Proble Solving

1Six Sigma Central Coordination Group

For Internal Circulation Only

TATA MOTORS LIMITED

Property of Tata Motors Ltd. All right reserved . No part of this manual may be reproduced ,stored in a retrieval system,or transmitted ,in any form or means without the prior written permission from Six Sigma Cell , Tata Motors ,Jamshedpur

Root cause Analysis And Problem Solving

Manual

Wave –1 For Variable ( Continuous Data )



TATA MOTORS LIMITEDIndexContents Page No

1. Concept & guidelines

2. Distribution of data

What is distribution

What is sample

Type of variable

Type of distribution

Property of normal distribution

3. Stability of process

4. Causes of variation

5. Tool of detecting assignable cause

6. Statistical process control

Purpose of control Chart

Control chart for variable

Control chart interpretation

Other control condition



TATA MOTORS LIMITED Page No7. Hypothesis Testing

Steps in hypothesis testing Power of hypothesis testingTesting main Effect due to testing Levels

2t TestPaired T TestOne Way ANOVANested Anova With Fixed Factor

Comparison of varianceANOVA With random factorF TestLevene’s test

8. Variation analysis by Shanin range Method9. Regression analysis

Simple Regression Multiple Regression Regression By Subset Method

10. Design Of ExperimentFactorial DesignAnalyze factorial designAnalyze factorial PlotScreening Factorial Design Prediction model for standard deviation Variable search



TATA MOTORS LIMITED

Any problem solving in a manufacturing process involves the identification of Assignable causes.Though the initial identification of factors is done through Tools like cause and effect and FMEA this volume stresses on the validation tools.

This volume aims at familiarizing users with some of the more Powerful six sigma tools and shanin tools used for root cause Identification .The six sigma tools operate through Detection of mean effects (through tools like 2t,one way ANOVA),variance analysis of a group of factors with multiple levels and interactions( through tools like nested ANOVA and ANOVA General linear model) and through establishing relationship between input and output variable when both are of continuous nature(throgh regression analysis).The Shanin tools for assignable causes are useful for detecting assignable input parameters on material and process parameter through use of tools like paired comparison and process parameter search both of which have been dealt in this volume.

Often once the assignable causes are improved, one comes up with the uphill task of improving the common causes affecting part to part variation when one has to interfere with the process to derive the optimum combination of process parameters and input material parameters for improving mean effect and variation.The six sigma tool of DOE and the shanin tool of variable search help in the above analysis.



TATA MOTORS LIMITED

CONCEPTS AND GUIDELINES



TATA MOTORS LIMITED

What is a distribution?



TATA MOTORS LIMITED

A distribution is a population of individual observations each with the probability of occurrence defined by a probability density function.



TATA MOTORS LIMITED

WHAT IS A SAMPLE?



TATA MOTORS LIMITED

A sample is a part of a population drawn in a random manner so as to enable one to make inference about population.



TATA MOTORS LIMITED

TYPES OF VARIABLE



TATA MOTORS LIMITED

Variables are classified as continuous and discrete.

A continuous Variable is one for which all values in some range are possible i.e. values are observed in a continuous scale.

A discrete variable is one where values are with gaps and not on Continuous scale.



TATA MOTORS LIMITED

TYPES OF DISTRIBUTION



TATA MOTORS LIMITED

Distributions for continuous variable.i)Normalii)exponentialiii)Weibull

Distribution for discrete variablei)Poisonii)Binomial.



TATA MOTORS LIMITEDNormal DistributionNormal Distribution

• A dimension on a part is critical. This critical dimension is measured daily on a random sample of parts from a large production process. The measurements on any given day are noted to follow anormal distribution.

• A customer orders a product. The time it takes to fill the order was noted to follow a normal distribution.

Area

-2 0 Z 1.645 2

f(x)

P = 0V = 1

D = 0.05

Z = (X - P) / V



TATA MOTORS LIMITEDAreas under normal curve - Normal distribution table

z Area z Area z Area z Area0.00 0.500000 0.20 0.420740 0.40 0.344578 0.60 0.2742530.01 0.496011 0.21 0.416834 0.41 0.340903 0.61 0.2709310.02 0.492022 0.22 0.412936 0.42 0.337243 0.62 0.2676290.03 0.488033 0.23 0.409046 0.43 0.333598 0.63 0.2643470.04 0.484047 0.24 0.405165 0.44 0.329969 0.64 0.2610860.05 0.480061 0.25 0.401294 0.45 0.326355 0.65 0.2578460.06 0.476078 0.26 0.397432 0.46 0.322758 0.66 0.2546270.07 0.472097 0.27 0.393580 0.47 0.319178 0.67 0.2514290.08 0.468119 0.28 0.389739 0.48 0.315614 0.68 0.2482520.09 0.464144 0.29 0.385908 0.49 0.312067 0.69 0.2450970.10 0.460172 0.30 0.382089 0.50 0.308538 0.70 0.2419640.11 0.456205 0.31 0.378281 0.51 0.305026 0.71 0.2388520.12 0.452242 0.32 0.374484 0.52 0.301532 0.72 0.2357620.13 0.448283 0.33 0.370700 0.53 0.298056 0.73 0.2326950.14 0.444330 0.34 0.366928 0.54 0.294599 0.74 0.2296500.15 0.440382 0.35 0.363169 0.55 0.291160 0.75 0.2266270.16 0.436441 0.36 0.359424 0.56 0.287740 0.76 0.2236270.17 0.432505 0.37 0.355691 0.57 0.284339 0.77 0.2206500.18 0.428576 0.38 0.351973 0.58 0.280957 0.78 0.2176950.19 0.424655 0.39 0.348268 0.59 0.277595 0.79 0.214764

Area





Area





Area





5.00 0.00000029

Area




z Area5.10 0.000001805.20 0.000001075.30 0.000000635.40 0.000000375.50 0.000000215.60 0.000000125.70 0.0000000705.80 0.0000000405.90 0.0000000226.00 0.000000012

Area

Note: Only absolute values of z should be taken while referring to this table.



TATA MOTORS LIMITEDExample : Normal DistributionExample : Normal Distribution

• The diameter of bushings is µ = 50 mm with a standard deviation of σ = 10 mm. Estimate the proportion of the population of bushings that have a diameter equal to or greater than 57 mm.

Ans. 0.2420



TATA MOTORS LIMITED

Binomial distribution. A random variable X is defined to have a binomial distribution if the discrete density function of X is given by

fX(x) = fX(x;n,p)nCx px(1-p)1-x for x= 0,1,2,…,n

=0 otherwise

where the two parameters n and p satisfy 0 ≤ p ≤1, n ranges over the positive integers and q=1-p. A distribution defined by the above density function is called a Binomial distribution.

E(X)=np, Var(X)=npq

Binomial DistributionBinomial Distribution



TATA MOTORS LIMITEDBinomial DistributionBinomial Distribution

• Useful when there are only two results in a random experiment e.g., pass/ fail, compliance/ noncompliance, yes/ no

• A dimension on a part is critical. This critical dimension is measured daily on a random sample of parts from a large production process. To expedite the inspection process, a tool is designed to either fail or pass a part that is tested. The output now is no longer continuous. The output is now binary; hence the binomial distribution applies in this case.



TATA MOTORS LIMITED… Binomial Distribution… Binomial Distribution

• Product either passes or fails test; determine the number of

defective units

• Light bulbs work or do not work; determine the number of

defective light bulbs

• People respond yes or no to a survey question; determine the

proportion of people who answer yes to the question

• Purchase order forms are either filled out incorrectly or

correctly; determine the number of transactional errors.



TATA MOTORS LIMITED

binomial distributionn = 8 p =0.1

0

0.2

0.4

0.6

x

p(x

)

0 1 2 3 4 5 6


0

0.1

0.2

0.3

x

p(x

)

0 1 2 3 4 5 6


0

0.2

0.4

x

p(x)

0 1 2 3 4 5 6


0

0.2

0.4

0.6

x

p(x)

0 1 2 3 4 5 6

Remember : binomial has two parameters ‘n’ and ‘p’



TATA MOTORS LIMITEDExample : Binomial DistributionExample : Binomial Distribution

• A part is said to be defective if a hole that is drilled into it is less or greater than specifications. A supplier claims a failure rate of 1 in 100. I f this failure rate were true, find the probability of observing one defective part in 10 samples.

Ans. 0.091



TATA MOTORS LIMITED

Poisson Distribution A random variable X is defined to have a Poisson distribution if the density of X is given by

e-λ λx/x! for x=0,1,….fX(x) = fX(x; λ) =

0 otherwisewhere the parameter λ satisfies λ >0.E(X)= λ and Var(X)= λ

The Poisson distribution provides a realistic model for many random phenomena. Since the values of a Poisson random variable are the non-negative integers, any random phenomenon for which a count of some sort is of interest is a candidate for modeling by assuming a Poisson distribution. Such a count might be the number of fatal traffic accidents per week in a given state, the radioactive particle emissions per unit of time, the number of telephone calls per hour coming into the switchboard of a large business, the number of organisms per unit volume of some fluid, the number of defects per unit of some material, the number of flaws per unit of some material etc.

Poisson DistributionPoisson Distribution



TATA MOTORS LIMITEDPoisson DistributionPoisson Distribution

• There are a large number of dimensions on a part that are critical. Dimensions are measured on a random sample of parts from a large production process. The number of ‘out-of-specification’ conditions are noted on each sample. This collective “number-of-failure’ information from the samples can be modeled using a Poisson distribution.

• Estimating the number of cosmetic non-conformance when painting an automobile

• Projecting the number of industrial accidents for next year

• Estimating the number of un-popped kernels in a batch of popcorn



TATA MOTORS LIMITED

Though there are many distributions for Continuous variable, normal distribution is the most common distribution in any manufacturing process.Therefore in our coverage of tools emphasis is given on tools applicable for normal distribution.In case of non-normality the data should be transformed and tools applied.



