essentials of marketing research chapter 13: determining sample size
TRANSCRIPT
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Essentials ofEssentials of Marketing Research Marketing Research
Chapter 13:
Determining Sample Size
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WHAT DO STATISTICS MEAN?WHAT DO 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 PARAMETERPOPULATION PARAMETERVERSUSVERSUS
SAMPLE STATISTICSSAMPLE STATISTICS
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POPULATION PARAMETERPOPULATION PARAMETER
• VARIABLES IN A POPULATION
• MEASURED CHARACTERISTICS OF A POPULATION
• GREEK LOWER-CASE LETTERS AS NOTATION, e.g. etc.
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SAMPLE STATISTICSSAMPLE STATISTICS
• VARIABLES IN A SAMPLE
• MEASURES COMPUTED FROM SAMPLE DATA
• ENGLISH LETTERS FOR NOTATION– e.g., or SX
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MAKING DATA USABLEMAKING DATA USABLE
• Data must be organized into:– FREQUENCY DISTRIBUTIONS– PROPORTIONS– CENTRAL TENDENCY
• MEAN, MEDIAN, MODE
– MEASURES OF DISPERSION• range, deviation, standard deviation, variance
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Frequency Distribution of DepositsFrequency Distribution of Deposits
Amount Frequency Percent Probability
Under $3,000 499 16 .16
$3,000-$4,999 530 17 .17
$5,000-$9,999 562 18 .18
$10,000-$14,999
718 23 .23
$15,000 or more 811 26 .26
Total 3,120 100 1
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MEASURES OF CENTRAL MEASURES OF CENTRAL TENDENCYTENDENCY
• MEAN - ARITHMETIC AVERAGE
• MEDIAN - MIDPOINT OF THE DISTRIBUTION
• MODE - THE VALUE THAT OCCURS MOST OFTEN
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Number ofSalesperson Sales calls
Mike 4Patty 3Billie 2Bob 5John 3Frank 3Chuck 1Samantha 5
26
Number of Sales Calls Per Day Number of Sales Calls Per Day by Salespersonsby Salespersons
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Sales for Products A and B, Both Average 200Sales for Products A and B, Both Average 200
Product A Product B
196 150198 160199 176199 181200 192200 200200 201201 202201 213201 224202 240202 261
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MEASURES OF DISPERSIONMEASURES OF DISPERSION
• THE RANGE
• STANDARD DEVIATION
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150 160 170 180 190 200 210
5
4
3
2
1
Low Dispersion
Value on Variable
Fre
quen
cyLow Dispersion Versus High Low Dispersion Versus High
DispersionDispersion
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150 160 170 180 190 200 210
5
4
3
2
1
Fre
quen
cy High dispersion
Value on Variable
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S = S2
= (X - X) n - 1
2
Standard Deviation
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THE NORMAL DISTRIBUTIONTHE 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 DISTRIBUTIONNORMAL DISTRIBUTION
MEAN
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2.14%
13.59% 34.13% 34.13% 13.59%
Normal DistributionNormal Distribution
2.14%
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An example of the distribution of Intelligence Quotient (IQ) scores
2.14%
13.59% 34.13% 34.13% 13.59%
2.14%
70 85 100
IQ115 130
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STANDARDIZED NORMAL STANDARDIZED NORMAL DISTRIBUTIONDISTRIBUTION
• SYMMETRICAL 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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A STANDARDIZED NORMAL CURVEA STANDARDIZED NORMAL CURVE
01 -1-2 2
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STANDARDIZED STANDARDIZED SCORESSCORES
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•POPULATION DISTRIBUTION
•SAMPLE DISTRIBUTION
•SAMPLING DISTRIBUTION
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POPULATION DISTRIBUTIONPOPULATION DISTRIBUTION
x
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SAMPLE DISTRIBUTIONSAMPLE DISTRIBUTION
XS
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SAMPLING DISTRIBUTIONSAMPLING DISTRIBUTION
µX SX
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STANDARD ERROR STANDARD ERROR OF THE MEANOF THE MEAN
STANDARD DEVIATION OF THE SAMPLING DISTRIBUTION
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CENTRAL LIMIT THEOREMCENTRAL LIMIT THEOREM
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PARAMETER ESTIMATESPARAMETER ESTIMATES
• POINT ESTIMATES
• CONFIDENCE INTERVAL ESTIMATES
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RANDOM SAMPLING ERROR RANDOM SAMPLING ERROR AND SAMPLE SIZE ARE AND SAMPLE SIZE ARE
RELATEDRELATED
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SAMPLE SIZESAMPLE SIZE
• VARIANCE (STANDARD DEVIATION)
• MAGNITUDE OF ERROR• CONFIDENCE LEVEL
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Determining Sample Size
Recap
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Sample Accuracy
• How close the sample’s profile is to the true population’s profile
• Sample size is not related to representativeness,
• Sample size is related to accuracy
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Methods of Determining Sample Size
• Compromise between what is theoretically perfect and what is practically feasible.
