MARKETING MARKETING RESEARCHRESEARCH

Week 5Week 5Session ASession A

IBMS Term 2, 2008-09IBMS Term 2, 2008-09

Last Week:

•Lecture 1: Sampling Methods (Ch 12)

•Lecture 2: No lecture – project session and

business case time

•Projects: Project briefs submitted

This week:

Monday

•Lecture 1: Sample Size Calculation (Ch 13)

•Project Session 1

Tuesday

•Lecture 2: Data Analysis & Collection (Ch 14 &15)

•Project Session 1

Tomorrow’s Test:

30 x MC Questions

Chapter 12 – and PPT slides 4a

Determining the Size of a Sample

Today:

6

Remember!

• Population: Every unit of interest, every person relevant to your study i.e. All potential MBO students in NL

• Sample: A sample, a section drawn from the population, for analysis and research purposes.

7

Sample Accuracy

• Sample accuracy: refers to how close a sample’s statistic is to the true population’s value

• Important points:

– Sample size is not related to representativeness

– Sample size is related to accuracy

8

Sample Size and Accuracy

• Which is more accurate: – a large probability sample– a small probability sample?

• The larger a probability sample, the more accurate it is (less sample error).

• Representativeness is assumed with appropriate sampling methods

9

Accuracy vs. Representativeness

• Representativeness pertains to whether your sample reflects the features of the total population

• Accuracy reflects how closely your sample statistics are to the actual population statistic

• A sample can be representative, and it can be either accurate or inaccurate.

• A sample can be not representative, and it will rarely be accurate.

• Probability sampling ensures optimal representativeness

10

A Picture Says 1,000 Words

Sample Size and Accuracy

0%2%4%6%8%

10%12%14%16%

Sample Size

Accu

racy

±

Probability sample accuracy (error) can be calculated with a simple formula, and expressed as a ± % number.

n 550 - 2000 = 1,4504% - 2% = ±2%

11

How to Interpret Sample Accuracy

• From a report…

– The sample is accurate ± 7% at the 95% level of confidence…

• From a news article

– The accuracy of this survey is ± 7%…

12

How to Interpret Sample Accuracy

• From a news article survey on brand awareness– The accuracy of this survey is ± 7%…

• Interpretation – Finding: 60% are aware of our brand– So… between 53% (60%-7%) and 67%

(60%+7%) of the entire population is aware of our brand

13

Sample Size Axioms

To understand how to determine sample size, it helps to understand the following axioms…

• The only perfectly accurate sample is a census.

• A probability sample will always have some inaccuracy (sample error).

14

Sample Size Axioms

3. The larger a probability sample is, the more accurate it is (less error).

4. Probability sample accuracy (error) can be calculated with a simple formula

5. A probability sample can be a very tiny percentage of the population size and still be very accurate (have little sample error).

15

Sample Size and Population Size

• Does the size of the population, N, affect sample size or sample error?

A probability sample size can be a very tiny percentage of the population size and still be very accurate (have little sample error).

Population Size

e=±3% Sample Size

Sample as % of the Population

10,000 ____ ___%

100,000 ____ ___%

1,000,000 ____ ___%

100,000,000 ____ ___%

1,067 1%

1,067 .1%

1,067 .01%

1,067 .0001%

16

Errors

• Nonsampling Error: all sources of error that may occur through methods other than sampling. i.e. Incorrect problem definition, questionnaire errors, inaccurate analysis

• Sampling Error: a source of error that may occur through mistakes in sample selection and sample size calculations.

