion 0.' p-,at

$

99% 95% 90%

6-**.

@ p

n  20

0( C.I.

!on"iden#e $nterval 0.' & 0.0892 0.' & 0.0679 0.' & 0.0570

(  (  ( 

0.3108/ 0.'892 0.3321/ 0.'679 0.3'30/ 0.'570

! ! !

 13

1   6  ± 1.6*/      3  3   3  1

20  40.102/, 0.133$5

6.*/./( C.I. 40.0032, 0.015 &9emplate :stimating roportion.+ls'



$

99% 95%

!on"iden#e $nterval 0.01&0.0029 0.01&0.0022

(  ( 

0.0071/ 0.0129 0.0078/ 0.0122

6.*6.0( C.I. 40.31$, 0.0235 &9emplate :stimating roportion.+ls' Confidence Interval for Po01lation Pro0ortion


$

99% 95% 90%

!on"iden#e $nterval 0.85 & 0.0920 0.85 & 0.0700 0.85 & 0.0587

(  (  ( 

6-1

0.7580/ 0.9'20 ! 0.7800/ 0.9200 ! 0.7913/ 0.9087 !

! !


6.*3. n  / x  2 2( C.I.

6-*2.

n  /0

2 /

±

n  $*3

±

./36

&.*2* '&./16 '  /0

 40.*06, 0./6/*5

x  01

0( C.I.

6./0.

/  40.0$3/, 0.305

x  11

( C.I. 0.*2* 6-*.

 2   **       /   / 

.$6

01 $*3

± 1.6*/

 01   1*6       $*3   $*3 

$*3  40./$/3, 0.625

&9emplate  :stimating roportion.+ls' p  /$/0  0.1 0( C.I. 40.16/, 0./*/5



$

99% 95% 90%

6./1.

!on"iden#e $nterval 0.212&0.0666 0.212&0.0507 0.212&0.0'25

(  (  ( 

0.1'5'/ 0.2786 ! 0.1613/ 0.2627 ! 0.1695/ 0.25'5 !

&9emplate :stimating roportion.+ls' Confidence Interval for Po01lation Pro0ortion p  1$0/30  0.21 /( C.I. 40.1$3, 0.6/5 Confidence Interval for Po01lation Pro0ortion


$

99% 95%

!on"iden#e $nterval 0.2281&0.0'53 (  0.2281&0.03'' ( 

6-1$

0.1828/ 0.273' ! 0.1937/ 0.2625 !


6-/.

n   s   2 /( C.I. for

 &n − 1' s     X

,

& n − 1' s  X



σ





   41&2'$/.*2, 1&2'10.25  4*.3*, 16.$*5 

&9emplate :stimating Bariance.+ls, sheet sample stats'

Confidence Interval for Po01lation Variance A22130tion4

,e population i+ normall di+triuted $

Sample Size 22 n Sample /arian#e 8 s2

6-/$.

95% 95%

!on"iden#e $nterval '.73521/ 16.3378! '.73521/ 16.3378!

 

n  *1 s  10

( C.I. for

σ



 4*0&10'66.366, *0&10'0.3035  461.11, 13.0$5

&9emplate :stimating Bariance.+ls, sheet sample stats'

Confidence Interval for Po01lation Variance A22130tion4

,e population i+ normall di+triuted Sample Size '1 n Sample /arian#e 102 s2

6-/*.

$

99%



!on"iden#e $nterval 61.1089/ 197.039 !

n  60 s   1,2

ell use df  /, appro+imatel% 60, and look in the table. /( C.I. for 6-//.

σ



 4/&1,2'2$.$, /&1,2'*0.*2135  426.2, 1,32.35

σ



&using appro+imate df  $0'

4$$.$/&$1'/$.3, $$.$/&$1'1$.25  41./, 3*.5 6-/6.

6-1*


9emplate :stimating Bariance.+ls Confidence Interval for Po01lation Variance A22130tion4

,e population i+ normall di+triuted n $ Sample Size 16 Sample /arian#e 66.35663 s2 95%  95%  90% 

!on"iden#e $nterval 36.2098/ 158.9'8 ! 36.2098/ 158.9'8 ! 39.8207/ 137.083 !

