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A Marketing Example
Sample Size and Statistics
Dr. Richard Jerz
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Equations for Sample Size
How to Calculate Sample Size
z
n  p (1  p )

E
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 z 
n 

 E 
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Reference p316‐317
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Statistical Concepts
Sampling Concepts
Descriptive versus inferential statistics
Population versus sample
Discrete versus continuous data
Probability distributions
Probability concepts
Binomial, Normal, and student‐t distributions
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•
•
•
•
•
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• Larger is better
• Larger is usually more costly
• Central limit theorem – sample means are normally distributed
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M&amp;M Count
M&amp;M Percentages
30%
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26%
5
25%
5
4
3
16%
10%
Red
Blue
Yellow
5%
Red
Orange
Blue
Brown
Yellow
0%
Green
5%
0
Green
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1
Brown
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Probability of Green M&amp;M's in Bag
# of Green M&amp;M's in Bag
60%
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50.0%
50%
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Percent
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9
8
7
6
5
4
3
2
1
0
16%
15%
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Count
16%
Orange
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21%
20%
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3
Tossing 2 Dice
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2
1
40%
30%
20%
10%
0
16.7%
16.7%
4
5
11.1%
5.6%
0.0%
0.0%
0%
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2
3
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# in Bag
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6
1
2
3
6
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# in Bag
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Confidence Intervals –
Continuous Normal Data
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Another View of Normally Distributed Data
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A Marketing Example
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Confidence Interval Estimates for the Mean
Use Z‐distribution
Use t‐distribution
• If the population • If the population standard deviation is standard deviation is known or the sample is unknown and the greater than 30.
sample is less than 30.
+/‐ Error, E
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Comparing the z and t Distributions when n is small 13
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Calculate Sample Size
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Sample Size Determination for a Variable • To find the sample size for a variable:  zs 
n

 E 
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Problems with Sample Size
• Sigma is for population
• Need to know sigma for population
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•
•
•
where :
E - the allowable error
Solutions (From Lind, p315)
Use a comparable study
Use a range‐based approach (range/6)
Conduct a pilot study
z - the z - value corresponding to the selected
level of confidence
s - the sample deviation (from pilot sample)
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Calculating Sample Size
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One more problem
• Excel model
• We may not know the population sigma
• Solution: Student‐t distribution
• Problem: not explained by Lind (and others)
• Solution: Excel ‐ Goal Seek
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Why Dr. Shoemaker’s Solution?
Characteristics:
• There are only two possible outcomes on a particular trial of an experiment.
• The outcomes are mutually exclusive, The random variable is the result of counts.
• The random variable is the result of counts.
• Each trial is independent of any other trial
• Examples:
• p is for binomial data
z
n  p(1  p )

E
Binomial Probability Distribution (discrete)
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•
•
•
•
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Binomial Probability Formula
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Yes or no
True or false
On or off
Correct or incorrect
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Binomial Distribution: n, x, π
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Some Interesting Characterists
Binomial – Shapes for Varying 
(n constant)
• As n increases, binomial distribution starts looking normal
• As p gets close to .5, binomial distributions starts looking normal
• Given this, you can use the normal Given this you can use the normal
distribution to estimate the binomial
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Equations for Sample Size
Problems with this approach
z
n  p (1  p )

E
• What is E?
• E is the allowable error, expressed as a percent (a value between 0 and 1)
• What is p?
2
 z 
n 

 E 
2
• How much will you pay for this cookie?
• Will it sell if it’s price was \$1.00?
• p is the probability of success. Most conservative estimate is .5
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