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These materials contain information that is proprietary and confidential to Bank of America. These materials shall not be duplicated. 2005 Bank of America. All rights reserved.

January 3 2005 ver.4.4 - Action Legal Copy Service.

How to make sure that your true

process population dynamics arerepresented by your sample data.

How many datapoints do I

need?

Sampling

Concepts


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Objectives Be able to select the proper sampling

strategy.

Be able to calculate the proper sample

size for a given confidence level and

confidence interval.

Be able to define confidence level, power

factor, alpha risk and beta risk.

Be able to calculate the proper sample

size for a hypothesis test or DOE.


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How Much Data Do You Need?It depends on what questions you are trying to answer?For example

Determining Process Capability (Baselining)

Your focus should be to collect enough baseline data to capture an entireiteration or cycle of the process.

An iteration should account for the different types of

variation seen within the process.

Cycles, shifts, seasonal trends, product types,

volume ranges, cycle times, demographic mixes, etc.

If historical data are not available, a data collection plan should be

instituted to collect the appropriate data.

Hypothesis Testing (comparing means/variances)Later in this section we will show you that your focus should be on other

statistical characteristics of the sample, such as, mean, variance, risk and

the level of confidence desired for seeing differences.


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Sampling Strategy Review

What must a sample be?

Random.

Unbiased.

Representative.

What kind of sampling can we do?

PopulationItems exist and theircharacteristics are stable.

ProcessItems continue to be produced and

their characteristics may change as theprocess varies.


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Sampling Strategy Review

Sampling Strategies

Random Sample - Each unit has an equal probability of

being selected in a sample (typically used for populationstudies).

Rational Subgroup Each unit is collected at point A in aprocess everynth hour. Usually multiple sequential units arecollected (typically used for process studies).

Stratified Random Sampling - Randomly sample within astratified category or group. Sample sizes for each groupare generally proportional to the relative size of the group.

Systematic Sampling - Sample every nth one (Ex: collectingevery 4th unit).


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XXXX

XXX

XXXX

XXXX

XXXX

XXXXXX

XXXXXXXX

XXXX

XXXXXXXXXXXXXX

XXXX

XXXXX

Population

Sample

XXXX

XXX

Sampling Strategy NewCluster Sampling

The population is composed of small groups called clusters.

There is very li ttle variation in the demographics of the clusters. Data is gathered in detail within one (or a few) cluster(s).

i.e. all the tellers in one representative banking center or allcall representatives in one representative call center.


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Sampling Exercise 1

1. We want to monitor our item processing associates on aregular basis to see if errors are going up.

2. Without spending a lot of money, we want to know whetherthe people in Charlotte prefer Bank of America orWachovia.

3. A DFSS team needs to thoroughly understand our affluent

urban customers to better develop products for their needs.

4. For planning purposes our Supply Chain Managementneeds to know how long it takes to enter into contracts. Wedeal with many types of suppliers in many different areas ofthe country.

5. We want to know how accurate our bills are to ourcorporate customers. It will take about 15 minutes toreview each bill for accuracy. Bills are issued monthly, butevenly throughout the month (I.e. not all end-of-monthbilling).

Random

RationalSubgroup

StratifiedRandom

Systematic

Cluster

Match the sampling situation with the best sampling strategy.


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How Much Data Do You Need?How many actual data points do you need to collect for

your sample? Green belt class introduced the following

sample size formulas:

2s96.1

n

= n = sample size

s = standard deviation

p = proportion defective

= precision, , of theestimate at 95% confidence

)p1(p96.1n

2 =

Continuous data form

Discrete data form


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10These materials contain information that is proprietary and confidential to Bank of America. These materials shall not be duplicated. 2005 Bank of America. All rights reserved.


Sample Size Calculator

Open the Excel fi le called: SamplCal4.x

This application can be used effectively for determining ini tial sample sizes required for

process studies or descriptive statistics of process information. It should not be used

for hypothesis testing. Minitab is the recommended sample size calculator for

hypothesis testing or experimental studies.