9he /( C.I. 4$6.02, 1/2.*25 6-/3.

=

µ ± 1.6

/( C.I. for

6-/2.

6-60.

&$'

 $*$.6$, */*.$3

 $&$2'  $&$2'   ,     41,62.0$, 1,636.625 $*$.6$   */*.$3

σ



 &1'&.*'  &1'&.*'   ,     40.0/, 0.$*1$5 $ . 2/$ 2.06//  

B  0.0/

0( confidence. =se a guess of p  0./ n  &1.6*/'&./'&./'&.0/'  30.6 >ample at least 31 items. &9emplate >ample >i"e.+ls, sheet opulation mean' !on"iden#e evel e+ired al"-idt, e+ired Population Proportion

90% 0.05 0.5

Minimum Sample Size

271

B  /

/( confidence

n  &1.6' 100   6.0*

6-61.

± 1.6

6-/.

σ



&$'  $

B  ,000

σ



B

 100

>ample at least 3 runs.

/( confidence

σ



 *0,000,000

n  &1.6' *0,000,000&,000'   $2.*16

>ample at least $ e+ecutives.

6-1/


&9emplate >ample >i"e.+ls, sheet opulation mean' !on"iden#e evel e+ired 95% al"-idt, e+ired 2000 B Population Stdev. 632'.555 σ Minimum Sample Size

6-6.

B  0./

39 σ 

/( confidence 







n  &1.6' &' &./'  61.*66

>ample at least 6 returns. 6-6$.

An estimate of p is 0.1* 0( confidence. B  .0/ n  &1.6*/'&.1*'&.26'&.0/'  1$0.$ >ample at least 1$1 people.

6-6*.

B  10

σ 

0( confidence

/0

n  &1.6*/'&/0'&10'  63.6/

9herefore, sample at least 62 da%s. &9emplate >ample >i"e.+ls, sheet opulation mean' !on"iden#e evel e+ired 90% B al"- idt, e+ired 10 σ Population Stdev. 50 Minimum Sample Size

6-6/.

68

B  0.0

/( confidence . Euessed p  0.10  n  &1.6' &.10'&.0'&.0'  26*.$6 >ample at least 26/ accounts. &9emplate >ample >i"e.+ls, sheet opulation mean' 

6-66.

!on"iden#e evel e+ired al"-idt, e+ired Population Proportion

95% 0.02 0.1

Minimum Sample Size

865

n  10 s  6000

B

X  $,000

/( C.I. $,000 ± .6 &6000 10 '  43,302.16, $6,1.2*5 6-63.

n  * s  *.312

X  $./

/( C.I. $./ ± .06 &*.312

* '  41./03, /.*$5

6-16


6-62.

( CI 4F2.6;, F.**;5 &9emplate :stimating ;ean.+ls, sheet  sample stats' "n#nown

Population Normal? *e+ n Sample Size 200 x -ar Sample Mean 9.2 Sample Stdev. 1.1'02 s !on"iden#e $nterval --5

6-6.

-&( ) %&(%-6+ ,  +&--%*( / -&.%-6+ !

  .31 /( C.I. .31 ± 1.6 &

6-30.

&.31'&.'  100

'  4.611, .325

&9emplate :stimating ;eans.+ls, sheet sample stats' Confidence Interval for /( C.I. 4/0.03162, 60./2$5 "n#nown


99% 95%

6-31.

*e+ 68 55.3 21.6

n x -ar s

!on"iden#e $nterval 55.3 & 6.9''55 (  '8.355'5 ) 62.2''55 ! 55.3 & 5.22832 (  50.07168 ) 60.52832 !

&9emplate :stimating ;eans.+ls, sheet sample stats'

Confidence Interval for 0( C.I. 4$/.21*13, /.12/2$5 "n#nown


99% 95% 90%

*e+ 15 '' 18

n x -ar s

!on"iden#e $nterval '' & 13.8351 (  30.16'86 ) 57.8351' ! '' & 9.96808 (  3'.03192 ) 53.96808 ! '' & 8.18583 (  35.81'17 ) 52.18583 !

