ch17 sample size
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Business
Research Methods
William G. Zikmund
Chapter 17:
Determination of Sample Sie
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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
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Population Parameter VersusSample Statistics
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*opulation *arameter
" +aria&les in a population
" Measured characteristics of a population
" Greek lo,er-case letters as notation
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Sample Statistics
" +aria&les in a sample
" Measures computed from data
" n/lish letters for notation
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Makin/ Data 0sa&le
" re2uenc( distri&utions
" *roportions
" Central tendenc(
$ Mean
$ Median
$ Mode
" Measures of dispersion
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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
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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
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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
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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
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*opulation Mean
N X iΣ= µ
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n X X
i
Σ=
Sample Mean
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*umber of +alesperson +ales calls
ike 4atty 3-illie -ob "
.ohn 3Frank 3/huck %+amantha "
#
%um&er of Sales Calls *er Da(
&( Salespersons
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roduct A roduct -
%#%"0
%'%#0%%&#%
%'%00%00
0000
Sales for *roducts 3 and B5
Both 3#era/e 88
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Measures of Dispersion
" 'he ran/e
" Standard de#iation
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Measures of Dispersion
or Spread" Ran/e
"Mean a&solute de#iation
" +ariance
" Standard de#iation
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'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
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De#iation Scores
" 'he differences &et,een each o&ser#ation
#alue and the mean:
x xd ii −=
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%"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
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%"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
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3#era/e De#iation
8<= =∑ −
n
X X i
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Mean S2uared De#iation
n
X X i
∑ −
7
<=
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'he +ariance
7
7
S
Sample
Population
σ
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+ariance
1
<
−
−(Σ=n
X X S
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+ariance
" 'he #ariance is /i#en in s2uared units
" 'he standard de#iation is the s2uare root of
#ariance:
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Sample Standard De#iation
( )1
7
−
−Σ=
n
X X iS
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*opulation Standard De#iation
σ σ =
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Sample Standard De#iation
S S =
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Sample Standard De#iation
( )1
7
−
−Σ=
n
X X iS
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'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
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M3%
%ormal Distri&ution
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2.14%
13.59%34.13% 34.13% 13.59%
2.14%
%ormal Distri&ution
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85 115100 14570
%ormal Cur#e: )? @ample
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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
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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
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8 1-1-
3 Standardied %ormal Cur#e
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'he Standardied %ormal is the
Distri&ution of Z
–z +z
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σ µ −= x z
Standardied Scores
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σ
µ −= x
z
Standardied +alues
" 0sed to compare an indi#idual #alue to the
population mean in units of the standard
de#iation
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"Population distribution
"Sample distribution
"Sampling distribution
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µ @σ−σ
*opulation Distri&ution
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_ Χ FS
Sample Distri&ution
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X S X X
µ
Samplin/ Distri&ution
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Standard rror of the Mean
" Standard de#iation of the samplin/
distri&ution
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Central imit 'heorem
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Standard rror of the Mean
n
S x
σ =
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Distri&ution Mean StandardDe#iation
*opulation µ σ Sample
X S
Samplin/ X
µ X
S
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*arameter stimates
" *oint estimates
" Confidence inter#al estimates
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error samplin/smalla X ±= µ
Confidence )nter#al
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X cl S Z =.RRR S3M*:)%GSM3::
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X cl S Z =.
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.±= X µ
ti ti th St d d f
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stimatin/ the Standard rror of
the Mean
nS S
x =
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n
S Z X cl ±= µ
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Random Sampling Error and
Sample Size are Related
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Sample Sie
" +ariance =standard
de#iation<
" Ma/nitude of error " Confidence le#el
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Sample Sie ormula
= E zsn
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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.
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7
s
n
= ( ) ( )
7
88.7
88.7DDA.1
=
7
88.7
BI.EA
= ( ) 7I7.7B= B8B=
Sample Sie ormula - @ample
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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.
Sample Sie ormula - @ample
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7
sn
= ( ) ( )
7
88.I
88.7DDA.1
=
7
88.IBI.EA
= ( ) 771.1I= 787=
Sample Sie ormula - @ample
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99% Confidence
Calculatin/ Sample Sie
[ ]%3'= #"3&
=
"3&4
=
))("&(n
=
]34&=[ #3"%'
=
4"3&4
=
4))("&(n
=
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n
p p
or
n
pq p s
<1=
−
=
Standard rror of the *roportion
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pcl S Z p ±
Confidence )nter#al for a
*roportion
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E pq Z n =
Sample Sie for a *roportion
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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.
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Calculatin/ Sample Sie
at the E Confidence e#el
&"3= 00%"
=
00%"
)4)('4%#3(=)03"(
)4)(#()#%(n
4q
#p
=
=
=