1. Essentials of Marketing Research
Chapter 13:
Determining Sample Size

2. WHAT DO STATISTICS MEAN?
DESCRIPTIVE STATISTICS
NUMBER OF PEOPLE
TRENDS IN EMPLOYMENT
DATA
INFERENTIAL STATISTICS
MAKE AN INFERENCE ABOUT A POPULATION FROM A SAMPLE

3. POPULATION PARAMETER VERSUS SAMPLE STATISTICS

4. POPULATION PARAMETER VARIABLES IN A POPULATION
MEASURED CHARACTERISTICS OF A POPULATION
GREEK LOWER-CASE LETTERS AS NOTATION, e.g. m, s, etc.

5. SAMPLE STATISTICS VARIABLES IN A SAMPLE
MEASURES COMPUTED FROM SAMPLE DATA
ENGLISH LETTERS FOR NOTATION
e.g., or S

6. MAKING DATA USABLE Data must be organized into:
FREQUENCY DISTRIBUTIONS
PROPORTIONS
CENTRAL TENDENCY
MEAN, MEDIAN, MODE
MEASURES OF DISPERSION
range, deviation, standard deviation, variance

7. Frequency Distribution of Deposits

8. MEASURES OF CENTRAL TENDENCY
MEAN - ARITHMETIC AVERAGE
MEDIAN - MIDPOINT OF THE DISTRIBUTION
MODE - THE VALUE THAT OCCURS MOST OFTEN

9. Number of Sales Calls Per Day by Salespersons
Salesperson Sales calls
Mike
Patty
Billie
Bob
John
Frank
Chuck
Samantha 26

10. Sales for Products A and B, Both Average 200
Product A Product B

12. MEASURES OF DISPERSION
THE RANGE
STANDARD DEVIATION

13. Low Dispersion Versus High Dispersion5 4 3 2 1
Low Dispersion
Frequency
Value on Variable

14. 5 4 3 2 1
High dispersion
Frequency
Value on Variable

17. Standard Deviation 2 2
(X - X)
n - 1
S = S =

18. 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

19. NORMAL DISTRIBUTION MEAN Conventional Product Adoption Life Cycle:
Five types of customers who will end up adopting a product
INNOVATORS (2.5%): People who are the first to adopt a product. They are trend-setting, risk-taking, and are not typical consumers. Example: See a movie first weekend it’s out or in a preview.
EARLY ADOPTERS (13.5%): People who are among the first but not as risk-taking. They adopt ideas early but with consideration, and they enjoy roles as opinion leaders. They spread the word about the product. Example: See a movie the first week of its release.
EARLY MAJORITY (34%): Deliberate customers; adopt earlier than most customers but are not leaders. Example: See a movie after a few weeks, after reading all the reviews and getting recommendations from early adopters.
LATE MAJORITY (34%): Skeptical customers, will only adopt an idea if the majority of people have tried it. Example: See a movie after it has been nominated for an Oscar.
LAGGARDS (16%): Tradition-bound, suspicious of change; will adopt an idea only after it has been around long enough. Example: See a movie after it has come out on video.
MEAN

20. Normal Distribution 13.59% 13.59% 34.13% 34.13% 2.14% 2.14%
Conventional Product Adoption Life Cycle:
Five types of customers who will end up adopting a product
INNOVATORS (2.5%): People who are the first to adopt a product. They are trend-setting, risk-taking, and are not typical consumers. Example: See a movie first weekend it’s out or in a preview.
EARLY ADOPTERS (13.5%): People who are among the first but not as risk-taking. They adopt ideas early but with consideration, and they enjoy roles as opinion leaders. They spread the word about the product. Example: See a movie the first week of its release.
EARLY MAJORITY (34%): Deliberate customers; adopt earlier than most customers but are not leaders. Example: See a movie after a few weeks, after reading all the reviews and getting recommendations from early adopters.
LATE MAJORITY (34%): Skeptical customers, will only adopt an idea if the majority of people have tried it. Example: See a movie after it has been nominated for an Oscar.
LAGGARDS (16%): Tradition-bound, suspicious of change; will adopt an idea only after it has been around long enough. Example: See a movie after it has come out on video.
2.14%
2.14%

21. An example of the distribution of Intelligence Quotient (IQ) scores
13.59%
13.59%
34.13%
34.13%
2.14%
2.14% 70 85
100
115
130 IQ