6-13


6-3.

  *3  *$0  .10 /( C.I. .10 ± 1.6 &

'  4.03/, .1$25

&.10 '&.21'  *$0

&9emplate :stimating roportion.+ls' Confidence Interval for Po01lation Pro0ortion

Sample +ize '30 n Sample proportion 0.109 p-,at $

!on"iden#e $nterval 0.109 & 0.0387 (  0.109 & 0.0295 ( 

99% 95%

6-3$.

0.0703 / 0.0795 /

0.1'77 ! 0.1385 !

n  10 s  0. X  1.

0( C.I. 1. ± 1.2$$ &. 10 '  41.02*, 1.$165 6-3*.

n  /00

s  200

X  $600

/( C.I. $600 ± 1.6 &200 6-3/.

'  4$/.22, $630.15

  $62  /00  .3$6 /( C.I. .3$6 ± 1.6 &

6-36.

/00

&.3$6 '&.6* '  /00

'  4.63*, .33*65

&9emplate :stimating ;eans.+ls, sheet sample stats' Confidence Interval for /( C.I. 40.**23, 1.13$5 "n#nown


99% 95%

*e+ 9 0.833 0.5

n x -ar s

!on"iden#e $nterval 0.833 & 0.55923 (  0.27377 ) 1.39223 ! 0.833 & 0.38'33 (  0.''8666 ) 1.21733' !

6-33.

If 0.00 was in the confidence interval for 6-36, then defense attorne%s could argue that the banker had a /( probabilit% of making no mone% from his insider trading. 9herefore, he did benefit from insider information.

6-32.

  161  00  .20/ /( C.I. .20/ ± 1.6 &

&.20/ '&.1/ '  00

'  4.3/0, .2605

6-12


6-3.

&9emplate :stimating ;ean.+ls, >heet >ample ?ata' Confidence Interval for /( C.I.43.01, 2.$*15 "n#nown

Population Normal? *e+ n Sample Size 10 Sample Mean 7.681 x -ar Sample Stdev. 0.92261 s

99% 95%

6-20.

!on"iden#e $nterval 7.681 & 0.9'816 7.681 & 0.66

( (

 6.7328'2 ) 8.629158 !  7.02100' ) 8.3'0996 !

/( CI 40.*06, 0.31$25 &9emplate :stimating roportion.+ls' Confidence Interval for Po01lation Pro0ortion

Sample +ize '0 n Sample proportion 0.56 p-,at $

!on"iden#e $nterval 0.56 & 0.2022 ( 

99% -5

6-21.

%&6 )

,

0.7622 !

%&.%6( /

%&'$*+ !

n   s  ./ X  .60

0( C.I. .60 ± 1.31 &./ 6-2.

%&$*+

0.3578 /

 '  4./02, .65

  .20 /( C.I. .20 ± 1.6 & &.20'&.0'  00 '  4.3*/, .2//5

6-2$.

  .1 /( C.I. .1 ± 1.6 &

&. 1'&.0 '  //0

'  4.0, .125

6-1


&9emplate :stimating roportion.+ls' Confidence Interval for Po01lation Pro0ortion

Sample +ize 5250 n Sample proportion 0.91 p-,at $

!on"iden#e $nterval 0.91 & 0.0102 (  0.91 & 0.0077 ( 

99% 95%

6-2*.

0.8998 / 0.9023 /

0.9202 ! 0.9177 !

&9emplate :stimating ;eans.+ls, >heet >ample ?ata'

Confidence Interval for 0( C.I.416.$, $6.115 "n#nown

Population Normal? *e+ Sample Size 8 n Sample Mean 269.25 x -ar Sample Stdev. 1'9.0788 s

99% 95% 90%

6-2/.

!on"iden#e $nterval 269.25 & 18'.''8 269.25 & 12'.633 269.25 & 99.8581

( ( (

 8'.80171 ) '53.6983 !  1''.6171 ) 393.8829 !  169.3919 ) 369.1081 !

