usm 2016-vesperyino-marketing-inferencias de una muestra-inglés (1).pptx

7/26/2019 usm 2016-vesperyino-marketing-inferencias de una muestra-ingls (1).pptx

1/74

Copyright 2014, 2011, and 2008 Pearson Education,

Inc.6-1


2/74

Statistics for Business and

Economics


Inc.6-2

Chapter 6Inferences Based on a Single Sample


3/74

Content


Inc.6-3

1. Identifying and Estimating the TargetParameter

2. Confidence Intera! for a Popu!ation

"ean#

$orma! %z& 'tatistic

(. Confidence Intera! for a Popu!ation "ean#

'tudent)s t*'tatistic

4. +arge*'amp!e Confidence Intera! for aPopu!ation Proportion

. -etermining the 'amp!e 'ie

/. inite Popu!ation Correction for 'imp!eandom 'am !in


4/74

Learning Obecti!es


Inc.6-"

1. Estimate a popu!ation parameter %means,

proportion, or ariance& ased on a !arge

samp!e se!ected from the popu!ation

2. 5se the samp!ing distriution of a statistic to

form a confidence intera! for the popu!ation

parameter

(. 'ho6 ho6 to se!ect the proper samp!e siefor estimating a popu!ation parameter


5/74

#hin$ing Challenge

'uppose you)re

interested in the

aerage amount of

money thatstudents in this

c!ass %the

popu!ation& hae onthem. 7o6 6ou!d

you find out


Inc.6-%


6/74

Statistical &ethods

Statistical

Methods


Inc.6-6

EstimationHypothesis

Testing

Inferential

Statistics

Descriptive

Statistics


7/74

6'1


Inc.6-(

Identif)ing and Estimatingthe #arget *arameter


8/74

Estimation &ethods

Estimation


Inc.6-+

Interval

Estimation

Point

Estimation


9/74

#arget *arameter


Inc.6-,

The un9no6n popu!ation parameter %e.g., mean or

proportion& that 6e are interested in estimating is

ca!!ed the target parameter.


10/74

#arget *arameter

etermining the #arget *arameter

Parameter :ey ;ords of Phrase


Inc.6-1.

Type of -ata

"ean< aerage =uantitatie

p Proportion< percentage

fraction< rate =ua!itatie


11/74

*oint Estimator


Inc.6-11

> point estimator of a popu!ation parameter is aru!e or formu!a that te!!s us ho6 to use the samp!e

data to ca!cu!ate a single numer that can e usedas an estimate of the target parameter.


12/74

*oint Estimation

1. Proides a sing!e a!ue

? @ased on oserations from one samp!e

2. Aies no information aout ho6 c!osethe a!ue is to the un9no6n popu!ation

parameter

(. EBamp!e# 'amp!e meanx ( isthe point estimate of the un9no6npopu!ation mean


Inc.6-12


13/74

Inter!al Estimator


Inc.6-13

>n inter!al estimator %or confidence inter!al& isa formu!a that te!!s us ho6 to use the samp!e data

to ca!cu!ate an interval that estimates the targetparameter.


14/74

Inter!al Estimation


Inc.6-1"

1. Proides a range of a!ues

? @ased on oserations from one samp!e

2. Aies information aout c!oseness to un9no6n

popu!ation parameter

? 'tated in terms of proai!ity

:no6ing eBact c!oseness reDuires 9no6ing

un9no6n popu!ation parameter

(. EBamp!e# 5n9no6n popu!ation mean !ies et6een

0 and 0 6ith F confidence


15/74

6'2


Inc.6-1%

Confidence Inter!al fora *opulation &ean/

0ormal z Statistic


16/74

Estimation *rocess

Mean, ,is

unknown

Population

Sample

andom Sample

x ! "#

I am $"%

confident that

is &etween'# ( )#*

Mean


Inc.6-16


17/74

e) Elements ofInter!al Estimation

Sample statistic

+point estimate-onfidence interval

-onfidence limit

+lower

-onfidence limit

+upper

. confidence interval provides a range of

plausi&le values for the population

parameter* Copyright 2014, 2011, and 2008 Pearson Education,Inc.

