ch17 sample size

7/24/2019 Ch17 Sample Size

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Business

Research Methods

William G. Zikmund

Chapter 17:

Determination of Sample Sie



What does Statistics Mean!

" Descripti#e statistics

$ %um&er of people

$ 'rends in emplo(ment

$ Data

" )nferential statistics

$ Make an inference a&out a population from a

sample



Population Parameter VersusSample Statistics



*opulation *arameter

" +aria&les in a population

" Measured characteristics of a population

" Greek lo,er-case letters as notation



Sample Statistics

" +aria&les in a sample

" Measures computed from data

" n/lish letters for notation



Makin/ Data 0sa&le

" re2uenc( distri&utions

" *roportions

" Central tendenc(

$ Mean

$ Median

$ Mode

" Measures of dispersion



Frequency (number of

people making deposits Amount in each range)

less than $3,000 4$3,000 ! $4, "30

$",000 ! $, "#$%0,000 ! $%4, &%'$%",000 or more '%%

3,%0

re2uenc( Distri&ution of

Deposits



Amount

ercentless than $3,000 %#$3,000 ! $4, %&$",000 ! $, %'$%0,000 ! $%4, 3

$%",000 or more #%00

*ercenta/e Distri&ution of

3mounts of Deposits



Amount robability

less than $3,000 %#$3,000 ! $4, %&

$",000 ! $, %'$%0,000 ! $%4, 3

$%",000 or more #

*ro&a&ilit( Distri&ution of

3mounts of Deposits



Measures of Central 'endenc(

" Mean - arithmetic a#era/e

$ 45 *opulation6 5 sample

" Median - midpoint of the distri&ution

" Mode - the #alue that occurs most often

X



*opulation Mean

N X iΣ= µ



n X X

i

Σ=

Sample Mean



*umber of +alesperson +ales calls

ike 4atty 3-illie -ob "

.ohn 3Frank 3/huck %+amantha "

#

%um&er of Sales Calls *er Da(

&( Salespersons



roduct A roduct -

%#%"0

%'%#0%%&#%

%'%00%00

0000

Sales for *roducts 3 and B5

Both 3#era/e 88



Measures of Dispersion

" 'he ran/e

" Standard de#iation



Measures of Dispersion

or Spread" Ran/e

"Mean a&solute de#iation

" +ariance

" Standard de#iation



'he Ran/e

as a Measure of Spread" 'he ran/e is the distance &et,een the smallest

and the lar/est #alue in the set.

" Ran/e 9 lar/est #alue $ smallest #alue



De#iation Scores

" 'he differences &et,een each o&ser#ation

#alue and the mean:

x xd ii −=



%"0 %#0 %&0 %'0 %0 00 %0

"

4

3

%

o1 2ispersion

alue on ariable

F r e q u

e n c y

o, Dispersion +erses ;i/h

Dispersion



%"0 %#0 %&0 %'0 %0 00 %0

"

4

3

%

F r e q u

e n c y

igh dispersion

alue on ariable

o, Dispersion +erses ;i/h

Dispersion



3#era/e De#iation

8<= =∑ −

n

X X i



Mean S2uared De#iation

n

X X i

∑ −

7

<=



'he +ariance

7

7

S

Sample

Population

σ



+ariance

1

<

−

−(Σ=n

X X S



+ariance

" 'he #ariance is /i#en in s2uared units

" 'he standard de#iation is the s2uare root of

#ariance:



