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Determining the Sizeof a Sample

Dr Mahfuz Kabir

Course Instructor

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Few basics

How much sample size is enough?

- its never enough

Sample size is typically a result of budget or timing considerathere are more subtle and much more important factors to c

One issue to think about is the difference between the intervcollect and the interviews you will use to make the final deci

Good research, therefore, must anticipate possible types of make provisions to ensure there will be more than never en

Overall, it is probably better to conduct fewer studies with mbehind them, while developing more meaningful results

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Sample accuracy

Instead of determining representativeness, the size of the sathe sample accuracy of results

Sample accuracy refers to how close a random samples staexample, mean of the responses to a particular question) is populations value it represents

Sample size will have a direct bearing on how accurate the sfindings are relative to the true values in the population

Larger random samples are more accurate than smaller randHowever, 5 is not 5 times more accurate than 1, and 10 is noaccurate as 5

Sample size is not fully related to representativeness, but reto accuracy

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SAMPLE SIZE How to determine the number of respondents in a parti

actually one of the simplest decisions within the markprocess

In reality, a sample size decision is usually a compromise btheoretically perfect and what is practically feasible

Two good reasons a marketing researcher shouldunderstanding of sample size determination:

First, many practitioners have a large sample size bias,belief that sample size determines a samples represen

Second, a marketing manager should have a firm unsample size determination because the size of the samajor cost factor, particularly for personal interviewstelephone surveys

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The xioms of Sample Size and Sample

1. The only perfectly accurate sample is a census.

2. A probability sample will always have some inaccuracy (samp

3. The larger a probability sample is, the more accurate it iserror).

4. Probability sample accuracy (error) can be calculated wformula, and expressed as a % number.

5. You can take any finding in the survey, replicate the survey w

probability sample size, and you will be very likely to ffinding within the range of the original finding.

6. In almost all cases, the accuracy (sample error) of a probabindependent of the size of the population.

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

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THE CONFIDENCE INTERVAL METHODOF DETERMINING SAMPLE SIZE

The most correct method of determining sample sconfidence interval approach, which applies the coaccuracy (sample error), variability, and confidenceto create a correct sample size

Because it is the theoretically most correct method

one used by national opinion polling companies amarketing researchers

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Sample Size and AccuracyAxiom 1: The only perfectly accurate sample is a census

Accuracy is referred to as sample error.

A survey has two types of error: non-sampling error and serror.

Non-sampling error pertains to all sources of error other sample selection method and sample size

includes problem specification mistakes, question bias, and inco

Sampling error is the difference between the sample findtrue population value due to the fact that a sample was ta

Because a census accounts for every single individual, it isaccurate, meaning that it has no sample error whatsoeve

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Sample Size and Accuracy

Axiom 2: A probability sample will always inaccuracy (sample error).

No random sample is a perfect representatpopulation.

However, a random sample is nonetheless arepresentation of the population, even if it is n

accurate.

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Sample Size and AccuracyAxiom 3: The larger aprobability sample is, the

more accurate it is (lesssample error)

There is a relationshipbetween sample size and theaccuracy of that sample

As the sample size increasesthe sample error decreases

Doubling sample size doesnot result in halving thesample error

The Relationship Between

and Sample Erro

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Sample Size and AccuracyAxiom 4: Probability sample accuracy (error) canbe calculated with a simple formula, andexpressed as a percent number

This formula is simple as n is the sample size,and there is a constant, 1.96. But what arep andq?

p and q: The Notion of Variability

p = percent (probability of success)q = 100% p

p and q will always sum to 100 percent as in thecases of 90% + 10% and 50% + 50%.

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The Notion of a Confidence IntervAxiom 5: You can take any finding inthe survey, replicate the survey withthe same probability sample size, and

you will be very likely to find the samefinding within the percent range ofthe original finding

This axiom is based on the idea of aconfidence interval.

A confidence interval is a range whose

endpoints define a certain percentageof the responses to a question

The properties of the normal curve aresuch that 1.96 times the standarddeviation theoretically defines theendpoints for 95% of the distribution

A Normal Curve wi

Properties Iden

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The Notion of a Confidence IntervAxiom 6: In almost all cases, the accuracy (sample error) of a probaindependent of the size of the population

Sample error formula with 50%, 50%, and 100

Confidence interval =p sample error

The limits of the 95 percent confidence interval in our example is 50%40.2% to 59.8%

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Confidence Interval

How can a researcher use the confidenceinterval?