Estimated Sample Sizes for Continuous Data at 99%, 95% and 90% Conf idence Levels

Enter Population Size Here* 1,000,000 Precision Sample Size Sample Size Sample Size

(d)99% 95% 90%1 167 97 68

Enter Estimated Standard Deviation Here 5 2 42 25 17

(If unknown, use 1/6 of the known range of the data) 3 19 11 8

4 11 7 5

* For process sampling use the total number of 5 7 4 3

items produced in the time period you wish 6 5 3 2

to characterize. The popu lation size is used 7 4 2 2

to adjust sample size with the Finite Population 8 3 2 2Correction Factor 9 3 2 1

10 2 1 1

11 2 1 1

12 2 1 1

13 1 1 1

14 1 1 1

15 1 1 1


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Sampling Exercise 2 Each month we want to monitor the errors made on

deposits in each region. We want to know where we are

within 1%. We believe the error rate is 11.5%. How manysamples should we take in each region, each month?

After sampling for a few months, we have found that the

error rate is only 8.1%. Additionally, our boss wants us toreport it in Sigma Level to the closest tenth. At this level

we calculated that 0.1 Sigma Level is about 1.4%, so that is

the most precision we will need. Now how many samples

should we take?


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Sample Sizes for

Hypothesis Testing

his isthe newstuffwithinit b

Right, this

is the newstuff with

Minitab.


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Sample Size Concepts

Smaller sample sizes:

Less Cost

Quicker data collection

Higher risks

- chance of missing animportant effect (false

negative)

- chance of declaring an

effect important whenit is not (false positive)

Wider confidence

intervals

Larger sample sizes:

Higher costs

Longer time to get data

Lower risks (but not zero)

- smaller real effects and

more likely to be declared

significant

Tighter confidence intervals


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Risks and Power

Alpha (): The risk associated with f inding that something is

significant when in reality it is not. The risk of chasing anunimportant X. The risk of doing the wrong thing.

Beta (): The risk associated with stating that an input or process is

not different when it is. This is the risk of missing animportant X. The risk of doing nothing.

Power (1-): The chance of correctly rejecting an X when indeed it should

be rejected. Power is the likelihood that you wil l identify a

significant difference when one exists.


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Alpha/Beta Risks

Alpha, , is the risk of f inding a difference when there really isnt one.Beta, , is the risk ofnotnot finding a difference when there really is one.

Truth is:Truth is:

Testsays:

Test

says:H0H0 HaHa

H0H0

HaHa

Type IIError

Type IError

CorrectCorrect

DecisionDecision

CorrectCorrect

DecisionDecision Risk also called:

Type II error

Risk - also called:Type I error

A fire alarm sounds,

but there is no fire.

Or we deny a loan to

a credit worthyperson.The fire alarm is silent, but

there is a fire. Or we approve

a loan for a non-credit worthy

person.


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Alpha Risk Graphically

Truth is: Both samples are from the same population.

If X2 here, conclude

one population;

correct!

Sample 1

Risk Area


two populations;

Type 1 Error.


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Beta Risk Graphically

Truth is: Samples are from different populations.

Power (1-) is theprobability of detecting

the dif ference, , and isrepresented by the

area in Population 2

less the beta risk area.


one population;

Type 2 Error.

Sample 1


two populations;

correct!

Population 2 Risk AreaRisk Area


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Power (1-) is theprobability of detecting

the dif ference, , and isrepresented by the

area in Population 2

less the beta risk area.

Beta Risk Graphically

Truth is: Samples are from different populations.


one population;

Type 2 Error.


two populations;

correct!

Sample 1

Population 2Risk Area Risk Area


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Sampling Terminology

n: The number of units making up the sample size. May be expressed

differently depending on the situation. For a DOE, n may be the number

of experimental runs. In a two sample t-test, n could represent the

number of observations for each group.: (alpha) Your chance of a false positive, which is the p-value at whichyou start calling things statistically significant.

: (beta) Your chance of a false negative.: (delta) The size of the real effect you want to be sure to detect if in fact it

is there. Often expressed as a multiple of .

: (sigma) The standard deviation of the noise variation when factors areheld fixed.