22. STANDARDIZED NORMAL DISTRIBUTION
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

23. A STANDARDIZED NORMAL CURVE
Conventional Product Adoption Life Cycle:
Five types of customers who will end up adopting a product
INNOVATORS (2.5%): People who are the first to adopt a product. They are trend-setting, risk-taking, and are not typical consumers. Example: See a movie first weekend it’s out or in a preview.
EARLY ADOPTERS (13.5%): People who are among the first but not as risk-taking. They adopt ideas early but with consideration, and they enjoy roles as opinion leaders. They spread the word about the product. Example: See a movie the first week of its release.
EARLY MAJORITY (34%): Deliberate customers; adopt earlier than most customers but are not leaders. Example: See a movie after a few weeks, after reading all the reviews and getting recommendations from early adopters.
LATE MAJORITY (34%): Skeptical customers, will only adopt an idea if the majority of people have tried it. Example: See a movie after it has been nominated for an Oscar.
LAGGARDS (16%): Tradition-bound, suspicious of change; will adopt an idea only after it has been around long enough. Example: See a movie after it has come out on video. 1 2 -2 -1

24. STANDARDIZED SCORES

26. POPULATION DISTRIBUTION
SAMPLE DISTRIBUTION
SAMPLING DISTRIBUTION

27. -s s m x POPULATION DISTRIBUTION
Conventional Product Adoption Life Cycle:
Five types of customers who will end up adopting a product
INNOVATORS (2.5%): People who are the first to adopt a product. They are trend-setting, risk-taking, and are not typical consumers. Example: See a movie first weekend it’s out or in a preview.
EARLY ADOPTERS (13.5%): People who are among the first but not as risk-taking. They adopt ideas early but with consideration, and they enjoy roles as opinion leaders. They spread the word about the product. Example: See a movie the first week of its release.
EARLY MAJORITY (34%): Deliberate customers; adopt earlier than most customers but are not leaders. Example: See a movie after a few weeks, after reading all the reviews and getting recommendations from early adopters.
LATE MAJORITY (34%): Skeptical customers, will only adopt an idea if the majority of people have tried it. Example: See a movie after it has been nominated for an Oscar.
LAGGARDS (16%): Tradition-bound, suspicious of change; will adopt an idea only after it has been around long enough. Example: See a movie after it has come out on video. -s s m x

28. SAMPLE DISTRIBUTION _ C X S Conventional Product Adoption Life Cycle:
Five types of customers who will end up adopting a product
INNOVATORS (2.5%): People who are the first to adopt a product. They are trend-setting, risk-taking, and are not typical consumers. Example: See a movie first weekend it’s out or in a preview.
EARLY ADOPTERS (13.5%): People who are among the first but not as risk-taking. They adopt ideas early but with consideration, and they enjoy roles as opinion leaders. They spread the word about the product. Example: See a movie the first week of its release.
EARLY MAJORITY (34%): Deliberate customers; adopt earlier than most customers but are not leaders. Example: See a movie after a few weeks, after reading all the reviews and getting recommendations from early adopters.
LATE MAJORITY (34%): Skeptical customers, will only adopt an idea if the majority of people have tried it. Example: See a movie after it has been nominated for an Oscar.
LAGGARDS (16%): Tradition-bound, suspicious of change; will adopt an idea only after it has been around long enough. Example: See a movie after it has come out on video. _ C X S

29. SAMPLING DISTRIBUTION
Conventional Product Adoption Life Cycle:
Five types of customers who will end up adopting a product
INNOVATORS (2.5%): People who are the first to adopt a product. They are trend-setting, risk-taking, and are not typical consumers. Example: See a movie first weekend it’s out or in a preview.
EARLY ADOPTERS (13.5%): People who are among the first but not as risk-taking. They adopt ideas early but with consideration, and they enjoy roles as opinion leaders. They spread the word about the product. Example: See a movie the first week of its release.
EARLY MAJORITY (34%): Deliberate customers; adopt earlier than most customers but are not leaders. Example: See a movie after a few weeks, after reading all the reviews and getting recommendations from early adopters.
LATE MAJORITY (34%): Skeptical customers, will only adopt an idea if the majority of people have tried it. Example: See a movie after it has been nominated for an Oscar.
LAGGARDS (16%): Tradition-bound, suspicious of change; will adopt an idea only after it has been around long enough. Example: See a movie after it has come out on video. C µX SX

30. STANDARD ERROR OF THE MEAN
STANDARD DEVIATION OF THE SAMPLING DISTRIBUTION

31. CENTRAL LIMIT THEOREM

32. PARAMETER ESTIMATES
POINT ESTIMATES
CONFIDENCE INTERVAL ESTIMATES

38. RANDOM SAMPLING ERROR AND SAMPLE SIZE ARE RELATED

39. SAMPLE SIZE VARIANCE (STANDARD DEVIATION) MAGNITUDE OF ERROR
CONFIDENCE LEVEL

41. Determining Sample Size Recap

42. 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

43. 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?

44. 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

45. 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

46. Notion of Variability
Little variability
Great variability
Mean

47. 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?

48. 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

49. Normal Curve and Standard Deviation

50. 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.

51. Normal, bell-shaped curve
Midpoint
(mean)

52. 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

53. 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

54. 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

55. 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

56. 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

57. 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

58. 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?

59. Practical Considerations in Sample Size Determination
How to calculate the level of confidence desired
risk
normally use either 95% or 99%

60. 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

61. 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!