Sample Standard De#iation

( )1

7

−

−Σ=

n

X X iS



*opulation Standard De#iation

σ σ =




S S =




( )1

7

−

−Σ=

n

X X iS



'he %ormal Distri&ution

" %ormal cur#e

" Bell shaped

" 3lmost all of its #alues are ,ithin plus or

minus > standard de#iations

" ).?. is an e@ample



M3%

%ormal Distri&ution



2.14%

13.59%34.13% 34.13% 13.59%

2.14%

%ormal Distri&ution



85 115100 14570

%ormal Cur#e: )? @ample



Standardied %ormal Distri&ution

" S(metrical a&out its mean

" Mean identifies hi/hest point

" )nfinite num&er of cases - a continuousdistri&ution

" 3rea under cur#e has a pro&a&ilit( densit( 9 1.8

" Mean of ero5 standard de#iation of 1



Standard %ormal Cur#e

" 'he cur#e is &ell-shaped or s(mmetrical

" 3&out A of the o&ser#ations ,ill fall ,ithin

1 standard de#iation of the mean" 3&out E of the o&ser#ations ,ill fall ,ithin

appro@imatel( =1.A< standard de#iations of

the mean" 3lmost all of the o&ser#ations ,ill fall ,ithin

> standard de#iations of the mean



8 1-1-

3 Standardied %ormal Cur#e



'he Standardied %ormal is the

Distri&ution of Z

–z +z



σ µ −= x z

Standardied Scores



σ

µ −= x

z

Standardied +alues

" 0sed to compare an indi#idual #alue to the

population mean in units of the standard

de#iation



"Population distribution

"Sample distribution

"Sampling distribution



µ @σ−σ

*opulation Distri&ution



_ Χ FS

Sample Distri&ution



X S X X

µ

Samplin/ Distri&ution



Standard rror of the Mean

" Standard de#iation of the samplin/

distri&ution



Central imit 'heorem



Standard rror of the Mean

n

S x

σ =



Distri&ution Mean StandardDe#iation

*opulation µ σ Sample

X S

Samplin/ X

µ X

S



*arameter stimates

" *oint estimates

" Confidence inter#al estimates



error samplin/smalla X ±= µ

Confidence )nter#al



X cl S Z =.RRR S3M*:)%GSM3::



X cl S Z =.



.±= X µ

ti ti th St d d f



stimatin/ the Standard rror of

the Mean

nS S

x =



n

S Z X cl ±= µ



Random Sampling Error and

Sample Size are Related



Sample Sie

" +ariance =standard

de#iation<

" Ma/nitude of error " Confidence le#el



Sample Sie ormula

= E zsn



Sample Sie ormula - @ample

Suppose a sur#e( researcher5 stud(in/

e@penditures on lipstick5 ,ishes to ha#e a

E percent confident le#el =Z< and a

ran/e of error =< of less than H.88. 'he

estimate of the standard de#iation is

H.88.



7

s

n

= ( ) ( )

7

88.7

88.7DDA.1

=

7

88.7

BI.EA

= ( ) 7I7.7B= B8B=




Suppose5 in the same e@ample as the one

&efore5 the ran/e of error =< is

accepta&le at HI.885 sample sie is

reduced.




7

sn

= ( ) ( )

7

88.I

88.7DDA.1

=

7

88.IBI.EA

= ( ) 771.1I= 787=




99% Confidence

Calculatin/ Sample Sie

[ ]%3'= #"3&

=

"3&4

=

))("&(n

=

]34&=[ #3"%'

=

4"3&4

=

4))("&(n

=



n

p p

or

n

pq p s

<1=

−

=

Standard rror of the *roportion



pcl S Z p ±

Confidence )nter#al for a

*roportion



E pq Z n =

Sample Sie for a *roportion



5pq6

n =

Where:

n 9 %um&er of items in samples

Z 9 'he s2uare of the confidence inter#al

in standard error units.

p 9 stimated proportion of success

2 9 =1-p< or estimated the proportion of failures

9 'he s2uare of the ma@imum allo,ance for error

&et,een the true proportion and sample proportion

or s p s2uared.



Calculatin/ Sample Sie

at the E Confidence e#el

&"3= 00%"

=

00%"

)4)('4%#3(=)03"(

)4)(#()#%(n

4q

#p

=

=

=

ch17 sample size

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