6-1(


18/74

Confidence Inter!al

>ccording to the Centra! +imit Theorem, thesamp!ing distriution of the samp!e mean is

approBimate!y norma! for !arge samp!es. +et us

ca!cu!ate the intera! estimator#

x 1.96 x= x

1.96 n

That is, 6e form an intera! from 1./ standard

deiations e!o6 the samp!e mean to 1./ standarddeiations aoe the mean. Prior to dra6ing the

samp!e, 6hat are the chances that this intera! 6i!!

enc!ose , the popu!ation mean


Inc.6-1+


19/74

Confidence Inter!al

If samp!e measurements yie!d a a!ue of x that fa!!s

et6een the t6o !ines on either side of , then the

intera! x 1.96 x 6i!! contain .

The area under the

norma! cure et6een

these t6o oundaries

is eBact!y .. Thus,the proai!ity that a

random!y se!ected

intera! 6i!! contain

is eDua! to .. Copyright 2014, 2011, and 2008 Pearson Education,Inc.

6-1,


20/74

Confidence Coefficient


Inc.6-2.

The confidence coefficient is the proai!ity thata random!y se!ected confidence intera! enc!oses

the popu!ation parameter * that is, the re!atie

freDuency 6ith 6hich simi!ar!y constructedintera!s enc!ose the popu!ation parameter 6hen

the estimator is used repeated!y a ery !arge

numer of times. The confidence le!el is the

confidence coefficient eBpressed as a percentage.


21/74

,%4 Confidence Le!el

If our confidence level is 95%, then in the long run,95% of our confidence intervals will contain and 5%

will not.

or a confidence coefficient of F, the area in the

t6o tai!s is .0. To choose a different confidencecoefficient 6e increase or decrease the area %ca!! it

& assigned to the tai!s. If 6e p!ace G2 in each

tai!

and z G2 is the z*a!ue,the confidence intera!

6ith coefficient %1 H &

is


Inc.6-21


22/74

Conditions 5euired for a 7alidLarge-Sample

Confidence Inter!al for

1. > random samp!e is se!ected from the target

popu!ation.

2. The samp!e sie n is !arge %i.e., n (0&. -ue tothe Centra! +imit Theorem, this condition

guarantees that the samp!ing distriution of x isapproBimate!y norma!. >!so, for !arge n, s 6i!! e

a good estimator of .


Inc.6-22


23/74

Large-Sample 1 8 4

ConfidenceInter!al for

6here z G2 is the z*a!ue 6ith an area G2 to its

right and in the standard norma! distriution. Theparameter is the standard deiation of the

samp!ed popu!ation, and n is the samp!e sie.

$ote# ;hen is un9no6n and n is !arge %n

(0&, the confidence intera! is approBimate!y

eDua! to

6here s is the samp!e standard deiation.

x (z 2 ) x = x z 2

n

x

z 2


Inc.6-23

s n


24/74

#hin$ing Challenge

Jou)re a =GC inspector for

Aa!!o. The for 2*!iter

ott!es is '.% !iters. >

random samp!e of 1.. ott!essho6edx 1',, !iters. ;hat

is the ,.4 confidence

intera! estimate of the true

mean amount in 2*!iterott!es

// lliitteerr 1984-1994 T/Maker Co.


Inc.6-2"


25/74

Confidence Inter!alSolution9

x z

/ 2

n x

+z

/2

n

1.99 1.645

.05

100 1.99+ 1.645

.05

100

1.982 1.998


Inc.6-2%


26/74

6'3


Inc.6-26

Confidence Inter!al fora *opulation &ean/

Student:s t-Statistic


27/74

Small Sample

;n$no


28/74

Student:s t-Statistic

The t*statistic has a samp!ing distriution erymuch !i9e that of the z*statistic# mound*shaped,

symmetric, 6ith mean 0.

The primary

difference et6een

the samp!ing

distriutions of t and

z is that the t*statisticis more aria!e than

the z*statistic.