• Remember, the larger the sample size, the more costly the research.
• Why sample one more person than necessary?
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Methods of Determining Sample Size
• Arbitrary– Rule of Thumb (ex. A sample should be at least 5%
of the population to be accurate– Not efficient or economical
• Conventional– Follows that there is some “convention” or number
believed to be the right size– Easy to apply, but can end up with too small or too
large of a sample
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Methods of Determining Sample Size
• Cost Basis– based on budgetary constraints
• Statistical Analysis– certain statistical techniques require certain
number of respondents
• Confidence Interval– theoretically the most correct method
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Notion of Variability
Great variabilit
y
Little variability
Mean
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Notion of Variability
• Standard Deviation– approximates the average distance away from
the mean for all respondents to a specific question
– indicates amount of variability in sample– ex. compare a standard deviation of 500 and
1000, which exhibits more variability?
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Measures of Variability
• Standard Deviation: indicates the degree of variation or diversity in the values in such as way as to be translatable into a normal curve distribution
• Variance = (x-x)2/ (n-1)• With a normal curve, the midpoint (apex) of the
curve is also the mean and exactly 50% of the distribution lies on either side of the mean.
i
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Normal Curve and Standard Deviation
Number ofstandard
deviationsfrom the
mean
Percent ofarea underthe curve
Percent ofarea to theright or left
+/- 1.00 st dev 68% 16%
+/- 1.64 st dev 90% 5%
+/- 1.96 st dev 95% 2.5%
+/- 2.58 st dev 99% 0.5%
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Notion of Sampling Distribution
• The sampling distribution refers to what would be found if the researcher could take many, many independent samples
• The means for all of the samples should align themselves in a normal bell-shaped curve
• Therefore, it is a high probability that any given sample result will be close to but not exactly to the population mean.
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Midpoint
(mean)
Normal, bell-shaped curve
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Notion of Confidence Interval
• A confidence interval defines endpoints based on knowledge of the area under a bell-shaped curve.
• Normal curve– 1.96 times the standard deviation theoretically defines
95% of the population
– 2.58 times the standard deviation theoretically defines 99% of the population
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Notion of Confidence Interval
• Example– Mean = 12,000 miles– Standard Deviation = 3000 miles
• We are confident that 95% of the respondents’ answers fall between 6,120 and 17,880 miles 12,000 + (1.96 * 3000) = 17,880 12,000 - (1.96 * 3000) = 6.120
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Notion of Standard Error of a Mean
• Standard error is an indication of how far away from the true population value a typical sample result is expected to fall.
• Formula– S X = s / (square root of n)
– S p = Square root of {(p*q)/ n}• where S p is the standard error of the percentage
• p = % found in the sample and q = (100-p)
• S X is the standard error of the mean
• s = standard deviation of the sample
• n = sample size
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Computing Sample Size Using The Confidence Interval Approach
• To compute sample size, three factors need to be considered: – amount of variability believed to be in the
population– desired accuracy– level of confidence required in your estimates
of the population values
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Determining Sample Size Using a Mean
• Formula: n = (pqz2)/e2
• Formula: n = (s2z2)/e2
• Where– n = sample size
– z = level of confidence (indicated by the number of standard errors associated with it)
– s = variability indicated by an estimated standard deviation
– p = estimated variability in the population
– q = (100-p)
– e = acceptable error in the sample estimate of the population
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Determining Sample Size Using a Mean: An Example
• 95% level of confidence (1.96)
• Standard deviation of 100 (from previous studies)
• Desired precision is 10 (+ or -)
• Therefore n = 384– (1002 * 1.962) / 102
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Practical Considerations in Sample Size Determination
• How to estimate variability in the population– prior research– experience– intuition
• How to determine amount of precision desired– small samples are less accurate– how much error can you live with?
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Practical Considerations in Sample Size Determination
• How to calculate the level of confidence desired– risk– normally use either 95% or 99%
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Determining Sample Size
• Higher n (sample size) needed when:– the standard error of the estimate is high
(population has more variability in the sampling distribution of the test statistic)
– higher precision (low degree of error) is needed (i.e., it is important to have a very precise estimate)
– higher level of confidence is required
• Constraints: cost and access
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Notes About Sample Size
• Population size does not determine sample size.
• What most directly affects sample size is the variability of the characteristic in the population.– Example: if all population elements have the
same value of a characteristic, then we only need a sample of one!