Download Determining Sample Size

Exploring Marketing
Research
William G. Zikmund
Chapter 17:
Determining 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
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Population Parameter Versus
Sample Statistics
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Population Parameter
• 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 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
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
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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 x x
i 
i 
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Low Dispersion Verses High Dispersion
5
Low Dispersion
4
3
2
1
150
160
170 180
190
Value on Variable
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200
210
5
4
High dispersion
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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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
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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 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
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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
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x

•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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CENTRAL LIMIT THEORM
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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
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 
 1.9629.00
n    

4.00 
E

2
2
2
56.84
2




14
.
21

 4.00 
 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
sp
pq
n

or
p (1 p )
n
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Confidence Interval for a Proportion
p  ZclSp
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Sample Size for a Proportion
2
Z pq
n
E
2
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