MARKETING RESEARCH PART 13 ppsx

Sample Size Axioms• A probability sample can be a very tiny percentage of the population size and still be very accurate have little sample error... Ch 13 11Population Size e=±3% Sample

Determining the Size of

a Sample

Sample Accuracy

• Sample accuracy: refers to how close

a random sample’s statistic is to the true population’s value it represents

Sample Size and Accuracy

• Intuition: Which is more accurate: a large probability sample or a small

probability sample?

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

error)

A Picture Says 1,000 Words

Sample Size and Accuracy

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%…

How to Interpret Sample

Accuracy

• 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

Sample Size Axioms

• To properly understand how to

determine sample size, it helps to

understand the following axioms…

Sample Size Axioms

• The only perfectly accurate sample is a census

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

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

• Probability sample accuracy (error) can

be calculated with a simple formula,

and expressed as a +- % number

Sample Size Axioms

• You can take any finding in the

survey, replicate the survey with the same probability sample size, and

you will be “very likely” to find the

same finding within the +- range of

the original finding

• In almost all cases, the accuracy

(sample error) of a probability sample

is independent of the size of the

Sample Size Axioms

• A probability sample can be a very

tiny percentage of the population size and still be very accurate (have little sample error)

Ch 13 11

Population

Size

e=±3% Sample Size

e=±4% Sample Size

In almost all cases, the accuracy (sample error) of a

probability sample is independent of the size of the

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

Population

Size

e=±3% Sample Size

Sample as % of the Population

Sample Size Axiom

• The size of the probability sample

depends on the client’s desired

accuracy (acceptable sample error) balanced against the cost of data

collection for that sample size

Putting It All Together

• MR – What level accuracy do you want?

• MM – I don’t have a clue.

• MR – National opinion polls use 3.5%.

• MR – Okay, that means we need a sample of

1,200.

• MM – Gee Whiz That small?

• MR – Yup, and at a cost of $20 per completion, it will be $24,000.

• MR – I could do 500 for $10,000, and that would

be 4.4% accurate, or 300 for $6,000 at 5.7%.

The size of a probability sample depends on the client’s desired accuracy (acceptable sample error) balanced against the cost of data collection for that sample size

• There is only one method of

determining sample size that allows the researcher to PREDETERMINE the accuracy of the sample results…

The Confidence Interval Method of Determining Sample

Size

The Confidence Interval Method

of Determining Sample Size

• This method is based upon the

Confidence Interval and the Central

Limit Theorem…

• Confidence interval: range whose

endpoints define a certain percentage

of the response to a question

The Confidence Interval Method

of Determining Sample Size

• Confidence interval approach: applies the concepts of accuracy, variability,

and confidence interval to create a

“correct” sample size

• Two types of error:

– Nonsampling error: pertains to all

sources of error other than sample selection method and sample size

– Sampling error: involves sample

selection and sample size

The Confidence Interval Method

of Determining Sample Size

• Sample error formula:

The Confidence Interval Method

of Determining Sample Size

• The relationship between sample size and sample error:

Computations Help Page

n

pq z

Ch 13 21

And the answers are…

n

pq z

Review: What does sample

accuracy mean?

• 95% Accuracy

– Calculate your sample’s finding, p%

– Calculate your sample’s accuracy, ±

e%

– You will be 95% confident that the

population percentage (π) lies between p% ± e%

Review: What does sample

The Confidence Interval Method

of Determining Sample Size

• Variability: refers to how similar or

dissimilar responses are to a given

higher the sample size needed to

achieve a stated level of accuracy

• With nominal data (i.e yes, no), we can conceptualize variability with bar charts…the highest variability is

50/50

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

THEOREM…regardless of the shape

of the population’s distribution, the

distribution of samples (of n at least

=30) drawn from that population will form a normal distribution

Central Limits Theorem

• The central limits theorem allows us

to use the logic of the normal curve

• If we conducted our study over and over, 1,000 times, we would expect our result to fall within a known

range Based upon this, we say that

we are 95% confident that the true

population range value falls within

this range

The Confidence Interval Method

of Determining Sample Size

• 1.96 x s.d defines the endpoints of

the distribution

• We also know that, given the amount

of variability in the population, the

sample size will affect the size of the confidence interval

Ch 13 31

So, what have we learned thus

far?

• There is a relationship between:

– The level of confidence we wish to

have that our results would be repeated within some known range if we were to conduct the study again, and…

– Variability in the population and…

– The amount of acceptable sample error

(desired accuracy) we wish to have and…

– The size of the sample!

Sample Size Formula

• Fortunately, statisticians have given us a formula which is based upon these

relationships.

– The formula requires that we

• Specify the amount of confidence we wish

• Estimate the variance in the population

• Specify the amount of desired accuracy

we want.

– When we specify the above, the formula tells us what sample we need to use…n

Sample Size Formula

• Standard sample size formula for

estimating a percentage:

Practical Considerations in

Sample Size Determination

• How to estimate variability (p times q)

in the population

– Expect the worst cast (p=50; q=50)

– Estimate variability: Previous

studies? Conduct a pilot study?

Ch 13 35

Practical Considerations in

Sample Size Determination

• How to determine the amount of

desired sample error

– Researchers should work with

managers to make this decision How much error is the manager willing to tolerate?

– Convention is + or – 5%

– The more important the decision, the more (smaller number) the

Practical Considerations in

Sample Size Determination

• How to decide on the level of

confidence desired

– Researchers should work with

managers to make this decision The more confidence, the larger the sample size

– Convention is 95% (z=1.96)

– The more important the decision, the more likely the manager will want more confidence 99%

Example

Estimating a Percentage in the Population

• What is the required sample size?

– 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?

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: (assume results show that 55% are aware)

– “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

N=374

Estimating a Mean

• Estimating a mean requires a

different formula (See MRI 13.2, p

attitudes on a 1-7 scale, we may want

error to be no more than ± 5 scale units

• S is a little more difficult to estimate…

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

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 the value you are estimating can take on (minimum and maximum value) and divide the range

Estimating s

– Why divide the range by 6?

• The range covers the entire distribution and ± 3 (or 6) standard deviations cover 99.9% of the area under the normal curve Since we are estimating one standard

deviation, we divide the range by 6.

Ch 13 44

Example

Estimating the Mean of a Population

– 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 wan the allowed error to be more than ±.5 scale points.

– What is n?

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

N=77

Other Methods of Sample Size

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

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

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

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

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

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

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

2100 x

3.84

=

5

70) x (30 96

1.

=

e

(pq) z

= n

2 2 2 2

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

6.66

=

5

40) x (60 58

2.

=

e

(pq) z

= n

2 2 2 2

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

2500 x

3.84

=

5

50) x (50 96

1.

=

e

(pq) z

= n

2 2 2 2