TATA MOTORS LIMITED

PROPERTIES OF NORMAL DISTRIBUTION



TATA MOTORS LIMITED

A process which produces a response which is Normally Distributed is characterized by its mean and standard deviation.

As per Normal Distribution theory :

A observation for response sample static will fall within the limits of +/- 3σ from the process center 99.73% of the time if the process is under statistical control.

Therefore if a point falls outside this limit, process is out of control and there is an assignable cause of variation.



TATA MOTORS LIMITED

(�[ ��σ ��σ

99.73 % area covered

K value of 3 implies that there is a probability of only 0.0027 of a sample statistic falling outside the control limits in spite of the process being in control.

We can choose value of k based on the desired probability of sample statistic falling outside when process is in control

Such a limit is known as probability limit

Ex.: if k=3.09, probability reduces to 0.002.



TATA MOTORS LIMITEDCentral Limit Theorem In general all processes are not normal.

But for large or small sample sizes with a population distribution that is unimodal and close to symmetric, the Central Limit theorem states that :

If the plotted statistic is a sample average, it will tend to have a normal distribution

Thus even if the parent population is not normally distributed ,control charts for averages and related statistics are based on normal distribution. However for highly skewed distribution it is safe to have sample size of 30.

The standard deviation for the averages of mean for random samples of size n taken from a normal population are

/Sqrt(n) (where n is the sample size and is the standard deviation of parent population).



TATA MOTORS LIMITED

Variances of random samples of size n taken from a normally distributed Universe have a skewed distribution withy heaping of variances to the Left of their mean and an attenuation to the right.

However the average of such sample variances approaches the population Variance for large number of such samples.

The sample mean and variance of sample therefore is used to estimate thevariance and mean of population.

.



TATA MOTORS LIMITED

STABILITY OF A PROCESS



TATA MOTORS LIMITED

Process Stability

Stability : Measure of consistency of a process in terms of its mean & variation. It involves a

time factor.

Tim

e

Process Mean(stable)

Process Spread= 6σ (stable)Stable Process

Mean & Variation Stable over time



TATA MOTORS LIMITED

Tim

e

Process Mean unstable

Process Spread(stable)

Unstable Process(Mean inconsistent with time)

Mean unstable

Variation Stable over time



TATA MOTORS LIMITED

Tim

e

Process Mean(stable)

Process Spread (unstable)

Unstable Process(Spread inconsistent with time)

Mean Stable over time

Variation Unstable



TATA MOTORS LIMITED

CAUSES OF INCONSISTENCY INMEAN AND VARIATION



TATA MOTORS LIMITED

Variation & the need for Control

VARIATION

Assignable / Special Causes

Inherent / Common Causes

(Specific external change / event) (Inherent in the process)



TATA MOTORS LIMITED

Assignable Causes of variation¾ Occurrence of a qualitative external event

¾ Variation controllable / immediate action possible

¾Two types stream to stream,time to time

¾ In a X bar and R chart it is reflected as out of control points in the X bar chart if the event falls between subgroups and will be reflected as out of control points in R chart if it falls within subgroup.Example :

¾ Tool change

¾ Setup change / depth adjustment

¾ Shift change

¾ Tool wear



TATA MOTORS LIMITED

STREAM TO STREAM VARIATION (SSV).

2 TYPES

1. Process Stream

2. Product Stream

PROCESS STREAM

Example : A multi spindle m/c producing the same product parameter

• 3 different spindles

• 3 different tool sets Contributes to SSV.

• 3 different fixtures

PRODUCT STREAMExample

: Variation in dia of holes drilled on a circular plate

: Taper on a shaft – variation in dia across cross-section



TATA MOTORS LIMITED

Variation in parts produced over a period of time.

Hour to hour variation

Shift to shift variation

Event to event variation.

¾ Controllable

¾ Assignable / special causes

Causes :

Tool wear

Qualitative events – tool change, setup change etc.

Time to Time variation



TATA MOTORS LIMITED

Common Causes of variation(Reflected in the R chart of X bar and R chart if found stable over time)

¾ Inherent process variation

¾ Uncontrollable ( immediate action/ correction not possible)

¾ Process characteristic

1. Man 4. Measurement

2. Machine 5. Method

3. Material 6. Mother Nature

Occurring due to variation in 6 M’s:



TATA MOTORS LIMITEDAssignable Causes

Common Causes

TIME

PR

OC

ES

S R

ES

PO

NS

E

For the above chart, the haggard pattern shows the inherent variation due to common causes

The trend of this inherent variation over time shows the assignable causes of variation



TATA MOTORS LIMITED

TOOLS FOR DETECTING ASSIGNABLE CAUSES.•SPC

•Test of hypothesis for detecting mean and variance shifts•Variance analysis

•Paired comparision and process parameter Search of SHANIN

•Regression analysis to establish relationship between input and Output both being continuous



TATA MOTORS LIMITED

SPC(STATISTICAL PROCESS CONTROL)

Detects shifts in mean and spread through useOf eight rules.



TATA MOTORS LIMITED

The purpose of a Control Chart

“The function of a control chart is to minimizethe net economic loss from over-adjustmentand under-adjustment.”

Dr. Deming

Rs.



TATA MOTORS LIMITED

Control Charts...* Provide a common language for

communications about the performance of a process.

* Give a good indication ofwhether any problems arelikely to be correctable locallyor will require action on the system.



TATA MOTORS LIMITED

Control Charts will not :

\ Make processes capable

\ Solve process problems

. . . but they can give clues to possible causes



TATA MOTORS LIMITED

Preparation and Use of X and R Charts

1. Make sure the gauge is capable and accurate. Verify that it is calibrated.

2. Decide on the proper sample size.3. Determine the sampling strategy.4. Measure and record each piece of the subgroup on the

control chart.5. Calculate the X and range for each subgroup.6. After 20 subgroups are collected, the central lines (X and R)

can be calculated and control limits are established.7. After control limits are established, analyze range and X

chart, locating special causes.8. If no special causes exist, continue to run and chart the

process.

Control Charts for Variables



TATA MOTORS LIMITED

Sample Size :

¾It is normally between 4 to 10

¾A size of 5 is followed as a standard

¾Larger the sample size, better the chances of detecting small shifts

Choice of measuring instrument :

¾The least count of the instrument should be 1/10th of the tolerance of the parameter.

¾An MSA should be conducted to check the accuracy and precision of the instrument

¾Instrument should be calibrated.



TATA MOTORS LIMITED

Selection of Samples

Thumb rule :

Maximize differences between subgroups

Minimize differences within subgroups

Selecting samples within a subgroup : should include only common causes.

¾ Selection to be one in order of production

¾ To be done within short span of time

¾ No qualitative events should occur within a subgroup



TATA MOTORS LIMITED

Spacing subgroups : Sampling frequency

¾ Subgroups should be spaced evenly across a shift /day.

¾ Should be at sufficiently large time intervals to include variation in process over time.

¾ If a known event occurs, the subgroup readings should be taken before or after the event.

¾ If an event occurs after every job, then consider the variation due to it as a common cause.

The sampling frequency depends on cost of obtaining information compared to cost of not detecting a nonconforming item.



TATA MOTORS LIMITEDConstructing an X – R chart : The Trial Run

Step 1.

Record selected parameter as per sampling scheme

Step 2.

For each sub group, calculate

Subgroup mean = X

Subgroup range = R

Step 3

Calculate average

Subgroup mean Center line of Xbar chart

Subgroup range Center line of R chart



TATA MOTORS LIMITED

Step 4.

Calculate Control Limits

For X bar Chart : UCLx = X + A2R

LCLx = X – A2R

For R chart : UCLr = D4R

LCLr = D3R

Where A2, D4 and D3 are statistical constants depending on sample size n



TATA MOTORS LIMITED

Step 5.

Plot the control limits for Range chart

Plot the values of subgroup Range

Check for points out of control limits

If so investigate causes associated with out of control points and take remedial action

R chart should be analyzed before Xbar chart

It shows stability of process variation.

This variation is included in the control limits of Xbar chart. Therefore if r chart shows out of control, limits of Xbar chart may not be meaningful.



TATA MOTORS LIMITED

Step 6

Delete the out of control points(on both R and Xbar chart) for which remedial actions have been taken to remove special causes of variation

Recalculate center line and control limits for the revised set of data

This should be an ongoing process leading to continuous improvement.

Step 7

Implement the control chart for future observation and monitoring.



TATA MOTORS LIMITED

n2345678910

A21.881.0230.7290.5770.4830.4190.3730.3370.308

D300000

0.0760.1360.1840.223

D43.2672.5742.2822.1152.0041.9241.8641.8161.777

d21.1281.6932.0592.3262.5382.7042.8472.9703.078

CONSTANTS

X = (X1 + X2 + X3+.......+ Xn)/ n

X = (X1 + X2 + X3+.......+ Xn)/ n

σ = R/d2XUCL = X + A2R

XLCL = X - A2R

RUCL = RxD4

RLCL = RxD3

FORMULAE USED IN X AND R CHART



TATA MOTORS LIMITED

How a x bar and R chart looks



TATA MOTORS LIMITED

\ Control Chart Interpretation



TATA MOTORS LIMITED

The R chart is a plot of inherent process variation(spread) and it shows the stability of process variation over a period of time.

It should be examined first.

An out of control condition(due to special causes) should be eliminated after finding the cause.

The X bar chart is a plot of process average and shows its stability over a period of time.

A gradual trend in X bar chart indicates a shift in process average.

It might be a result of change in controllable parameters such as tool sharpness, depth of cut, feed etc.



TATA MOTORS LIMITED

These four test hold for both xbar and R Chart



TATA MOTORS LIMITED

Only for X bar



TATA MOTORS LIMITED

Natural Pattern

\ When points fall within control limits, also referred to as common cause variation.