17

• There is only one method of determining sample size that allows the researcher to PREDETERMINE the accuracy of the sample results…

The Confidence The Confidence Interval Method of Interval Method of

Determining Sample Determining Sample SizeSize

18

The Confidence Interval Method of Determining Sample Size

• The relationship between sample size and sample error:

19

The Confidence Interval Method of Determining Sample Size

• Sample error formula:

• n= sample size• p= probability of outcome a• q= probability of outcome b

20

Computations Help Page

n

pqze

Let’s try 3 n’s

1000

500

100

1.9650 times 50

Answers this way…

21

And the answers are…

n

pqze

Let’s try 3 n’s

1000 ±3.1%

500 ±4.4%

100 ±9.8%

1.9650 times 50

22

The Confidence Interval Method of Determining Sample Size

• Variability: expressed via P & Q

• Can be expressed as– A measure of probability that P or Q will occur

• Before data is captured, an estimate of sample error

– A measure of what was observed for P & Q• After data is captured, an estimate of variation in the

responses received

23

The Confidence Interval Method of Determining Sample Size

• Variability: refers to how similar or dissimilar responses are to a given question

P: percent

Q: 100%-P

• Important point: the more variability in the population being studied, the higher the sample size needed to achieve a stated level of accuracy.

High variability = Need for larger sample size

24

• With nominal data (i.e. yes, no), we can conceptualize variability with bar charts– The highest variability is 50/50

– Why?• Have most respondents indicated that they agree –

LOW VARIABILITY• Have respondents indicated a variation in responses

(no response has majority) – HIGH VARIABILITY

25

Confidence Interval Approach

• The confidence interval approach is based upon the normal curve distribution.

• We can use the normal distribution because of the CENTRAL LIMITS

26

The Confidence Interval Method of Determining Sample Size

• 1.96 x s.d. defines the endpoints of the distribution.

27

• We also know that, given the amount of variability in the population, the sample size will affect the size of the confidence interval.

28

So, what have we learned thus far?

• There is a relationship between:– Variability in the population

– The amount of acceptable sample error (desired accuracy) we wish to have and…

– The size of the sample!

29

Break

• 15 minutes

30

Sample Size Formula

• There is a given formula to calculate Sample Size.

– To compute you need:1. Confidence Level (95% or 1.96)

2. Variance (an estimation of p & q)

3. Sample Error ( + %)

– You are calculating for … n

31

Sample Size Formula

• Standard sample size formula for estimating a percentage:

32

Practical Considerations in Sample Size Determination

• How to estimate variability (p x q) in the population

– Expect the worst cast (p=50; q=50)

– Estimate variability: Previous studies? Conduct a pilot study?

33

Practical Considerations in Sample Size Determination

• How to determine the amount of desired sample error

– Negociated between researchers and managers. How much error is the manager willing to tolerate?

– Convention is + or – 5%

– The more important the decision, the smaller the sample error.

34

Practical Considerations in Sample Size Determination

• How to decide on the level of confidence desired

– Negociated between researchers and managers. The more confidence, the larger the sample size.

– Convention is 95% (z=1.96)– The more important the decision,

the more a manager will want more confidence. 99% confidence, z=2.58.

35

ExampleEstimating he Population, given a %

statistic• What is the required sample size?

1. Five years ago a survey showed that 42% of consumers were aware of the company’s brand (Consumers were either “aware” or “not aware”)

After an intense ad campaign, management wants to conduct another survey and they want to be 65% confident that the survey estimate will be within ±5% of the true percentage of “aware” consumers in the population.

– What is n?

36

Estimating a Percentage: What is n?

• Z=1.96 (95% confidence)

• p=42

• q=100-p=58

• e=5

• What is n?

37

Estimating a Percentage: What is n?

• What does this mean?– It means that if we use a sample size of

374, after the survey, we can say the following of the results:

– “Our most likely estimate of the percentage of consumers that are ‘aware’ of our brand name is 55%. In addition, we are 95% confident that the true percentage of ‘aware’ customers in the population falls between 50% and 60%.”

N=374

38

Estimating Sample Size, given a Mean

• Estimating the Sample Size, when given a mean statistic requires a different formula (See MRI 13.2, p. 378)

• Z represents the confidence interval (1.96 or 2.58)• E represents the acceptable sample error (± 5%)• S represents the standard deviation - a little more

difficult to estimate…

39

Estimating s

• Since we are estimating a mean, we can assume that our data are either interval or ratio.

