Business
Research Methods
William G. Zikmund
Chapter 17:
Determination of Sample Size
What does Statistics Mean?
• Descriptive statistics
– Number of people
– Trends in employment
– Data
• Inferential statistics
– Make an inference about a population from a
sample
Population Parameter Versus
Sample Statistics
Population Parameter
• Variables in a population
• Measured characteristics of a population
• Greek lower-case letters as notation
Sample Statistics
• Variables in a sample
• Measures computed from data
• English letters for notation
Making Data Usable
• Frequency distributions
• Proportions
• Central tendency
– Mean
– Median
– Mode
• Measures of dispersion
Frequency Distribution of
Deposits
Amount
less than $3,000
$3,000 - $4,999
$5,000 - $9,999
$10,000 - $14,999
$15,000 or more
Frequency (number of
people making deposits
in each range)
499
530
562
718
811
3,120
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
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
Measures of Central Tendency
• Mean - arithmetic average
– µ, Population;
X
, sample
• Median - midpoint of the distribution
• Mode - the value that occurs most often
Population Mean
X

N
i
Sample Mean
 Xi
X
n
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
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
Measures of Dispersion
• The range
• Standard deviation
Measures of Dispersion
or Spread
•
•
•
•
Range
Mean absolute deviation
Variance
Standard deviation
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
Deviation Scores
• The differences between each observation
value and the mean:
di  xi  x
Low Dispersion Verses High
Dispersion
5
Low Dispersion
4
3
2
1
150
160
170 180
190
Value on Variable
200
210
Low Dispersion Verses High
Dispersion
5
4
High dispersion
3
2
1
150
160
170
180
190
Value on Variable
200
210
Average Deviation
(X i  X )
0
n
Mean Squared Deviation
 ( Xi  X )
n
2
The Variance
Population

2
Sample
S
2
Variance
 X  X )
S 
n 1
2
2
Variance
• The variance is given in squared units
• The standard deviation is the square root of
variance:
Sample Standard Deviation
S
2
 Xi X 

n 1
Population Standard Deviation
 
2
Sample Standard Deviation
S S
2
Sample Standard Deviation
S
2
 Xi X 

n 1
The Normal Distribution
• Normal curve
• Bell shaped
• Almost all of its values are within plus or
minus 3 standard deviations
• I.Q. is an example
Normal Distribution
MEAN
Normal Distribution
13.59%
2.14%
34.13%
34.13%
13.59%
2.14%
Normal Curve: IQ Example
70
85
100
115
145
Standardized Normal Distribution
• Symetrical about its mean
• Mean identifies highest point
• Infinite number of cases - a continuous
distribution
• Area under curve has a probability density = 1.0
• Mean of zero, standard deviation of 1
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
A Standardized Normal Curve
-2
-1
0
1
2
z
The Standardized Normal is the
Distribution of Z
–z
+z
Standardized Scores
z
x

Standardized Values
• Used to compare an individual value to the
population mean in units of the standard
deviation
z
x

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

•Population distribution
•Sample distribution
•Sampling distribution
Population Distribution



x
Sample Distribution
_
C
S
X
Sampling Distribution
X
SX
X
Standard Error of the Mean
• Standard deviation of the sampling
distribution
Central Limit Theorem
Standard Error of the Mean
Sx 

n
Distribution
Mean
Population

Sample
X
S
X
SX
Sampling
Standard
Deviation

Parameter Estimates
• Point estimates
• Confidence interval estimates
Confidence Interval
  X  a small sampling error
SMALL SAMPLING
ERROR  Z cl S X
E  Z cl S X
 X E
Estimating the Standard Error of
the Mean
S
x

S
n
  X  Z cl
S
n
Random Sampling Error and
Sample Size are Related
Sample Size
• Variance (standard
deviation)
• Magnitude of error
• Confidence level
Sample Size Formula
zs 

n 
E
2
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.
Sample Size Formula - Example
 zs 
n  
E
2
 1.9629.00 


2.00


2
2
 56.84 
2




28
.
42

 2.00 
 808
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.
Sample Size Formula - Example
 zs 
 1.9629.00
n    

4.00 
E

2
2
2
56.84
2




14
.
21

 4.00 
 202
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
2
Standard Error of the Proportion
s
p

or
p (1 p )
n
pq
n
Confidence Interval for a
Proportion
pZ S
cl
p
Sample Size for a Proportion
2
Z pq
n
E
2
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.
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