The confidence interval approach allows the

researcher to predict what would be found ifa survey were replicated many, many times

No client would agree to the cost of 1,000replications, but the researcher can say, Ifound that 50% of the sample intends toorder Dominos the next time. I am very

confident that the true populationpercentage is between 40.2% and 59.8%; infact, I am confident that if I did this surveyover 1,000 times, 95% of the findings will fallin this range.

What if the confidence interval was too

wide?

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Sample Size and AccuracyAxiom 7: A probability sample size can be a very tiny percepopulation size and still be very accurate (have little sample

National opinion polls tend to use sample sizes ranging fr1,200 people, meaning that the sample error is around 3highly accurate.

A sample size of 5,000 yields an error of 1.4 percent, whsmall error level

Yet 5,000 is less than 1% of 1 million, and a great mamarketscola drinkers, condominium owners, debit cardsufferers, home gardeners, Internet surfers, and so ocomprised of many millions of customers

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THE S MPLE SIZE FORMUL

Determining Sample Size via the Confidence Interval Fo

Standard sample size formula for a mean

where

n = the sample size

z = standard error associated with the chosen level of confp = estimated percent in the population

q = 100 p

e = acceptable sample error

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Sample Size FormulaVariability: p Times q

This formula is used if we are focusing on some nom

question in the survey. With little variation in the population, we can take sma

because this is accommodated by the formula byp times q.

The estimated percentage in the population, p, is the meperforms this translation along with q, which is always dete

as q = 100% p

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Sample Size FormulaDesired Accuracy: e

The formula includes another factoracceptable sample er

Acceptable error is the very same concept that was introduearlier. That is, the term, e, is the amount of sample error thassociated with the survey.

It is used to indicate how close your sample percentage findthe true population percentage if you were to report the stu

many times.

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Sample Size FormulaLevel of Confidence: z

The percentage of area under the normal curve described by ou

confidence intervals Thus far, we used the constant, 1.96, because 1.96 is the z value

to 95 percent confidence intervals

Researchers typically only worry about the 95 percent or 99 percconfidence

The 95% level of confidence is by far the most commonly used o

used 1.96 in the examples earlier and referred to it as a constantthe chosen z in most cases

Also, 1.96 rounds up to 2

Market researchers almost never deviate from 95%, but if they dlikely level to be used.

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Sample Size Formula: Example 1

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Sample Size Formula: Example 2

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Sample Size Formula: Example 3

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DeterminingSampleSizeEstimatingaM

1.7 is the largest variability estimate possible (1-1

with mid-point not 5)

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How to Estimate Variability in the Popul Two alternatives:

1. expect the worst case or

2. estimate what the actual variability is.

The worst case, or most, variability is 50 percent/50 percen

This assumption is the most conservative one, and it will rescalculation of the largest possible sample size.

On the other hand, a researcher may want to estimatep, or

percentage, in order to lower the sample size Remember that anyp/q combination other than 50%/50% w

in a lower calculated sample size.

A lower sample size means less effort, time, and cost, so thegood reasons for a researcher to try to estimatep.

h f

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How to Determine the Amount of DesSample Error

Marketing researchers often must help decision makers un

sample size implications of their requests for high precisionallowable error

It is almost always up to the researcher to educate the manmight be acceptable or standard sample error

Translated in terms of precision, the more precise the mark

maker desires the estimate to be, the larger must be the sam

Thus, it is the task of the marketing research director to extmarketing decision maker the acceptable range of allsufficient for him or her to make a decision

The acceptable sample error is specified as a plus or minus

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How to Decide on the Level of ConfidDesired

The typical approach in marketing research is to use the

confidence interval of 95 percent.

As we have also indicated, this level translates into a z of 1.9

Any level of confidence between 1% and 99.9% is posmarketing researchers primarily use 95 or 99% level of conf

With the 99% level of confidence, the corresponding z value

The 99% level of confidence means that if the survey weremany, many times with the sample size determined by usthe sample size formula, 99% of the sample ps wouldsample error range.

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To be continued