A. Most of the points are near the grand average(centerline) - 68%

B. A few of the points spread out and approach the control limits.

C. Very few of the points (only 3 out of 1,000) exceed the control limits.

\ Some causes include :å Machine capabilityå Normal variation of incoming materialså Uniformity of raw parts

Control Chart Patterns and related causes



TATA MOTORS LIMITED

Single point out of control- 1 pt outside of control limits

Gradual Shift- Gradual change in process

parameter. Afterwards the process stabilizes.

In X bar chart due to : change in incoming quality of raw material, change in maintenance program

In r chart due to : new operator, decrease in worker skill due to fatigue

UNSTABLE CONDITION AND THEIR PROBABLE CAUSE



TATA MOTORS LIMITED

Trend- gradual shift in level which

does not stabilize.(eg. Six points in a row on rising or declining trend or seven points on one side of center line)Causes

In X bar chart : tool wear, die wear, gradual deterioration of equipment, build up of debris in jigs/fixtures.

In R chart : gradual improvement in operator skill due to training, deterioration due to fatigue, deterioration in accuracy of m/c elements due to wear and tear(e.g..Spindle play)



TATA MOTORS LIMITED

Cycle- Short trends in the data whichoccur in repeated patterns.

Causes :

Rotation in operator

Periodic changes in temperature

Seasonal variation of incoming components



TATA MOTORS LIMITED

Freaks- A single point or groups of points being

greatly different from the others

Wild Patterns

Causes:External disturbances influencing 1 or more samples.Cluster(eg. 9 points in a row on same side

Of center line)

Cluster of several observations definitely different from others

Cause :

Use of a new vendor for a short period

Use of new tool for a short period

Measurement & Sampling problem



TATA MOTORS LIMITED

Stratification(eg. 15 points in a row within 1 sigma from Center line on either side)

- All points very close to the center line.

The output is combined and each distribution negates the other thus giving most of the points near the center line.

It might also be due to charting the output of a process after gating.Remedial action is to have separate control charts.

occurs when the output of several parallel (and assumed identical) processes into a single sample for charting the combined process. Typically a single sample is taken from each machine and included in the subgroup



TATA MOTORS LIMITED

Mixtures - Opposite of stratification, a combination of two different patterns caused by presence of 2 or more population in the sample.(8 points in a row more than one sigma from center line on either side)

Absence of points near the center line.

Causes :

Difference in quality of incoming material(different suppliers)

Over control

Representing 2 or more m/c on same chart

Is similar to stratification, except the output of several parallel machines is mixed and the periodic sample is drawn from the mixture.



TATA MOTORS LIMITED

Instability- Identify by a person havinglarge fluctuations and erraticups and downs.

Totally unpredictable!

Systematic Variables- Predictable fluctuation or a “saw tooth” pattern.(14 points in

a row oscillating up and down)



TATA MOTORS LIMITED

Other Control Conditions] Control chart tests are based on where points

fall with regards to the 3 sigma limits.

Upper control limitZone A - (3 sigma) 2 out of 3 in zone or aboveZone B - (2 sigma) 4 out of 5 in zone or aboveZone C - (1 sigma) 7 in a row in zone or aboveZone C - (1 sigma) 7 in a row in zone or aboveZone B - (2 sigma) 4 out of 5 in zone or aboveZone A - (3 sigma) 2 out of 3 in zone or above

Center line

Lower control limit



TATA MOTORS LIMITED

Once the final (modified if required) control limits are decided based on the trial run, these limits are used to monitor the process in the future.

If any point is found out of limits or if any of the special cause id detected , action is taken to eliminate the special cause associated with it and then proceed.

only after stability is achieved on X bar and R chart is One allowed to find out capability of a process through Cp=(USL-LSL)/(6*(R/d2))

Cpk=lower of( X-LSL)/(3*(R/d2)) &( USL-X)/(3*(R/d2))



TATA MOTORS LIMITED

TEST OF HYPOTHESIS

Conducts test through use of test statisticto accept or reject a alternate hypothesis



TATA MOTORS LIMITED

For Internal Circulation Only Six Sigma Central Coordination Group

Key Symbols:

Population

Mean : PStandard Deviation :VProportion : P

Size : N

Sample

Mean : xStandard Deviation :s

Proportion : p

Size : n

Test statistic-Z,t,F



TATA MOTORS LIMITED

A test statistic is used to find out the confidence Interval in which the parameter being Investigated will lie with a100 (1- ) % confidence level. is the probability of type 1 error(normally assumed 0.05).Alternatively if the value of the parameter is given, test statistic value with the parameter value is computed to find out if it lies in the acceptance region of Test statistic 100(1- )% confidence interval(found out from table of distribution of the test statistic).



TATA MOTORS LIMITED

Comparison Testing:

9 A statistical hypothesis is a claim about the value of a population characteristic such as the population mean or variance.

9 In any hypothesis testing situation, there are always two contradictory hypotheses (claims) under consideration. The objective of hypothesis testing is to decide, based on information derived from a sample (because the population is either too large or does not exist), which of the two claims is correct.



TATA MOTORS LIMITED

The two competing statements or claims are called

¾ Null hypothesis (H0) and

¾ Alternative hypothesis(H1) or (Ha).



TATA MOTORS LIMITED

A null hypothesis (Ho) hypothesis is a statement about the population(s) parameters.

Usually, the null hypothesis represents the “status quo” or “nodifference” statement

Null Hypothesis(Ho)

• Usually describes a status Quo

• The one you assume unless otherwise shown

• Sign used are = or ≥, ≤

.



TATA MOTORS LIMITED

Alternative hypothesis (H1)

It represents all possible values for the population parameter except that in the Null Hypothesis. It is the statement that must be true if the null hypothesis is false.

The Alternate hypothesis (Ha)

• Usually describe the difference

• The one you accept or reject based on the evidence

• Sign used are ≠or ,< or ,>



TATA MOTORS LIMITED

Decision MatrixDecision Matrix

CorrectDecision

(1- β)= power of the test.

Type I errorP(type I) = α

Producer’s riskReject H0

Type II errorP(type II)= β

Consumer's risk

CorrectDecisionAccept H0

H1 is true

H0 is false

H0 is trueReality

Decision



TATA MOTORS LIMITED

Errors in Hypothesis Testing

¾ Type , Error : The null hypothesis is rejected when it is actually true. The probability of type Ι error is indicated by D, the level of significance of the test.

¾ Type ,, Error : The null hypothesis is not rejected even though it is false. The probability of a type ΙΙ error is denoted by E.



TATA MOTORS LIMITED

The power of a hypothesis testThe power of a hypothesis test is equal to (1- β). It is the probability of rejecting the null hypothesis when it is false. In general, the more powerful the test, the better it is.

The right way to perform a comparison test is to determine, before the test is conducted, how large of a difference is meaningful and then ensure that the sample size is chosen such that the test has sufficient power (say 80 or 90 percent) to detect that difference.



TATA MOTORS LIMITED

Six steps in hypothesis testingHypothesis testing usually involves the following six action steps:

Step 1:

State the hypothesis. This could be a specific statement about a population parameter.

Step 2:

Choose (D). The alpha factor designates the risk of rejecting the hypothesis if it is actually true. Common values are 0.05 or 0.01



TATA MOTORS LIMITED

Step 3:

Choose the test statistic for testing the hypothesis. This could be a Z-statistic, t-statistic, an F-statistic, or a Chi-square statistic. The statistic used depends on the type of test being conducted.

Step 4:

Determine the acceptance region for the test i.e. the range of values of the test statistic which results in a decision to accept the hypothesis. Define pass/fail criteria of the test.



TATA MOTORS LIMITED

Step 5:

Draw a representative sample from the product population, calculate sample statistics such as the samplemean, sample variance or, in case of discrete values, proportion defectives and apply a hypothesis test using these statistics. Compute the test statistic and compare the obtained value with the acceptance region to make a decision to accept or reject the hypothesis.

Step 6:

Draw an engineering conclusion.



TATA MOTORS LIMITED

Interval estimation consists of finding an interval defined by two end points--- say, L and U --- such that the probability of the parameter θ being contained in the interval is some value (1-α).

P( Ld TdU ) = (1-D)

This represents a two sided confidence interval, with L representing the lower confidence level and U representing the upper confidence level.

Confidence Interval of test statistic



TATA MOTORS LIMITED

¾ Confidence intervals can also be one sided .An interval of the type

Ld T, such that P(L d T) = 1-Dis a one sided lower 100(1- α) % confidence interval for θ .

ORAn interval of the type

T dU, such that P( TdU ) = 1-Dis an upper 100(1-α ) % confidence interval for θ.

Example for Z statistic is shown on next slide



TATA MOTORS LIMITED

Z Curve ( Probability distribution of test statistic z with one tailed

And two tailed rejection region based on α )

Shaded Area = α = P (type 1 error )

Shaded Area = α/2

Shaded Area = α/2

00 0zα -zα -zα/2 zα/2Rejection region:z≥zα Rejection

region:z≤-zα

Rejection region:either z ≥zα/2 or z ≤-zα/2

(a) (b)©

Rejection region for z tests:(a) upper- tailed test; (b) lower-tailed test; © two-tailed test



TATA MOTORS LIMITED

TESTING MEAN EFFECTS DUE TOFACTOR LEVELS

2t test- Mean effect of 2 levels of assignable factorPaired t test-Mean effect of 2 levels of assignable factorOne way ANOVA-Mean effect of multi levels of assignable factor



TATA MOTORS LIMITED

2t test

Fixture1Sample size n1Mean= 1Std.deviation= 1

Fixture2Sample size n2Mean= 2Std.deviation= 2

Is difference between 1 and 2 of the dependentVariable a result of Sampling error or do they come from Different population signifying effectDue to fixture?