Inc.6-2+


29/74

egrees of =reedom


Inc.6-2,

The actua! amount of ariai!ity in the samp!ing

distriution of t depends on the samp!e sie n. >

conenient 6ay of eBpressing this dependence is

to say that the t*statistic has %nH 1& degrees offreedom df.


30/74

z

t

Student:s t istribution

0

t %df &

'tandard

$orma!

t %df 1(&

@e!!*'haped

'ymmetric

Katter)

Tai!s


Inc.6-3.


31/74

t - #able


Inc.6-31


32/74

t-!alue

If 6e 6ant the t*a!ue 6ith an area of .02 to itsright and 4 df, 6e !oo9 in the ta!e under the

co!umn t.02 for the entry in the ro6 corresponding

to 4 df. This entry is t.02 2./. The

corresponding standard norma! z*score is z.02

1./.


Inc.6-32


33/74

Small-Sample


6here taG2 is ased on %nH 1& degrees of freedom.

x

t 2


Inc.6-33

s n


34/74

Conditions 5euired for a7alid Small-Sample



Inc.6-3"

1. > random samp!e is se!ected from thetarget popu!ation.

2. The popu!ation has a re!atie freDuency

distriution that is approBimate!y norma!.


35/74

Estimation E>ample

&ean

;n$no


36/74

#hin$ing Challenge

Jou)re a time study ana!yst

in manufacturing. Jou)e

recorded the fo!!o6ing tas9

times %min.'6? "'2? "'.? 3'%? 3'+? 3'1.;hat is the ,.4 confidenceintera! estimate of the

popu!ation mean tas9 time


Inc.6-36


37/74

Confidence Inter!al Solution9

? x (.

? s (.88

? n /, df nH 1 / H 1

? t.0 2.01

3.7 2.015 .38987 3.7 +

2.015 .38987

6Copyright 2014, 2011, and 2008 Pearson Education,Inc. 6-3(


38/74

6'"


Inc.6-3+

Large-Sample ConfidenceInter!al for a *opulation

*roportion


39/74

Sampling istribution of pL


Inc.6-3,

1. The mean of the samp!ing distriution of p*

isp samp!e sie is

considered !arge if oth np* 15 andnq* 15.

is pq n < that is,

p*

= pq n


40/74

Large-Sample ConfidenceInter!al for

6here p*

= x

n

p* z 2 p* =

p*

z 2

pq

n p*

z 2

p*

q*

n

and q* = 1

p*.


Inc.6-".

$ote# ;hen n is !arge,p* can approBimate

the a!ue ofp in the formu!a for p* .

pL


41/74

1. > random samp!e is se!ected from the targetpopu!ation.

2. The samp!e sie n is !arge. %This condition 6i!! e

satisfied if oth np* 15 and nq* 15 . $otethat np* and nq* are simp!y the numer

of successes and numer of fai!ures,

respectie!y, in the samp!e.&.Copyright 2014, 2011, and 2008 Pearson Education,

Inc.6-"1

Conditions 5euired for a

7alid Large-SampleConfidence Inter!al forp

E ti ti E l


42/74

Estimation E>ample*roportion

> random samp!e of 400 graduates

sho6ed (2 6ent to graduate schoo!. 'et

up a F confidence intera! estimate for

p.

.

08(.92)


Inc.6-"2

.

08(.92)

/ 2

/ 2 p*q*

p p* + Z

p*

q*p* = 32 = 0.08

p*

Z400

.08 1.96

p .08+ 1.96 40

0

40

0

.053 p

.107

n n


43/74

#hin$ing Challenge

Jou)re a production

manager for a ne6spaper.

Jou 6ant to find the F

defectie. Mf 2..ne6spapers, 3% haddefects. ;hat is the ,.4confidence intera! estimate

of the popu!ationproportion defectie


Inc.6-"3

C fid I t l


44/74


/2


Inc.6-""

/2 p* q*

p p* + z

p*

q*

p*

z

.175+.825,

p

.