• When we have interval or ratio data, the standard deviation, s, may be used as a measure of variance.

40

Estimating s

• How to estimate s?– Use standard deviation from a previous

study on the target population.

– Conduct a pilot study of a few members of the target population and calculate s.

– Estimate the range of the value you are estimating and divide the range by 6.

41

Estimating s

–Why divide the range by 6?• The range covers the entire

distribution• ± 3 (or 6) standard deviations cover

99.9% of the area under a normal curve.

• If you have 10 options available, calculate s by 10/6

42

ExampleEstimating the Mean of a Population

• What is the required sample size?– Management wants to know customers’ level of

satisfaction with their service.

– They propose conducting a survey and asking for satisfaction on a scale from 1 to 10. (since there are 10 possible answers, the range=10).

– Management wants to be 99% confident in the results and they do not want the allowed error to be more than ±.5 scale points.

– What is n?

43

Estimating a Mean: What is n?

• S=10/6 or 1.7

• Z=2.58 (99% confidence)

• e=.5 scale points

• What is n?

44

Estimating a Percentage: What is n?

• What does this mean?– After the survey, management may make

the following statement: (assume satisfaction mean is 7.3)

– “Our most likely estimate of the level of consumer satisfaction is 7.3 on a 10-point scale. In addition, we are 99% confident that the true level of satisfaction in our consumer population falls between 6.8 and 7.8 on a 10-point scale”

N=77

45

Other Methods of Sample Size Determination

• Arbitrary “percentage of thumb” sample size:

– Arbitrary sample size approaches rely on erroneous rules of thumb.

– Arbitrary sample sizes are simple and easy to apply, but they are neither efficient nor economical.

46

Other Methods of Sample Size Determination

• Conventional sample size specification:

– Conventional approach follows some convention: or number believed somehow to be the right sample size.

– Using conventional sample size can result in a sample that may be too large or too small.

– Conventional sample sizes ignore the special circumstances of the survey at hand.

47

Other Methods of Sample Size Determination

• Statistical analysis requirements of sample size specification:

– Sometimes the researcher’s desire to use particular statistical technique influences sample size

48

Other Methods of Sample Size Determination

• Cost basis of sample size specification:

– “All you can afford” method

– Instead of the value of the information to be gained from the survey being primary consideration in the sample size, the sample size is determined by budget factors that usually ignore the value of the survey’s results to management.

49

Special Sample Size Determination Situations

• Sampling from small populations:– Small population: sample exceeds

5% of total population size– Finite multiplier: adjustment factor for

sample size formula– Appropriate use of the finite

multiplier formula will reduce a calculated sample size and save money when performing research on small populations.

50

Special Sample Size Determination Situations

• Sample size using nonprobability sampling:

– When using nonprobability sampling, sample size is unrelated to accuracy, so cost-benefit considerations must be used.

51

Practice Examples

• We will do some examples from the questions and exercises at the end of the chapter on sample size…question 5 on page 386.

52

Practice Examples

• 5a. Using the formula provided in your text, determine the approximate sample sizes for each of the following cases, all with precision (allowable error) of ±5%:

– Variability of 30%, confidence level of 95%

(323) 322.6 =25

8064 =

25

2100 x 3.84 =

5

70) x (30961. =

e

(pq)z = n

2

2

2

2

53

Practice Examples

• 5b. Using the formula provided in your text, determine the approximate sample sizes for each of the following cases, all with precision (allowable error) of ±5%:

– Variability of 60%, confidence level of 99%

(639) 639.4 =25

15,984 =

25

2400 x 6.66 =

5

40) x (60582. =

e

(pq)z = n

2

2

2

2

54

Practice Examples

• 5c. Using the formula provided in your text, determine the approximate sample sizes for each of the following cases, all with precision (allowable error) of ±5%:

– Unknown variability, confidence level of 95%

384 =25

9600 =

25

2500 x 3.84 =

5

50) x (50961. =

e

(pq)z = n

2

2

2

2