TATA MOTORS LIMITED

• Variances of the two populations assumed equal:

We assume that each population is normally distributed. If we assume that the population variances, though unknown are equal, i.e. σ2

1= σ22

An F test needs to be done to verify equality of variance

Hypothesis H0: µ1- µ2= µ0 H0: µ1- µ2 ≤ µ0 H0: µ1- µ2 ≥ µ0

H1: µ1- µ2 ≠µ0 H1: µ1- µ2 > µ0 H1: µ1- µ2 < µ0

Rejection Region |t0|>tα/2,n1+n2-2, t0 >tα,n1+n2-2 t0 < - tα,n1+n2-2



TATA MOTORS LIMITED

(X1- X2) - µ0

( degrees of freedom = n1+ n2 – 1 )

If the distribution is non normal then take n1 & n2 > 30

Test Statistic t0 =

Depending on whether calculated t0 lies in the acceptance or Rejection Region from students t table alternate hypothesis is rejected or accepted respectively.

sp √(1/ n1+ 1/ n2 )

Sp2=(n1-1)s21+(n2-1)s2

2 / n1+ n2-2



TATA MOTORS LIMITED

If the variance of populations cannot be assumed to be equal:

Hypothesis H0: µ1- µ2= µ0 H0: µ1- µ2 ≤ µ0 H0: µ1- µ2 ≥ µ0

H1: µ1- µ2 ≠µ0 H1: µ1- µ2 > µ0 H1: µ1- µ2 < µ0

Rejection Region |t0|>tα/2,ν, t0 >tα, ν t0 < - tα, ν

Test Statistic t0 =(X1- X2) - µ0

√s21/n1+s2

2/n2

Q = (s21/n1+s2

2/n2 )2 (s21/n1

)2+ (s22/n2 )2

n1-1 n2-1(degree of freedom)



TATA MOTORS LIMITED

Example of two sample t test :A large corporation is interested in determining whether the average days of sick leave taken annually is more for night shift employees than for day shift employees.It is assumed that the distribution of the days of sick leave is normal for both shifts and that the variances of sick leave taken are equal for both shifts. A random sample of 12 employees from the night shift yields an average sick leave X1 of 16.4 days with a standard deviation of 2.2 days. A random sample of 15 employees from the day shift yields an average sick leave X2 of 12.3 days with a std. dev. Of 3.5 days. At a level of significance of .05, can we conclude that the average sick leave for the night shift exceeds that for the day shift?

Solution: The hypotheses are H0 :µ1− µ2 ≤ 0 ; H1 : :µ1− µ2 >0

The pooled estimate of the variance,

Sp2 = = 8.990

So Sp = 2.998 25

15/√20Test Statistic t0=

(16.4-12.3) –0 =1.491

t.05,25=1.708. Since the test statistic is less than 1.708 and does not fall in the rejection region, we cannot reject the null hypothesis .Therefore we cannot conclude that there is evidence of average sick leave for the night shift exceeding that for the day shift.

11(2.2)2 + 14(3.5)2



TATA MOTORS LIMITED

A study was performed in order to evaluate the effectiveness of two devices for improving the efficiency of

gas home-heating systems. Energy consumption in houses was measured after each of the devices was

installed. The two devices were an electric vent damper (Damper=1) and a thermally activated vent damper

(Damper=2). The energy consumption data (BTU.In) are stacked in one column with a grouping column

(Damper) containing identifiers or subscripts to denote the population. After a variance test no evidence was

found for variances being unequal. Now we want to compare the effectiveness of these two devices by

determining whether or not there is any evidence that there is significant difference between the means of the

two devices ie testing alternate hypothesis 1- 2 is not equal to zero.BTU.In Damper BTU.In Damper BTU.In Damper BTU.In Damper BTU.In Damper

7.87 1 4 1 6.62 1 6.72 2 7.69 29.43 1 8.58 1 5.2 1 10.21 2 12.19 27.16 1 8 1 12.28 2 8.61 2 5.56 28.67 1 5.98 1 7.23 2 11.62 2 9.76 212.31 1 15.24 1 2.97 2 11.21 2 7.15 29.84 1 8.54 1 8.81 2 10.95 2 12.69 216.9 1 11.09 1 9.27 2 7.62 2 13.38 210.04 1 11.7 1 11.29 2 10.4 2 13.11 212.62 1 12.71 1 8.29 2 12.92 2 10.5 27.62 1 6.78 1 9.96 2 15.12 2 14.35 211.12 1 9.82 1 10.3 2 13.47 2 13.42 213.43 1 12.91 1 16.06 2 8.47 2 6.35 29.07 1 10.35 1 14.24 2 11.7 2 9.83 26.94 1 9.6 1 11.43 2 7.73 2 12.16 210.28 1 9.58 1 10.28 2 8.37 29.37 1 9.83 1 13.6 2 7.29 27.93 1 9.52 1 5.94 2 10.49 213.96 1 18.26 1 10.36 2 8.69 26.8 1 10.64 1 6.85 2 8.26 2

Example



TATA MOTORS LIMITED

Using Minitab on 2 sample T Test Use menus:Stat / Basic statistics /2 Sample T test ...

If data is stacked in one column

Response data

Data population code

Response data of first population

Response data of Second population

If data is two column

Test the equality of variance & then select the required setting



TATA MOTORS LIMITED

Sd. Of sample

Sd/Sqrt(N)

confidence interval is a range of likely values for the difference, µ1 - µ 2

If reference value (0 in this case ) lies outside the confidence interval then reject Null Hypothesis

T= (µ1- µ 2)/(pooled SE Mean)

Note -If P<0.05 reject null hypothesis and accept alternate hypothesisThis means that in this example difference of mean not equal to zero

Minitab Output



TATA MOTORS LIMITED

PAIR1

PAIR2...PAIR n

MATL.HARDNESS MATL.HARDNESS BA B

X1A X1B X1d

X2A X2B X2d

XnA XnB Xnd

DIFF.

Is the average of differences between observations of the dependent variable on the two Material types large enough to indicate that observations are from two different populationsignifying the effect of hardness?

PAIRED t test



TATA MOTORS LIMITED

� Paired t- test :

Often a test can be designed so that the observations on same sample are taken in pairs to increase the ability to detect small difference.

Data are taken in (n) pairs and the difference (d) within each pair is calculated for dependent samples i.e. effect of factor has to be found on same sample.

H0: P d = µ0 H0: P d ≤ µ0 H0: P d ≥ µ0

H1: P d ≠µ0 H1: P d > µ0 H1: P d < µ0

where P d is the population mean of the differences and P 0

is the hypothesized mean of the differences

Hypothesis for Paired T test

Rejection Region |t0|>tα/2,n-1, t0 >tα,n-1 t0 < - tα, n-1



TATA MOTORS LIMITED

t0 =

The null hypothesis is rejected if calculated to lies in the rejection region of t from table

dSd

� n

To calculate the test statistic t0, the mean of the differences Xd

and the sample standard deviation Sd are calculated.



TATA MOTORS LIMITED

A shoe company wants to compare two materials, A and B, for use on the soles of boys’ shoes. In

this example, each of ten boys in a study wore a special pair of shoes with the sole of one shoe

made from Material A and the sole on the other shoe made from Material B. The sole types were

randomly assigned to account for systematic differences in wear between the left and right foot.

After three months, the shoes are measured for wear.

Mat-A Mat-B13.2 148.2 8.8

10.9 11.214.3 14.210.7 11.86.6 6.49.5 9.8

10.8 11.38.8 9.3

13.3 13.6

Example



TATA MOTORS LIMITED

Using Minitab on Paired sample T Test Use menus:Stat / Basic statistics /Paired T test ...

Put the test condition not equal or less than or greater than as required

Confidence interval of test

Null hypothesis

Sample –1 data

Sample –2 data

Click Option



TATA MOTORS LIMITED

Sd. Of sample

Sd/Sqrt(N)

confidence interval is a range of likely values of µd

T= µd/ (pooled SE Mean)

Note -If P<0.05 Or if reference value in this case(0) does not lie in the intervalreject null hypothesis and accept alternate hypothesis

Minitab Output

µd = population mean of the differences



TATA MOTORS LIMITED

ONE WAY ANOVA

One-way analysis of variance tests the equality of population means when classification is by one variable. The classification variable, or factor, usually has three or more levels (one-way ANOVA with two levels is equivalent to a t-test).



TATA MOTORS LIMITED

Tool1Sample size nMean= 1Std.deviation= 1




Do the means of the observation on the dependent variableFor the four tools come from the same population Or are they from different populations signifying the presence

Of a effect?

One way ANOVA



TATA MOTORS LIMITED

The null hypothesis Ho for the test is that all population means ( All level) are the same.

The alternative hypothesis Ha is that one or more population means differ from the others.

Hypothesis for 1 way ANOVA



TATA MOTORS LIMITED

APPLICATION STEPS

Steps to consider when applying a single factor analysis of variance:

1. Describe the problem using a response variable that corresponds to the key process output variable or measured quality characteristic. Example of applications include the following:

a. Customer delivery time is sometimes too long.b. The dimension on a part is not meeting specification.

2. Describe the analysis. Example applications include the following:a. Determine if there is a difference in the mean delivery time of five

departments.b. Determine if there is a difference in the dimension of a part when a

particu1ar setting on a machine is changed to five different levels.3. Test the null and alternate hypotheses.

delivery time of department X.

4321: µµµµ ===oH4321: µµµµ ≠≠≠AH



TATA MOTORS LIMITED

Analysis of variance is a misleading name for a statistical method and model that deals with differences in the means of a variable across groups Since the methodsl employ ratios of variances in order to determine whether the means differ we call it analysis of variance.

Control µ1 , σ1

Controlµ2 , σ2

x1, s1 n= 10

x2, s2 n= 10



TATA MOTORS LIMITED

The following assumptions are made in ANOVA:

1. Independence of observations.

2. Means and variances of population should be independent. I t turns out that this can be only be safely assumed only if population is normally distributed.(or choose n > 30)

3. For an analysis of variance hypothesis test. model errors are assumed lo be normally and independently distributed random variables with mean zero and variance σ2. This variance is assumed constant for all factor levels.