175+.825,.175

1.645 .175 + 1.645 200 200

.1308 p .2192

n n


45/74

@dusted 1 8

1..4Confidence Inter!al for a*opulation *roportion?p

6herep =

n

+ 4

is the adNusted samp!e proportion

of oserations 6ith the characteristic of interest,xis the numer of successes in the samp!e, and n is

the samp!e sie.

p (1 p)

p z 2n + 4x +

2


Inc.6-"%


46/74

6'%


Inc.6-"6

etermining the Sample SiAe


47/74

Sampling Error


Inc.6-"(

In genera!, 6e eBpress the re!iai!ity associated

6ith a confidence intera! for the popu!ation mean y specifying the sampling error 6ithin 6hich6e 6ant to estimate 6ith 100%1 H &Fconfidence. The samp!ing error %denoted 'E&,then, is eDua! to the ha!f*6idth of the confidenceintera!.


48/74

Sample SiAe eterminationfor 1..1 8

4Confidence Inter!al forIn order to estimate 6ith a samp!ing error %'E&

and 6ith 100%1 H &F confidence, thereDuiredsamp!e sie is found as fo!!o6s#

z 2

n

= #

The so!ution for n is gien y the eDuation

n = (z 2 )

( )

2 2

(#)2Copyright 2014, 2011, and 2008 Pearson Education,

Inc.6-"+


49/74

Sample SiAe E>ample

;hat samp!e sie is needed to e 0F

confident the mean is 6ithin > pi!otstudy suggested that the standard

deiation is 4.

+z 2

,

2

2

+SE,2 (1.645)

2

(45)2(5)2

2 9.2 220


Inc.6-",


50/74

Sample SiAe eterminationfor 1..1 8

4Confidence Inter!al forpIn order to estimatep 6ith a samp!ing error 'E and

6ith 100%1 H &F confidence, the reDuired

samp!e sie is found y so!ing the fo!!o6ing

eDuation forn#

z

2

pq

n= #

The so!ution for n can e 6ritten as fo!!o6s#

n = (z 2 )

(pq)

2

(#)2

Note# >!6ays round nup to the nearestinteger a!ue.


Inc.6-%.


51/74

Sample SiAe E>ample

;hat samp!e sie is needed to estimate

p

6ithin .0( 6ith 0F confidence

SE = width = .03= .015 2 2

+Z 2 ,

(pq)

2

+SE,2

(1.645 ) (.5 .5)

(.015)2


Inc.6-%1

2

3 06.69 3007


52/74

#hin$ing Challenge

Jou 6or9 in 7uman

esources at "erri!! +ynch.

Jou p!an to surey emp!oyees

to find theiraerage medica! eBpenses.

Jou 6ant to e ,%4 confidentthat the samp!e mean is

6ithin %..> pi!ot study sho6ed that

6as aout "... ;hatsample siAe do you use


Inc.6-%2


53/74

Sample SiAe Solution9

n =+z

2,

2 2

+SE,2

= (1.96)2

(400)2

(50)2

= 245.86 246


Inc.6-%3


54/74

6'6

=inite *opulationCorrection

for Simple 5andom Sample


Inc.6-%"


55/74

=inite *opulation Correction =actor


Inc.6-%%

In some samp!ing situations, the samp!e sie n

may represent F or perhaps 10F of the tota!

numer N of samp!ing units in the popu!ation.

;hen the samp!e sie is !arge re!atie to the

numer of measurements in the popu!ation %see

the neBt s!ide&, the standard errors of the

estimators of andp shou!d e mu!tip!ied y afinite popu!ation correction factor.

5 l f #h b f =i i


56/74

5ule of #humb for =inite*opulation Correction =actor


Inc.6-%6

5se the finite popu!ation correction

factor 6hen nGN O .0.

Si l 5 d S li ith


57/74

Simple 5andom Sampling pproBimate F confidence intera!# x

2 *x

* x

=s N nn N


Inc.6-%(

Si l 5 d S li ith


58/74

Simple 5andom Sampling pproBimate F confidence intera!#p*

2 *p*

*p*

= p* +1 p* ,N

nn N


Inc.6-%+

=inite *opulation Correction


59/74

=inite *opulation Correction=actor E>ample

Jou 6ant to estimate a popu!ation mean,, 6here

x 11, s 18, N 00, and n /0. ind an

approBimate F confidence intera! for.

