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Business
Research Methods
William G. Zikmund
Chapter 17:
Determination of Sample Size
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32887-6777.
WHAT DOES STATISTICS
MEAN?
• DESCRIPTIVE STATISTICS
– NUMBER OF PEOPLE
– TRENDS IN EMPLOYMENT
– DATA
• INFERENTIAL STATISTICS
– MAKE AN INFERENCE ABOUT A
POPULATION FROM A SAMPLE
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POPULATION PARAMATER
• VARIABLES IN A POPULATION
• MEASURED CHARACTERISTICS OF A
POPULATION
• GREEK LOWER-CASE LETTERS AS
NOTATION
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SAMPLE STATISTICS
• VARIABLES IN A SAMPLE
• MEASURES COMPUTED FROM
SAMPLE DATA
• ENGLISH LETTERS FOR NOTATION
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MAKING DATA USABLE
• FREQUENCY DISTRIBUTIONS
• PROPORTIONS
• CENTRAL TENDENCY
– MEAN
– MEDIAN
– MODE
• MEASURES OF DISPERSION
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Frequency Distribution of Deposits
Frequency (number of
Amount
less than $3,000
$3,000 - $4,999
$5,000 - $9,999
$10,000 - $14,999
$15,000 or more
people making deposits
in each range)
499
530
562
718
811
3,120
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Percentage Distribution of Amounts
of Deposits
Amount
less than $3,000
$3,000 - $4,999
$5,000 - $9,999
$10,000 - $14,999
$15,000 or more
Percent
16
17
18
23
26
100
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Probability Distribution of Amounts
of Deposits
Amount
less than $3,000
$3,000 - $4,999
$5,000 - $9,999
$10,000 - $14,999
$15,000 or more
Probability
.16
.17
.18
.23
.26
1.00
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MEASURES OF CENTRAL
TENDENCY
• MEAN - ARITHMETIC AVERAGE
– µ, population;
X
, sample
• MEDIAN - MIDPOINT OF THE
DISTRIBUTION
• MODE - THE VALUE THAT OCCURS
MOST OFTEN
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POPULATION MEAN
Xi
 
N
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SAMPLE MEAN
 Xi
X
n
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Number of Sales Calls Per Day
by Salespersons
Salesperson
Mike
Patty
Billie
Bob
John
Frank
Chuck
Samantha
Number of
Sales calls
4
3
2
5
3
3
1
5
26
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Sales for Products A and B, Both Average 200
Product A
196
198
199
199
200
200
200
201
201
201
202
202
Product B
150
160
176
181
192
200
201
202
213
224
240
261
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MEASURES OF DISPERSION
• THE RANGE
• STANDARD DEVIATION
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Measures of Dispersion or Spread
•
•
•
•
Range
Mean absolute deviation
Variance
Standard deviation
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THE RANGE AS A MEASURE
OF SPREAD
• The range is the distance between the smallest
and the largest value in the set.
• Range = largest value – smallest value
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DEVIATION SCORES
• the differences between each observation
value and the mean:
d i  xi  x
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Low Dispersion Verses High Dispersion
5
Low Dispersion
4
3
2
1
150
160
170 180
190
Value on Variable
200
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210
High dispersion
5
4
3
2
1
150
160
170
180
190
200
Value on Variable
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210
AVERAGE DEVIATION
(X
i
n
X)
0
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MEAN SQUARED DEVIATION
(X
i
X)
2
n
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THE VARIANCE
Population

2
Sample
S
2
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VARIANCE
 X  X )
S 
n 1
2
2
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• The variance is given in squared units
• The standard deviation is the square root of
variance:
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SAMPLE
STANDARD DEVIATION
Sx 
 X i  X 
n 1
2
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POPULATION STANDARD
DEVIATION
 
2
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SAMPLE STANDARD
DEVIATION
S S
2
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SAMPLE STANDARD
DEVIATION
S
(X