TATA MOTORS LIMITED

2

1 1)(∑ ∑ −=

= =

a

i

n

jijtotal yySS

The overall variation of the data is given by:

Total variability in data as measured by

….(2)

Within-group variability measures the variation about the mean of each

group(level) and is used to estimate the variation within each group or

category in the research problem. Sometimes called "unexplained" or "residual"

variation.I ts is assumed equal for all levels as it reflects part to part variation.

Between-group variability focuses on how the sample mean of each

group(level) differs from the overall or grand mean. Sometimes called

"explained variation” because it is due to group structure.

SS represents Sum of squares which when divided by degree of freedom gives mean square which is a indicatorof variance



TATA MOTORS LIMITED

∑=

−=a

iilsfactorleve yynSS

1

2)(

2

1 1)(∑ ∑ −=

= =

a

i

n

jiijerror yySS

The first element includes a measure of the variance due to differences between the means due to effect of levels as well as any random difference, whereas the second element is due to random error.

errorlsfactorlevetotal SSSSSS +=

yi= mean of individual subgroup

Y = mean of all observationsYij = individual observationsi=sub group levelsj=observation no within subgroup



TATA MOTORS LIMITED

The main goal in ANOVA is to see whether or not the variation betweensample means representing various levels of a factor is significantly greater than that within the samples themselves.A greater variation between the samples than within the samples would suggest that the two groups came from different populations (i.e., a significant difference exists).

1−=

a

SSMS lsfactorleve

lsfactorleve

anSS

MS errorserrors −

=

When divided by the appropriate number of DOF(degree of freedom), these sum of squares gives good estimate of the total variability.

….(6)

….(7)



TATA MOTORS LIMITED

The null hypothesis that there is no difference in factor levels is tested by calculating the F -test statistic:

error

lsfactorleve

MS

MSF =0

Using an F table, we should reject the null hypothesis and conclude there are differences in treatment means if

anaFF −−> ,1,0 α

Minitab also does comparision of mean difference to find significant difference between levels



TATA MOTORS LIMITED

Data shows durability of four types of carpet.To examine whether means of all four carpets are same or mean of at-least one carpet is significantly different



TATA MOTORS LIMITED

Using Minitab on 1way ANOVA test Use menus:Stat / ANOVA /One way ...

Response Variable

Factor Level of data

Click Ok



TATA MOTORS LIMITED

No of observation of

each level Mean of Each level

Common Sd of all level

If the intervals for two means do not

overlap, it suggests that the population means are different

Mean of Each level

Note -If P<0.05 reject null hypothesis and accept alternate hypothesisThere is no difference in mean in given example as P>0.05.

Minitab Output



TATA MOTORS LIMITED

Stat-ANOVA-General Linear Model-comparisons

COMPARISON OF MEAN THROUGH MINITAB



TATA MOTORS LIMITED

Since all intervals contain zero none of the differences are significant.If an interval does not contain zero the difference is significant and must be investigated.

Interval for Diff. Of mean of carpet1-mean ofcarpet2



TATA MOTORS LIMITED

Comparison of mean effect of Nested factors (fixed)

A factor B is said to be nested within A if all levels of B can only existas a sublevel of A



TATA MOTORS LIMITED

machine

Spindle 1 Spindle 2

tool1 tool2 tool1 tool2

Is the mean effect of spindle and tool significant?

All the factors are fixed and one has to find whether significant difference between the two spindles or tools exist



TATA MOTORS LIMITED

The above structure shows the readings of a dependent variable for 6 spindles of a machine with two tools per spindle for 4 time periods.To investigate whether differences between spindle and tool are significant



TATA MOTORS LIMITED

Stat-ANOVA-General Linear Model

Shows toolNested withinSpindle and partNested within tool

Time is assumedRandom factor



TATA MOTORS LIMITED

The p value of <0.05 for spindle denote that the one of the spindle means is Different from the others and requires investigation.



TATA MOTORS LIMITED

COMPARISON OF VARIANCEANOVA with random factors(nestedFor hierarchical structure,SHANIN range method)

F testLevene’s test( Minitab)



TATA MOTORS LIMITED

Nested ANOVA with Random factorsIs used in the initial stages of a exercise to find Out variance due to identified assignable factors with multiple levels mostly nested in a family tree.The selection of levels is a random selection from population.



TATA MOTORS LIMITED

For the random factor model as discussed in the previous slide the among levels mean square for a factor estimates the Variability of the levels of a factor and any variability in this estimate comes as a result of variability in the estimate of 2 and sample to sample differences in levels selected at random from a population. If the factors are nested we use nested ANOVA with all treatments(levels) for a factor asrandom and the P value <0.05for each factor indicates variability of treatment means>0.



TATA MOTORS LIMITED

Example An engineer trying to understand the sources of variability in the manufacture casting. The process of making the casting requires mixing materials in small furnaces for which the temperature setting is to be 475° F. Company has a number of plants where the Casting are made, Engineer select four as a random sample. He conduct an experiment and measure furnace temperature three times during a work shift for each of four operators from each plant over four different shifts

Plant 1 Plant 2 Plant 3 Plant 4

Operator -1

Batch 1

Operator -2 Operator -3 Operator -4

Shift –11 Shift –12 Shift –13 Shift –14

Batch 2 Batch3

Df = (N-1)=(4-1)=3

Df = (N-1)*Nplnat =(4-1)*4=12

Df = (N-1)*N(ope)r*N(plant)=(4-1)*4*4=48

Df = (N-1)*N(shift)*N(oper)*N(plant)=(3-1)*4*4*4=128

(factors chosen at random from a population)

Nested structure



TATA MOTORS LIMITED

Using Minitab to obtain nested ANOVA Use menus:Stat / ANOVA / Fully Nested ANOVA ...

Response Variable

Factors in hierarchical order ( Higher To lower order based on Nested Diagram )

Hypothesis for Nested ANOVA

H0 (the null hypothesis): That the Variances for the factors is Zero

H1 (the alternative hypothesis): That the Variances for the factors is significantly different from Zero



TATA MOTORS LIMITED

Minitab Output Degree of Freedom for the each factor

Sequential Sum of square for the factor

P value is based on the following hypothesis test Ha = Variance of the factor is not ZeroHo = Variance of the factor is ZeroIf P value is <0.05 for α =0.05reject H0

Variance of the Component

% of total Variance

“B” Vs. “C”

OBJECTIVE :

• VALIDATES A BETTER OR A SUPERIOR PRODUCTOR PROCESS (B) OVER CURRENT ONE (C) WITH ADESIRED STATISTICAL CONFIDENCE.(UAUALLY 95%)

• EVALUATES ENGINEERING CHANGES

PROCEDURE FOR “ B” Vs “ C”

STEP 1 : CHOOSE AN ACCEPTABLE LEVEL OF “ D” RISK(ASSUMING IMPROVEMENT WHEN NO IMPROVEMENT EXISTS)

STEP 2 : DECIDE ON SAMPLE SIZES FOR “ B” AND “ C”TESTS

STEP 3 : RANDOMIZE AND CONDUCT THE TESTSTEP 4 : RANK ORDER THE RESULTSSTEP 5 : DECISION RULESSTEP 6 : SEPARATE THE MEANS - THE “ ß” RISK

(ASSUMING NO IMPROVEMENT WHEN IMPROVEMENT EXISTS)

B Vs. CDECISION RULES

• No overlap End count techniques:B B B C C C

Best WorstB end count = 3 ; C end count = 3Total end counts = Total no. of rankings ( No overlap )• The Overlap End count techniques:B B B C B C C C C

Best WorstB end count = 3 ; C end count = 4Total end counts = 7Confidence Level End Counts >

90 695 7

99 10• Separate the meansX - bar ( B) - X - bar ( C ) > K V



TATA MOTORS LIMITED

SHANIN RANGE METHOD(Previous example evaluated)



TATA MOTORS LIMITED

plant1 operatorshift 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4Batch

1 17 18 12 22 11 19 10 20 15 0 17 16 5 17 21 1014.46 2 12 15 15 17 14 22 17 13 12 9 23 20 4 15 17 15

3 21 14 8 14 10 17 23 18 10 12 15 11 11 11 15 14

Range batch 9 4 7 8 4 5 13 7 5 12 8 9 7 6 6 5avg.shift 16.7 15.7 11.7 17.7 11.7 19.3 16.7 17 12.3 7 18.3 15.7 6.67 14.3 17.7 13range of mean of shift w ithin operatoraverage of operatorrange of operator w ithin plant

plant2 1 24 17 19 17 12 12 12 22 25 22 17 19 17 16 15 2117.29 2 17 22 17 10 15 15 17 17 21 23 16 16 15 11 15 10

3 21 21 22 19 15 10 15 23 17 25 21 25 16 12 12 12

Range batch 7 5 5 9 3 5 5 6 8 3 5 9 2 5 3 11avg.shift 20.7 20 19.3 15.3 14 12.3 ‘ 20.7 21 23.3 18 20 16 13 14 14.3range of mean of shift w ithin operatoraverage of operatorrange of operator w ithin plant

Plant3 1 15 17 9 17 10 11 18 17 6 13 11 15 9 13 17 311.98 2 10 11 13 15 6 13 20 11 5 15 9 17 11 15 10 11

3 9 14 8 13 8 16 14 9 11 18 11 12 8 13 9 9

Range batch 6 6 5 4 4 5 6 8 6 5 2 5 3 2 8 8avg.shift 11.3 14 10 15 8 13.3 17.3 12.3 7.33 15.3 10.3 14.7 9.33 13.7 12 7.67range of mean of shift w ithin operatoraverage of operatorrange of operator w ithin plant

plant4 1 24 16 15 21 9 22 17 12 10 15 9 25 9 17 10 2315.79 2 17 15 10 16 15 23 19 16 21 10 17 19 13 13 8 17

3 20 14 9 12 19 19 15 19 21 15 22 14 15 11 14 16

Range batch 7 2 6 9 10 4 4 7 11 5 13 11 6 6 6 7avg.shift 20.3 15 11.3 16.3 14.3 21.3 17 15.7 17.3 13.3 16 19.3 12.3 13.7 10.7 18.7range of mean of shift w ithin operatoraverage of operatorrange of operator w ithin plant

range of plantmax range of operatormax range of shiftmax range of batch

11.3313

3.25

2.08

5.316.25

15.75 17.08 16.50 13.839 7 6 8

12.58 12.75 11.92 10.67

14.336.25

5.00 9.33 8.00 6.00

18.83 15.42 20.58

3.25

5.33 8.33 5.33 3.00

11.0015.42 16.17 13.33 12.926.00 7.67 11.33

1 2 3 4

Ran

ge M

e tho

d o f

Sha

n in



TATA MOTORS LIMITED

F test



TATA MOTORS LIMITED

� Hypothesis Testing for the ratio of two variances: (F-test)

We assume that both populations are normally distributed.