'ince

n

N

= 60700

= .086

is greater than .0 use the finite correction

factor


Inc.6-%,

=inite *opulation Correction


60/74

=inite *opulation Correction=actor E>ample

Jou 6ant to estimate a popu!ation mean,, 6here

x 11, s 18, N 00, and n /0. ind an

approBimate F confidence intera! for.

x 2s N n

= 115 2 18

n

60

N

700 60

700= 115 4.4

= (110.6

119.4

)Copyright 2014, 2011, and 2008 Pearson Education,Inc. 6-6.


61/74

6'(

Confidence Inter!al fora

*opulation 7ariance


Inc.6-61

Confidence Inter!al for a


62/74

Confidence Inter!al for a*opulation 7ariance


Inc.6-62

C diti 5 i d f


63/74

Conditions 5euired for a7alid Confidence Inter!al

for 2


Inc.6-63

1.> random samp!e is se!ected from the

target popu!ation.

2.The popu!ation of interest has a re!atiefreDuency distriution that is

approBimate!y norma!.


64/74

#hin$ing Challenge

Jou)re a mar9eting

manager for a : race. Jou

ta9e a random samp!e of

the times of 2,2 runnersfrom the !ast race, 6ithmean of 2+'% minutes andstandard deiation of +'3

minutes. ;hat is the ,%4confidence intera!

estimate

of the popu!ation !arianceCopyright 2014, 2011, and 2008 Pearson Education,

Inc.6-6"

Confidence Inter!al


65/74


(n

1)s2 (12

)

(292

1)(8.3)2

(292

1)(8.3)2

2

2

2

2

57.30 2

78.95

349.874 253.912

= .0252

(n 1)s2


Inc.6-6%

2

df 22 1 21 %use (00df&


66/74

e) Ideas

*opulation *arameters? Estimators?andStandard Errors

ropor$(onp

s n

p*q*

n

pq np

*

n

arame$er

( )


Inc.6-66

s$(ma$or

( *)#$andard

rror o

s$(ma$or

s$(ma$ed

#$d rror

Mean

(* )x

*

( *

)


67/74

e) Ideas


Inc.6-6(

*opulation *arameters? Estimators?andStandard Errors

Confidence Interval: >n intera! that enc!osesan un9no6n popu!ation parameter 6ith a certain

!ee! of confidence %1 H &

Confidence Coefficient: The proai!ity %1 H

& that a random!y se!ected confidence

intera! enc!oses the true a!ue of the

popu!ation parameter.


68/74

e) Ideas


Inc.6-6+

e) Dords for Identif)ing the #arget*arameter

H "ean, >erage

pH Proportion, raction, Percentage, ate,

Proai!ity

2 * 3ariance


69/74

e) Ideas


Inc.6-6,

Commonl) ;sed z-7alues for a Large-Sample Confidence Inter!al

90% CI +1 , ! .10 z.05 ! 1.64595% CI +1 , ! .05 z.025 ! 1.96

98% CI +1 , ! .02 z.005 ! 2.326

99% CI +1 , ! .01 z.005 ! 2.575


70/74

e) Ideas

etermining the Sample SiAen

Estimating # n = (z 2 )(

) (M )

2 22

Estimatingp# n =

(z 2

)(pq)

(M)


Inc.6-(.

2 2


71/74

e) Ideas


Inc.6-(1

=inite *opulation Correction=actor

eDuired 6hen nGN O .0


72/74

e) Ideas

Confidence Inter!al for*opulation7ariance

5ses chi*sDuare % 2& distriution

$eed to 9no6 and df.2


Inc.6-(2


73/74

e) IdeasIllustrating the 0otion of ,%4ConfidenceF


Inc.6-(3


74/74

e) Ideas

Illustrating the 0otion of ,%4ConfidenceF

Date post:	13-Apr-2018
Category:	Documents
Upload:	victor-sebastian-flores-onate
View:	217 times
Download:	0 times

usm 2016-vesperyino-marketing-inferencias de una muestra-inglés (1).pptx

Documents