&9emplate :stimating roportions.+ls' /( C.I. 4 0./6/, 0.6$0*5 Confidence Interval for Po01lation Pro0ortion


$

99% 95%

6-26.

!on"iden#e $nterval 0.6&0.0399 0.6&0.030'

(  ( 

0.5601/ 0.6399 ! 0.5696/ 0.630' !

&9emplate >ample si"e.+ls, sheet opulation ;ean'

6-0


a' sample si"e  13 Sa30le Si7e Deter3ination for Po01lation 8ean

!on"iden#e evel e+ired 95% al"-idt, e+ired 0.002 Population Stdev. 0.015 Minimum Sample Size 217

b' stdev n +t217 0.002 0.01 97 0.011 117 0.012 139 0.013 163 0.014 189 0.015 217 0.016 2'6 0.017 278 0.018 312 0.019 3'7 0.02 385 c' cost of sampling 1327 Cost 0.01 607 0.011 727 0.012 859 0.013 1003 0.014 1159 Popn. Std. 0.015 1327 evn. 0.016 1501 0.017 1693 0.018 1897 0.019 2107 0.02 2335 6-23.

n  0 s  $

X  2

3/( C.I. 2 ± 1.123 &$ 6-22.

0

'  43.36, 2.*5

&9emplate >ample >i"e.+ls, sheet opulation roportion'

6-1


a' sample si"e 10$ Sa30le Si7e Deter3ination for Po01lation Pro0ortion

!on"iden#e evel e+ired 90% al"-idt, e+ired 0.01 B Population Proportion 0.0' Minimum Sample Size 1039

b' op rop range >ample 1039 0.01 0.03 788 0.033 86' 0.036 939 0.039 1015 0.042 1089 Popn. 0.045 1163 Proportion 0.048 1237 0.051 1310 0.054 1383 0.057 1'55 0.06 1526 c' op rop range Cost 6259 0.01 0.03 '753 0.033 5209 0.036 5659 0.039 6115 0.042 6559 Popn. 0.045 7003 Proportion 0.048 7''7 0.051 7885 0.054 8323 0.057 8755 0.06 9181

6-


6-2.

&9emplate :stimating ;ean.+ls, sheet >ample stats' a' /( CI 40.2/16, 0.202*5 b' ( CI 40.2$62*, 1.000$165 "n#nown


*e+ 20 0.992 0.013

n x -ar s

!on"iden#e $nterval 99% 95%

6-0.

%&--( ) %&%%+*( %&--( ) %&%%6%+

, ,

 %&-+*6+. / $&%%%*$6 !  %&-+-$6 / %&--+%+. !

/( CI 40.062, 0.1$$5 &9emplate :stimating roportion.+ls'


Sample +ize 100 n Sample proportion 0.08 p-,at $

99% -5

!on"iden#e $nterval 0.08 & 0.0699 (  %&%+ )

%&%*(

,

0.0101 /

0.1'99 !

%&%(6+

%&$**( !

6-1

(σ n ) = $.3* then (σ n ) = $.3* " n ) = 1.6( .3$/6) = *.*/6

>ince " 0 " /

(σ

0

= $.3*1.6*/ = .3$/6

9he /( C.I. 13./2 ± *.*/6

6-$

/


6- &9emplate found in >ample >i"e.+ls, sheet opulation roportion' Sa30le Si7e Deter3ination for Po01lation 8ean a. minimum sample si"es Ta91lation of

al" idt, e+ired

Popn. Std. evn.

'3

2

3

4

5

6

14 15 16 17 18 19 20 21 22 23 24

189 217 2'6 278 312 3'7 385 '2' '65 509 55'

8' 97 110 12' 139 155 171 189 207 226 2'6

'8 55 62 70 78 87 97 106 117 128 139

31 35 '0 '5 50 56 62 68 75 82 89

21 25 28 31 35 39 '3 '8 52 57 62

7 16 18 21 23 26 29 32 35 38 '2 '6

b. >ampling costs Co2t Anal:2i2

Samplin4 !o+t 

522

Ta91lation of


Popn. Std. evn.