Hypotheses H0: σ2= σ20 H0: σ2≤ σ2

0 H0: σ2≥ σ20

H1: σ2≠σ20 H1: σ2>σ2

0 H1: σ2< σ2

0

Rejection Region F0> Fα/2,n1-1,n2-1, F0 > Fα,n1-1,n2-1 F0 < F(1-α),n1-1,n2-1

F0 < F(1-α/2 ,n1-1,n2-1

s21

s22

Test Statistic F0= ( d.o.f.: n1-1,n2-1)

(



TATA MOTORS LIMITEDExample of F-test :The variabilities of the service times of two bank tellers are of interest. We want to determine whether the variance of service time for the first teller is greater than that for the second. A random sample of 8 observations from the first teller yields a sample average of 3.4 min. with a standard deviation of 1.8 min. A random sample of 10 observations from the second teller yields a sample average of 2.5 min. with a standard deviation of 0.9 min. Can we conclude that the variance of the service time is greater for the first teller than for the second? Use a level of significance of 0.05.

Solution:

The hypotheses are H0: σ21≤ σ2

2 H1: σ21>σ2

2

The test statistic is F =

(1.8)(1.8)2

(0.9)2= 4.00

From table F0.05,7,9= 3.29. The test statistic lies in the rejection region, and so we reject the null hypothesis.



TATA MOTORS LIMITED

LEVENES TEST



TATA MOTORS LIMITED

In a experimental study conditions conducive to potato rot by injecting potatoes with bacteria that cause rotting and subjecting them to different temperature and oxygen regimes. Check the equal variance assumption.

Rot Temp13 1011 103 1010 104 107 1015 102 107 1026 1619 1624 1615 1622 1618 1620 1624 168 16

Example



TATA MOTORS LIMITED

Using Minitab on Variance test Use menus:Stat / ANOVA /Test for equal of variance test ...

Response Variable

Factor Level of data

Null hypothesis (Ho) is that the population variances under consideration (or equivalently, the population standard deviations) are equal.

Alternative hypothesis (H1) is that not all variances are equal.



TATA MOTORS LIMITED Minitab Output

Note -If P<0.05 reject null hypothesis and accept alternate hypothesis

This means that in this example variances are not equal to each others

Levene’s test is not sensitive to assumption of normality



TATA MOTORS LIMITED

SHANIN DOE



TATA MOTORS LIMITED

PAIRED COMPARISON

Are incoming parameters on product affecting the response?



TATA MOTORS LIMITED

Why is it required ?

1. To identify what suspect product characteristic or quality parameter is contributing to the problem.

2. This tool can be used only if there are suspect product parameters ( hardness, yield strength, ovality,etc.) that are contributing to the problem.



TATA MOTORS LIMITED

. Identify eight Very good parts and eight Very bad parts which are having the problem that is under analysis.

. List as many Product characteristics/parameters which may explain the difference between the Good and Bad parts (This list is based on the engineering judgment of the person/team, but later will be proved using this tool). List it in the descending order or importance to the problem .

. Measure all the Good parts and bad parts for the parameters identified above. There will then be totally 16 values.

. Arrange the 16 values in the rank order (start from the smallest to the largest or the largest to the smallest) irrespective of whether they are good or bad.

Steps followed in paired comparison :



TATA MOTORS LIMITED

. Write against each value whether that particular value corresponds to a bad or Good part. If it belongs to the good part write ‘G’ within bracket after the value. If it belongs to the bad part, write ‘B’ within bracket after the value.

. From the top, find out when for the first time, the Good changes to Bad (or) the Bad changes to Good.

. Draw a line at the change point (For e.g.., if the change occurs after the 5th data, draw a line after the fifth data).

. From the bottom, find out when for the first time, the Good changes to Bad (or) the Bad changes to Good.

. Draw a line at the change point (For e.g.., if the change occurs after the 15th value from the bottom, draw a line between 15 & 16).



TATA MOTORS LIMITED

. Find out the top count (Count the number of values till the line is drawn).

. Find out the Bottom count (Count the number of values till the line is drawn from the bottom).

. Add both these counts to find the Total count.

. If the Total count <=5, then that quality parameter or productcharacteristic is not the reason for the problem.

. If the Total count is = 6, it can be concluded that at 90% confidence level, it is this quality parameter or product characteristic that is leading to the problem.



TATA MOTORS LIMITED

Rules for Count1. If first & last parameter are of same family( Both Bad or both good ) then there is no significance .

2. At the point of transition ( when good converting to bad or Bad converting to good ) if same two values appeared then we subtract ½ from the count for the two value.

3. At the point of transition ( when good converting to bad or Bad converting to good ) if same three or more values appeared then we delete the entire block from count.



TATA MOTORS LIMITED

Confidence Level for other counts:

Confidence level90%95%99%99.90%

Number of Total count671013



TATA MOTORS LIMITED

The DOE team suspected the hardness of the shell as one of the product characteristics that could lead to crack . Pair comparison was done to find out whether this indeed was the parameter leading to the crack The eight good(Best of Best) & eight bad (Worse of Worse) are selected & it’s results are shown below

Hardness Value Component 55 B54 B

54.5 B49 B51 B50 B52 B

51.5 B46 G42 G42 G49 G

51.5 G50 G44 G

Example



TATA MOTORS LIMITED

The results after arranging in the rank order ( Lower To Higher)

Hardness Value Component 42 G42 G44 G46 G49 B49 G50 B50 G51 B

51.5 B51.5 G52 B54 B

54.5 B55 B

Top Count= 4

Bottom Count= 5-12 =4.5

Total Count =8.5 ,explains that the hardness is creating the problem at 95% confidence level



TATA MOTORS LIMITED

Process parameter search

Does a process parameter have significant impact on the output?



TATA MOTORS LIMITED

/ PROCESS SEARCH

OBJ ECTIVE :

�Process Search separates

important Process parameters from

unimportant ones.



TATA MOTORS LIMITED

Met hodology :

1. If process parameters variations are suspected as possible causes for good and bad finished products, make a list of these process parameters, in descending order of likelihood.

2. Determine how each process parameter is to be measured, who is to measure it, and where precisely it is to be measured.

3. Make sure that the accuracy of the measuring instrument is at least 5 times the accuracy ( tolerance) of the process parameter.



TATA MOTORS LIMITED

Met hodology : conti….

4. Make sure that the actual process parameters, not just mere settings, measured.

5. If a particular process parameter does not vary at all while being monitored, it can be eliminated from further consideration.

6. Run a 100 % sample of units or a multi-vari sample of units until :¾ A minimum of eight good units and eight bad units are collected

at the end of the process.

¾ The spread between the best unit and the worst unit is a minimum of 80% of the historic variation observed on the product produced by the process.



TATA MOTORS LIMITED

Met hodology :conti….

7. Measure all the designated process parameters associated with each unit.

8. As the units go through the process, a determination of whether the units are going to be good or bad cannot be made until the end of the process. This means that many potentially good units will have to be measured, along with many potentially bad units, in order to obtain a minimum of eight eventually bad units.

9. A Paired Comparison of the process parameters associated with the eight good and eight bad units is then run. And a Tukey test is performed on each parameter, with total end count calculation.



TATA MOTORS LIMITED

Met hodology :conti…..10. If there are several process parameters identified with

90% or greater confidence, run a “ B verses C” test to verify that the important parameters are truly captured.

11.Next, a variable search or a full factorial should be run to quantify the most important parameters and their interaction effect.

12.Further optimization, through Scatter Plots or Evolutionary Operation, followed by Positrol, Process Certification, and Pre-control should be conducted.

13. The tolerances of the unimportant parameters in step 8 can be expanded to reduce costs, although some experimentation may be necessary to determine how much to expand them.



TATA MOTORS LIMITED

Cooling Time Zone Temp. Molding TempSpec 3Sec 415-425 Deg. 100-110Good 3.1 423 106Good 3 415 103Good 3.1 422 105Bad 2.9 422 109Bad 2.8 419 101

Good 3 419 105Bad 2.9 417 102

Good 3 418 105Good 3 420 105Bad 3 420 108Bad 2.8 417 108Bad 2.8 414 104

Good 3 414 103Good 3 420 105Bad 2.8 415 102Bad 2.8 416 103

In a foundry process following process parameters where identified as the suspects for this problem.A/ Cooling timeB/ Zone temperatureC/Mould temperatureThe data collected for 8 good & 8 bad parts as given below.

Identify the parameter that is creating the problem.

Example



TATA MOTORS LIMITED

Spec 3SecCooling Time

Bad 2.8Bad 2.8Bad 2.8Bad 2.8Bad 2.9Bad 2.9

Good 3Good 3Good 3Good 3Bad 3

Good 3Good 3Good 3.1Good 3.1

Bad 2.8Bad 2.8Bad 2.9Bad 2.9

Good 3.1Good 3.1

Spec Cooling Time

The results of cooling time after arranging in the rank order ( Lower To Higher)

Based on Rule 4 of paired comparison Total block of data is removed.

Reduce matrix after Block removal

4

2

Total Count =6,explains that the cooling time is suspected parameterfor the problem at 90% confidence level



TATA MOTORS LIMITED

Spec 415-425 Deg.Zone Temp.