X
)
i
n 1
2
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THE NORMAL DISTRIBUTION
• NORMAL CURVE
• BELL-SHAPPED
• ALMOST ALL OF ITS VALUES ARE
WITHIN PLUS OR MINUS 3
STANDARD DEVIATIONS
• I.Q. IS AN EXAMPLE
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NORMAL DISTRIBUTION
MEAN
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Normal Distribution
13.59%
34.13%
34.13%
13.59%
2.14%
2.14%
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Normal Curve: IQ Example
70
85
100
115
145
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STANDARDIZED NORMAL
DISTRIBUTION
• SYMETRICAL ABOUT ITS MEAN
• MEAN IDENFITIES HIGHEST POINT
• INFINITE NUMBER OF CASES - A
CONTINUOUS DISTRIBUTION
• AREA UNDER CURVE HAS A PROBABLITY
DENSITY = 1.0
• MEAN OF ZERO, STANDARD DEVIATION
OF 1
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STANDARD NORMAL CURVE
• The curve is bell-shaped or symmetrical
• about 68% of the observations will fall within 1
standard deviation of the mean,
• about 95% of the observations will fall within
approximately 2 (1.96) standard deviations of
the mean,
• almost all of the observations will fall within 3
standard deviations of the mean.
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A STANDARDIZED NORMAL CURVE
-2
-1
0
1
2
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z
The Standardized Normal is the
Distribution of Z
–z
+z
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STANDARDIZED SCORES
z
x

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Standardized Values
• Used to compare an individual value to the
population mean in units of the standard deviation
z
x

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Linear Transformation of Any Normal
Variable into a Standardized Normal Variable




Sometimes the
scale is stretched
X
Sometimes the
scale is shrunk
z
-2
-1
0
1
2
x

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•Population Distribution
•Sample Distribution
•Sampling Distribution
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POPULATION DISTRIBUTION



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x
SAMPLE DISTRIBUTION
_
C
S
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X
SAMPLING DISTRIBUTION
X
SX
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X
STANDARD ERROR OF THE
MEAN
• STANDARD DEVIATION OF THE
SAMPLING DISTRIBUTION
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STANDARD ERROR
OF THE MEAN
Sx 

n
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PARAMETER ESTIMATES
• POINT ESTIMATES
• CONFIDENCE INTERVAL ESTIMATES
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CONFIDENCE INTERVAL
  x  a small sampling error
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SMALL SAMPLING ERROR  Z cl S X
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E  Z cl S X
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  X E
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ESTIMATING THE STANDARD
ERROR OF THE MEAN
S
x

S
n
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  X  Z cl
S
n
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RANDOM SAMPLING ERROR
AND SAMPLE SIZE ARE
RELATED
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SAMPLE SIZE
• VARIANCE
(STANDARD
DEVIATION)
• MAGNITUDE OF
ERROR
• CONFIDENCE
LEVEL
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Sample Size Formula
zs 

n 
E
2
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Sample Size Formula - example
Suppose a survey researcher, studying
expenditures on lipstick, wishes to have a
95 percent confident level (Z) and a
range of error (E) of less than $2.00. The
estimate of the standard deviation is
$29.00.
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Sample Size Formula - example
 zs 
n  
E
2
 1.96 29.00  


2.00


2
2
 56.84 
2




28
.
42

 2.00 
 808
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Sample Size Formula - example
Suppose, in the same example as the one
before, the range of error (E) is
acceptable at $4.00, sample size is
reduced.
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Sample Size Formula - example
 zs 
n  
E
2
 1.96 29.00  


4.00


 56.84 


 4.00 
2
 14.21
2
2
 202
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Calculating Sample Size
99% Confidence
(2.57)(29) 
n

2


74.53 


 2 
2
 [37.265]
 1389
2
2
(2.57)(29) 
n

4


74.53 


 4 
2
 [18.6325]
 347
2
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2
STANDARD ERROR OF THE PROPORTION
s
p 
pq
n
or
p ( 1 p )
n
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CONFIDENCE INTERVAL FOR A
PROPORTION
pZ S
cl
p
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SAMPLE SIZE FOR A
PROPORTION
2
Z pq
n
2
E
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The Sample Size Formula for a Proportion
z2pq
n
2
E
Where
n = Number of items in samples
Z2 = The square of the confidence interval
in standard error units.
p = Estimated proportion of success
q = (1-p) or estimated the proportion of failures
E2 = The square of the maximum allowance for error
between the true proportion and sample proportion
or zsp squared.
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Calculating Sample Size at the 95% Confidence Level
p  .6
q  .4
(1. 96 )2(. 6)(. 4 )
n
( . 035 )2
(3. 8416)(. 24)
001225
. 922

. 001225
 753

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