522

2

3

4

5

6

14 15 16 17 18 19 20 21 22 23 24

1106 1218 133' 1'62 1598 1738 1890 20'6 2210 2386 2566

686 738 790 8'6 906 970 103' 1106 1178 125' 133'

5'2 570 598 630 662 698 738 77' 818 862 906

'7' '90 510 530 550 57' 598 622 650 678 706

'3' '50 '62 '7' '90 506 522 5'2 558 578 598

6-*

7 '1' '22 '3' ''2 '5' '66 '78 '90 502 518 53'


c. 9otal costs Co2t Anal:2i2

Samplin4 !o+t  522 rror !o+t 1'')000 otal !o+t 1'')522

Ta91lation of

al" idt, e+ired 1''522

Popn. Std. evn.

14 15 16 17 18 19 20 21 22 23 24

2

3

4

5

6

11'6 1258 137' 1502 1638 1778 1930 2086 2250 2'26 2606

776 828 880 936 996 1060 112' 1196 1268 13'' 1'2'

702 730 758 790 822 858 898 93' 978 1022 1066

72' 7'0 760 780 800 82' 8'8 872 900 928 956

79' 810 822 83' 850 866 882 902 918 938 958

9he value of G that minimi"es the total costs when H1* is * 9he value of G that minimi"es the total costs when H* is / 6-$ &9emplate >ample >i"e.+ls, sheet opulation roportion' Sa30le Si7e Deter3ination for Po01lation Pro0ortion

minimum +ample +ize ( 162' !on"iden#e evel e+ired al"-idt, e+ired Population Proportion

90% 0.02 0.6

Minimum Sample Size

162'

6-/

7 90' 912 92' 932 9'' 956 968 980 992 1008 102'


6.*.

&9emplate :stimating roportion.+ls' /( C.I. 40.*26, 0./$$25 Confidence Interval for Po01lation Pro0ortion

Sample +ize 1700 n Sample proportion 0.51 p-,at !on"iden#e $nterval 0.51&0.0312 0.51&0.0238

$

99% 95%

6./.

(  ( 

0.'788/ 0.5'12 ! 0.'862/ 0.5338 !

/( C.I. 4/.1*3, 6.2/$5 Confidence Interval for "n#nown


99% 95%

*e+ 50 6 3

n x -ar s

!on"iden#e $nterval 6 & 1.13701 (  '.86299 ) 7.13701 ! 6 & 0.85259 (  5.1'7'1 ) 6.85259 !

Case 3 residential olling 1' sample si"e  $$ Sa30le Si7e Deter3ination for Po01lation Pro0ortion

!on"iden#e evel e+ired 95% al"-idt, e+ired 0.02 B Population Proportion 0.53 Minimum Sample Size 2393

6-6


' Ta91lation of


Popn. Proportion

2393

0.01

0.014

0.018

0.022

0.026

0.4 0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6

9220 9358 9'66 95'3 9589 960' 9589 95'3 9'66 9358 9220

'70' '775 '830 '869 '892 '900 '892 '869 '830 '775 '70'

28'6 2889 2922 29'6 2960 2965 2960 29'6 2922 2889 28'6

1905 193' 1956 1972 1982 1985 1982 1972 1956 193' 1905

136' 1385 1'01 1'12 1'19 1'21 1'19 1'12 1'01 1385 136'

0.03 1025 10'0 1052 1061 1066 1068 1066 1061 1052 10'0 1025

$' As the half-width increases, the sample si"e decreases. As the population proportion approaches /0( &from either direction', the sample si"e increases. *' /0( /' cost  F*/  F1.0&n' Ta91lation of

al" idt, e+ired 3296.6

Popn. Proportion

0.4 0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6

0.01

0.014

0.018

0.022

0.026

11'89 1165'.6 1178'.2 11876.6 11931.8 119'9.8 11931.8 11876.6 1178'.2 1165'.6 11'89