Bad 415Good 415Bad 416Bad 417Bad 417

Good 418Good 419Good 420Bad 420

Good 420Bad 422

Good 422Good 423

1/2 1

Spec 100-110Molding Temp

Bad 102Good 103Bad 103

Good 105Good 105Good 105Good 105Good 105Good 106Bad 108Bad 108Bad 109

2-1/2

3

The results of Zone Temp after arranging in the rank order ( Lower To Higher)

The results of Molding Temp after arranging in the rank order ( Lower To Higher)

Total Count =2,explains that the Zone temp is not suspected parameter

Total Count =4,explains that the Molding is not suspected parameter



TATA MOTORS LIMITED

Regression Analysis

ESTABLISHES THE RELATION SHIP BETWEEN A CONTINUOUS INPUT VARIABLE AND A CONTINUOUS OUTPUT VARIABLE AND ALSO HELPS IN FINDING OUTTHE TOLERANCE OF INPUT VARIABLE REQUIRED TO ACHIEVE OUTPUT SPECIFICATION



TATA MOTORS LIMITED

Simple Regression Analysis

The regression analysis is performed when

there is single X(Predictor)



TATA MOTORS LIMITED

What is Regression AnalysisRegression analysis is a statistical technique used to model & investigate or predict

the relationship between dependent variable & the independent variable.

Where Y is the response, X is the predictor, and are the regression coefficients,

e the error terms with normal distribution with mean equal to zero & standard deviation

equal to sigma

The regression analysis perform when the X &Y are continuous .

Regression Is a hypothesis test

Ha :The model is significant predictor of the response

Ho : The model is not a significant predictor of the response



TATA MOTORS LIMITED

Using Minitab to obtain regression equation

Use menus:

Stat / Regression / Regression ...

Dependent variable

Independent variable

(Mintab labels as ‘Predictor’)

Click Graph



TATA MOTORS LIMITED

Select following order

Analysis of residuals ( Difference between the observed value & corresponding fitted value )

Residual analysis investigate adequacy of fitted model & detecting

departures from the model

By residual analysis technique

1. Check the normality assumption through a normal probability plot and /or histogram of the residual

2. Check the correlation between residuals by plotting residuals in time sequence

3. Check the correctness of the model by plotting residual versus fitted values.



TATA MOTORS LIMITED

Analysis of Residuals

1. Normality Assessment : The histogram plot of residuals should look like a sample

from normal distribution which is the assumption for regression analysis.

2. Time Sequence :A plot of residuals against the observation order as a graphical

way of detecting a correlation that may exist between random errors.

3. Fitted Value : Residuals plotted against the fits should be randomly scatter &

having no pattern .

Check for outliers ,non constant variance ( Difference in the residual values either

increases or decreases for the increase in the fitted value ) & poor model fit

( Residual value tend to increase and then decrease with an increase in the fitted

value )



TATA MOTORS LIMITED

Regression Analysis: Drug_conc versus Height

The regression equation isDrug_conc = 79.3 - 6.30 Height

Predictor Coef SE Coef T PConstant 79.296 6.023 13.17 0.000Height -6.296 1.176 -5.35 0.000

S = 9.751 R-Sq = 68.8% R-Sq(adj) = 66.4%

Analysis of Variance

Source DF SS MS F PRegression 1 2726.0 2726.0 28.67 0.000Residual Error 13 1236.0 95.1Total 14 3962.0

Minitab output

Regression equation

Regression Analysis: Drug_conc versus Height

The regression equation isDrug_conc = 79.3 - 6.30 Height

Predictor Coef SE Coef T PConstant 79.296 6.023 13.17 0.000Height -6.296 1.176 -5.35 0.000

S = 9.751 R-Sq = 68.8% R-Sq(adj) = 66.4%

Analysis of Variance

Source DF SS MS F PRegression 1 2726.0 2726.0 28.67 0.000Residual Error 13 1236.0 95.1Total 14 3962.0

Minitab output

Regression equation

P value of predictor is based on the following hypothesis test Ha = Coefficient is not zero Ho = Coefficient is zero If P value is <0.05 for α =0.05 reject Ho or coefficient of predictor is not zero

P value of regression is based on the following hypothesis test Ha = The model is a significant predictor of response Ho = The model not a significant predictor of response If P value is <0.05 for α =0.05 reject Ho or the model having significant predictor of response

Value represent the proportion of variation in the response data explained by the predictorsIn general , the closer is to 1 ,The better the fit of the model

R2 Adj(R-Sq) is the % of variation explained by the model adjusted for the no. of terms in your model & no. of data points

R2

S represent the std.error of the prediction = Sigma of individual data point

P value of constant is based on the following hypothesis test Ha = Coefficient is not constant Ho = Coefficient is constant If P value is <0.05 for α =0.05reject Ho or coefficient is not constant



TATA MOTORS LIMITED

Using Minitab on Regression Analysis :Confidence & prediction Band Use menus:Stat / Regression / Fitted Line Plot ...

Dependent variable

Independent variable

(Minitab labels as ‘Predictor’)

Click Option

Click both the box

Click Ok



TATA MOTORS LIMITED

Minitab Output

A confidence band ( or interval ) is a measure of the certainty of the shape of the fitted line at 95% confidence interval .

A prediction band ( or interval ) is a measure of the certainty of the scatter of individual points about the regression line

The regression line always passes through the average value of the cause variable X and the average value

of the response variable Y.

The prediction interval & confidence interval are at a minimum width at X = X bar.



TATA MOTORS LIMITED

Use of Regression plot in Tolerancing The prediction interval is used in statistical tolerancing

USL

LSL

Note1 -In positive co-relation The USL should cut the upper prediction line & on negative co-relation vice versa Note2 – If the line do not cross in a negative relationship or line cross inpositive relationship, you can not use this relationship to control the output

Shows the Band for controllingX to meet tolerance



TATA MOTORS LIMITED

Multiple Regression Analysis

The regression analysis is perform , when

there is Multiple X(Predictor)



TATA MOTORS LIMITED

yi = β0 + β1x1i + β2x2i + ... + βkxki

where yi = a value of the dependent variable, y

β0 = the y-intercept

x1i, x2i, ... , xki = individual values of the independent variables, x1, x2, ... , xk

β1, β2 ,... , βk = the partial regression coefficients for the independent variables, x1, x2, ... , xk

Regression Is a hypothesis test

Ha :The model is significant predictor of the response

Ho : The model is not a significant parameter.

Multiple linear regression examines the linear relationships between one continuous response(Y) and two or more predictors (X).



TATA MOTORS LIMITED

Using Minitab on Multiple Regression Analysis :Using Subset Method

The Regression analysis using subset method is used to select or identify a group of predictor variable (X’s) which are important. This technique is used to funnel group of predictor to important predictors. These important predictors are used further in multiple regression analysis.



TATA MOTORS LIMITED

Using Minitab to obtain regression equation by best Sub –Set methodUse menus:

Stat / Regression / Best subset ...

Dependent variable

Independent variables

(Minitab labels as ‘Predictors)



TATA MOTORS LIMITED

Highest Sum of square

Lowest C-p(which is used to determine minimum no. of parameter which best fits the model . It measures Sum of square of biases + squares of random error in Y at all n data points)

Using given criteria, the

model (row 5) with four

predictors,( Score-2,

insulation ,East ,South &

north appears to be the best

among all the X’s

Small Sd of error in the model

Note : Start with the single factor & select the variable having highest R-sq(adj) & lowest C-P ( In single factor) . Then take two factor & so on.. . Stop When C-P value become minimum & then increases

Do multiple regression on these Selected Predictor variable

Minitab Output



TATA MOTORS LIMITED

Using Minitab to obtain regression equationUse menus:

Stat / Regression / Regression ...

Dependent variable

Independent variables

(Mintab labels as ‘Predictors)



TATA MOTORS LIMITED

Value represent the proportion of variation in the response data explained by the predictorsIn general , the closer R- Sq is to 1 ,The better the fit of the model

R2Adj(R-Sq) is the % of

variation explained by the model adjusted for the no. of terms in your model & no. of data points

P value of regression is based on the following hypothesis test Ha = The model is a significant predictor of response Ho = The model is not a significant predictor of response If P value is <0.05 for α =0.05 reject Ho or the model is a significant predictor & response

P value of predictor is based on the following hypothesis test Ha = Coefficient is not zero Ho = Coefficient is zero If P value is <0.05 for α =0.05 reject Ho or coefficient of predictors are not zero

Minitab Output Regression Equation P value of constant is based on the

following hypothesis test Ha = Coefficient is not constant Ho = Coefficient is constant If P value is <0.05 for α =0.05reject Ho or coefficient is not constant




Total Sum of all Seq SS is SS Regression given in previous table sum



TATA MOTORS LIMITED

Select following order

Analysis of residuals ( Difference between the observed value & corresponding fitted value )

Residual analysis investigate adequacy of fitted model & detecting

departures from the model

By residual analysis technique

1. Check the normality assumption through a normal probability plot and /or histogram of the residual

2. Check the correlation between residuals by plotting residuals in time sequence

3. Check the correctness of the model by plotting residual versus fitted values.



TATA MOTORS LIMITED

Design Of Experiment



TATA MOTORS LIMITED

Y=f(X1,X2,X3,X4)

Find out the optimum combination of levels of x1,x2,x3,x4 Which will optimize Y



TATA MOTORS LIMITED

The above table shows the durability values of pins found with combination of factors namely vendor size of clip and heat-treated state of clip.Each has been tried out at two levels.Are the three factors and their interaction Significant?



TATA MOTORS LIMITED

Each row in the previous slide is a experimental condition with a combination of various level of the factor coded as -1 and +1.Each experimental condition has a output that is measured during experimental run.The example in the previous slide is a 23 factorial experiment with two replication thus requiring 16 experimental runs.Once the output of all experimental condition using all possible combination of factor levels is obtained these are than analyzed to find out the mean effect of the factor and the interactions.