Power: Your chance of detecting a real effect, i.e., declaring it to be

statistically significant. You want this high. Power = 1-


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Using Minitab to Determine

Sample SizeStat>Power and Sample Size

Minitab can calculate power or sample sizes for:

1-sample t & 2-sample t1 Proportion & 2 Proportions

One-way ANOVA

2-level factorial designs

Enter and , plus two of n, , or (1-), andMinitab wil l solve for the third!Enter and , plus two of n, , or (1-), andMinitab will solve for the third!


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General Sample Size Formula

The sample size formula presented earlier in this module is

actually a simplified version of the general formula that is given

below:

datadiscretefor)p1(p)ZZ(

n

datacontinuousfors)ZZ(

n

2

2

=

=

/2

/2

Note: The formula used earlier assumed Z = 0 or = 0.50. Fora hypothesis test, a = 0.5 implies a power of 50%. You wouldhave a 50% chance of seeing a significant difference if it were

actually there.

Note: The formula used earlier assumed Z = 0 or = 0.50. Fora hypothesis test, a = 0.5 implies a power of 50%. You wouldhave a 50% chance of seeing a significant difference if it were

actually there.


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Sample Size Example 1

We want to see if the average time to process a second

mortgage application is the same for two banking centers. A

2-sample t-test was selected. Our best (planning) estimate

for the average time is around 15 days with a standarddeviation, = 2 days.The sample size must be large enough to provide a 95%

chance of detecting a difference (if it exists) in the average

processing times, as small as 3 days (because a 3 daydifference is of practical signif icance to us). Using an

alpha risk of 0.05, what sample size would you recommend?


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Minitab

1. Fill in two, leaving one empty.

2. Put in Standard

Deviation. 3. Click Options.

4. Choose Ha and enter .

5. OK.

6. OK.

Stat>Power and Sample Size>2-Sample t


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Minitab

Power and Sample Size

2-Sample t Test

Testing mean 1 = mean 2 (versus not =)

Calculating power for mean 1 = mean 2 + difference

Alpha = 0.05 Sigma = 2

Sample Target Actual

Difference Size Power Power

3 13 0.9500 0.9561

Power and Sample Size

2-Sample t Test

Testing mean 1 = mean 2 (versus not =)

Calculating power for mean 1 = mean 2 + difference

Alpha = 0.05 Sigma = 2

Sample Target Actual

Difference Size Power Power

3 13 0.9500 0.9561


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Comparison of proportions:

Card Services wants to compare the default rate of a private

labeled credit card (PLC) against Visa to see if the Visa default rate

is really larger. PLC has claimed a default rate of 1 percent or less.

The default rate for Visa is roughly 1.5 percent. We want to be at

least 90 percent sure (power) to find a significant difference if i t

exists and are willing to take a 5 percent risk ().What is the required sample size (for each credit card)?

What is your false negative risk ()?

Sampling Exercise 3


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Designed experiment:

A Six Sigma team want to conduct a 23 factorial with replicates.

Assuming an = .05, the team needs an 80 percent assuranceof detecting at least a 2 percent difference (effect) on Y .Typical experimental variation with factors held fixed(determined from an earlier pilot study) is about 1.5 percent ( =1.5%).

How many replicates will you have to run to achievethe 80% detection assurance?

Sampling Exercise 4


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ase power n

1

2

3

4

5

6

Co ns tant Co ns tant

Constant Constant

Constant Constant


Constant Constant


?

?

?

?

?

?

Fill in the relationships chart (hint: refer to the general

formula to determining the sample size).

Sampling Exercise 5


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Sampling Summary

Sample size depends on:

What level of risk youre willing to take.

What size difference you want to detect.

How powerful you want the test to be.

Before collecting data, you should think about the sampling

strategy and sample size requirements to ensure that you

have an appropriate amount of data for drawing

conclusions.

Choosing the right sample size allows us to better manage

our risks of making a wrong decision or missing an

important factor.

Sample size determination should include practicalconsiderations, like economics, as well.


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Black Belt Key Learnings

Does this tool have an application to my current project?

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This tool can help me answer the following questions:

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What are the key learnings about this tool and/or subject?________________________________________________________________________

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How comfortable will I be in training my team on this tool?

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