6069.8 6155 6221 6267.8 6295.' 6305 6295.' 6267.8 6221 6155 6069.8

38'0.2 3891.8 3931.' 3960.2 3977 3983 3977 3960.2 3931.' 3891.8 38'0.2

2711 27'5.8 2772.2 2791.' 2803.' 2807 2803.' 2791.' 2772.2 27'5.8 2711

2061.8 2087 2106.2 2119.' 2127.8 2130.2 2127.8 2119.' 2106.2 2087 2061.8

6-3

0.03 1655 1673 1687.' 1698.2 170'.2 1706.6 170'.2 1698.2 1687.' 1673 1655


6' Cost for reducing the error to 1( will increase sampling costs b% a factor of appro+imatel% $.60 from the current costs associated to a ( error. Ta91lation of

al" idt, e+ired 3296.6

Popn. Proportion

0.4 0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6

0.01

0.012

0.014

0.016

0.018

11'89 1165'.6 1178'.2 11876.6 11931.8 119'9.8 11931.8 11876.6 1178'.2 1165'.6 11'89

8108.6 8223.8 8313.8 8377.' 8'15.8 8'29 8'15.8 8377.' 8313.8 8223.8 8108.6

6069.8 6155 6221 6267.8 6295.' 6305 6295.' 6267.8 6221 6155 6069.8

'7'7.' '812.2 '862.6 '898.6 '920.2 '927.' '920.2 '898.6 '862.6 '812.2 '7'7.'

38'0.2 3891.8 3931.' 3960.2 3977 3983 3977 3960.2 3931.' 3891.8 38'0.2

0.02 3191 3233 3265.' 3288.2 3302.6 3306.2 3302.6 3288.2 3265.' 3233 3191

"a#tor in#rea+e 3.600'39 3.60'887 3.608807 3.611885 3.61285 3.61'361 3.61285 3.611885 3.608807 3.60'887 3.600'39

Case 2 rivac% roblem &=sing the template J>ample >i"e.+lsK, sheet Jopulation roportionK' 4#ote this case is dealing with the number of attempts to e+tract private information and not the number of calls being placed. >o the sample si"e reflects the number of attempts being made.5 1. minimum sample si"e &no. of attempts' is 66, which is not evenl% divisible b% $


95% 0.02 0.07

Minimum Sample Size

626

9otal Cost  $200.3/&66' F2/0

B

Co2t Anal:2i2

Samplin4 !o+t  850 rror !o+t otal !o+t  850

. minimum sample si"e &no. of attempts' is /01, which is also not evenl% divisible b% $


95% 0.01 0.07

Minimum Sample Size

2501

9otal Cost $200.*/&/01' F/6

6-2

Co2t Anal:2i2

Samplin4 !o+t  2256 rror !o+t otal !o+t  2256


6-


$. 9abulation of 9otal Cost Ta91lation of

al" idt, e+ired 2255.75

0.01

0.05 17'8.75 0.055 1877.75 0.06 2005.25 0.065 2131.25 0.07 2255.75 0.075 2379.5 0.08 2501 0.085 2621 0.09 27'0.25 0.095 2857.25 0.1 2973.5

Popn. Proportion

0.014

0.018

1078.25 803 11''.25 8'2.75 1209.5 881.75 127' 920.75 1337 959 1'00 997.25 1'62.25 103'.75 1523.75 1072.25 158'.5 1109 16''.5 11'5 1703 1181

0.022

0.026

663.5 689.75 716 7'2.25 767.75 793.25 818.75 8'3.5 868.25 892.25 916.25

582.5 602 620.75 639.5 657.5 676.25 69'.25 711.5 729.5 7'6.75 76'

0.03 532.25 5'6.5 560.75 575 588.5 602.75 616.25 629 6'2.5 655.25 668.75

Eraph 3000

2500

2000

Total Co2t

1500

1000

0.05 0.06

500 0.07 0.08 0

0.09 0.01

0.014

0.018

Po0n& Pro0ortion

0.1 0.022

0.026

Half ;idt<

0.03

4#ote in parts *' and /', the cost per attempt drops to F0.*/. 9herefore, the 9otal Cost will drop.5 *. Although the total number of attempts is not evenl% divisible b% /, the 9otal Cost drops to F66 Co2t Anal:2i2

Samplin4 !o+t  rror !o+t otal !o+t 

662 662

6-$0

Chap 006

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