TATA MOTORS LIMITED

What is a Interaction

•If we get a graph like the one shown above, there is an interaction between the engine and the carburetor.

•The engine contributes to an increase in mileage of 5 when the carburetor is bad, whereas it contributes to an increase in 13 when the carburetor is good.

•The carburetor contributes to an increase in mileage of 2 when the engine is bad, whereas it contributes to an increase of 10 when the engine is good.

0

5

10

15

20

25

30

E- E+

Mile

age

25

10 13



TATA MOTORS LIMITED

Before any formal data collection can start planning of experiment has to be done to ensure replication and randomization

• Replication

-- Multiple execution of all or part of the experimental process with the same factor setting.

-- It is done to measure the experimental variability. By doing this we can decide whether the difference in response is due to change in factor levels ( an induced special cause) or common cause variability i.e. noise factor ( which is uncontrollable)

-- to obtain the responses for each set of experimental conditions i.e.. Location & spread

• Properties of Replication

-- It estimates experimental error This estimate becomes the unit of measurement for determining whether observed difference are statistically significant.

-- It Yields more precise estimate of factor effects.



TATA MOTORS LIMITED

Randomization

• Randomization -- To assign the order in which the experimental trails will be run using a random

mechanism *It is not the the standard order * It is not running in an order that is convenient * Computer help or random table help is taken in randomization

-- this is done to average out the effect of lurking variables over all the factors of experiment & help to validate statistical conclusions made from the experiment

• Lurking Variable -- A variable that has an important effect & yet is not included among the factors

under consideration because* Its existence is unknown *. Its Influence is thought to be negligible * data on it are unavailable

• Experimental trial are randomized to protect the effect of lurking variable



TATA MOTORS LIMITED

.FortunatelyMinitab helps us formulating the structure.In this volume we will be limiting ourselves to creation of factorial design of Experiments with 2 levels through minitab.Any DOE experimentation has to gothrough the following steps of analysis.

•Perform analysis¾Identify factors¾Choose factor levels¾Select design¾Randomize runs¾Collect data¾Analyze data¾Draw conclusions¾Verify results

•Determine solutions•Record results•Standardization



TATA MOTORS LIMITED

The project is to increase the durability of clip. The durability of clip is measured by number of bends a paper clip will withstand before breaking .The vendors are claiming improvement in durability due to new heat treatment process.As heat treatment procedure was identified as a important parameter from cause andeffect along with clip size and vendor a DOE is planned with twovendors,two clip sizes for heat treated and non heat treated clips.

Identify factors

example



TATA MOTORS LIMITED

Creating factorial design Creating factorial design



TATA MOTORS LIMITED

Input factors Input factors



TATA MOTORS LIMITED

Random design Random design

Select randomize runsTo create run order and experimental conditions



TATA MOTORS LIMITED

Data Sheet

The order in which experiment was run after randomizing



TATA MOTORS LIMITED

Analyze data–Is data OK

•Plot the raw data in time order by factors-• Assess process stability over time to see the presence of

manufacturing trends and systematic influences.

•Do box plot to identify outliers.to focus quick improvement efforts on the outliers and reduce variation

•Do one way ANOVA to find out if any experimental level issignificant with respect to residuals

•Plot the residuals•Plot the residuals against time order to ensure the experimentalvariability has only common causes and absence of lurking variable (trends,outliers,or non-random patterns) that might influence conclusions.



TATA MOTORS LIMITED

Using Minitab on Time Order Plot Use menus:Stat / Quality Tools /Run Chart ...

Response Variable

Subgroup Size of data

If P<0.05 then reject the Null hypothesisIn this case P value is grater then 0.05 for Mixture Trend & oscillation & clustering

Check for any Non random Patterns



TATA MOTORS LIMITED Using Minitab on Box Plot Use menus:Stat / Graph /Box plot ...

Y Variable & X category

Display Type in the graph

The Box Plot is graphical tool to look at several aspect of data Such as spread , symmetry & outliers.

The left most edge of the box plot represent the first quartile (25% of points ) . The right most edge represents the third quartile (75% of points) The centre line represent the 50th

percentage or median . A outlier is any points at the distance grater then 1.5times the width of the box from the first or third quartile , plotted with Asterisks (*)

Click Option



TATA MOTORS LIMITED

Bend * Exp_Cond

Bend * Size

Bend * Vendor

Bend * HeatBox Plot of Bend

Check the highest variability Exp. Condition

There is difference between the heat treatment & Non

heat treatment

There is slight difference between the vendors

There is no outliers

Minitab Output There is slight difference

between Sizes



TATA MOTORS LIMITED

Using Minitab for 1 Way ANOVAUse menus:Stat /ANOVA /1 Way ANOVA ...

Response Variable

Use each condition of the experimental condition column as factors

Click Ok

The null hypothesis Ho for the test is that all means for all experimental condition are the same.

The alternative hypothesis Ha is that one or more experimental condition has different mean.

Hypothesis for 1 way ANOVA



TATA MOTORS LIMITED

Minitab Output Note -If P<0.05 reject null

hypothesis and accept alternate hypothesisThere is difference in mean of experimental condition.

If the Sp (pooled Std. Deviation ) is very large any factor effects may

appear insignificant



TATA MOTORS LIMITED

•Plot the residuals

•Plot the residuals against time order to ensure the experimental variability has only common causes and absence of lurking variable (trends,outliers,or non-random patterns) that might influence conclusions.

•Plot the residual against average fits on Minitab for each experimental condition-look for presence of trends like megaphone shape.Note:ignore the pattern indicated by symmetry of dots around 0.

•Plot residuals on normal probability scale-any non normal trend indicates the variation in the experiment is non random and assignable causes are present beyond ones identified for experiment.



TATA MOTORS LIMITED Using Minitab for Residual Plot Use menus:Stat /Doe /Factorial /Analyze Factorial Design ...

Response data

Click Graph

Click four residual Plot




Check for any non random patterns or

outliers

Check for any correlation between &

residual

Check for normal distribution of

residuals



TATA MOTORS LIMITED

Action For Residual Plot

• If there are outliers , try to find an explanation & correct the

data If not able to find an explanation , leave the data as it is.

• If there is trend in the residuals over time ,there may be lurking

variable . Look for the cause & correct the situation.

• If there is a correlation between the residuals & fits ,understand

the cause of relation .

• If the residual are not normally distributed ,there may be some

significant factors which are not accounted . Look for that factor

& include it in the experiment



TATA MOTORS LIMITED

Analysis –Make Inferences

The main aim of analysis is to understand which Xs and interaction affect the Y 1.Examine the factor Effect Using Minitab for Pareto Use menus:Stat /Doe /Factorial /Analyze Factorial Design ...

Response data

Click Graph



TATA MOTORS LIMITED

Click Pataro

Pareto chart of the effects to compare the relative magnitude and the statistical

significance of both main and interaction effects in decreasing order of the

absolute value of the standardized effects and draws a reference line on the chart.

Any effect that extends past this reference line is significant.



TATA MOTORS LIMITED

The null hypothesis Ho Factor has no effect on the results.

The alternative hypothesis Ha is that Factors has an effect on the result.

Analyze Factorial Design

Note -If P<0.05 reject null hypothesis and accept alternate hypothesisOnly Heat & vendor *Size Interaction have a significant affect on the number of bends

•Conform impressions with statistical Procedures

The sum of square for the main effects , 2-way interactions , &3-way interactions represent SSBThe residual Error represent SSW.Further exploration of sum of squares is done throughANOVA(GLM)

The Equation is Bend = 15.688-0.437V+0.563S+4.063H-2.563V*S



TATA MOTORS LIMITED

Analysis –confirm Inferences (Practical Significance) Practical Significance is evaluated by analyzing the sum of Square (SS) andsubsequently mean square.

Using Minitab for ANOVA GLM Use menus:Stat /ANOVA /General Linear model ...

Response Variable

Model contain the main effect & Interaction (Crossed ) for the factors



TATA MOTORS LIMITED

Minitab Output The null hypothesis Ho Factor has no effect on the results.

The alternative hypothesis Ha is that Factors has an effect on the result.

Note -If P<0.05 reject null hypothesis and accept alternate hypothesisOnly Heat & vendor *Size Interaction have a significant affect on the number of bends



TATA MOTORS LIMITED

Using Minitab for factorial Plot Use menus:Stat /Doe /Factorial /Factorial Plot ...

Select the type of plot required

The center value for the plot

Select the Factor for the Plots



TATA MOTORS LIMITED Use a main effects plot to determine which factors

influence the response and to compare the relative

strength of the effects

• If the line is horizontal (parallel to the x-axis), there is no main effect present. The response mean does not change depending on the factor level.

• If the line is not horizontal (parallel to the x-axis), there is a main effect present. The greater the degree of departure from being horizontal, the stronger the effect.

Use an interaction plot to determine if two factors interact in their effect on the response and to compare the relative strength of the effects.

• .If the lines are parallel to each other, there is no interaction present. The change in the response mean from the low to the high level of a factor does not depend on the level of a second factor.

• If the lines are not parallel to each other, there is an interaction present. The change in the response mean from the low to the high level of a factor depends on the level of a second factor. The greater the degree of departure from being parallel, the stronger the effect.



TATA MOTORS LIMITED

Suppose one finds seven significant factors.With 2 replications this will require 256 experiments.Conducting so many experiments will consume lot of time .In such cases minitab allows fractional factorial.The only disadvantage with fractional factorial is that the higher order interactions are Confounded.The confounding is determined by degree of resolution and fraction applied.

SCREENING



TATA MOTORS LIMITED

WITH SEVEN FACTORS 16 TRIALS CAN BE CONDUCTED WITH RESOLUTION IV32 TRIALS CAN BE CONDUCTED WITH RESOLUTION 1V64 TRIALS CAN BE CONDUCTED WITH RESOLUTION SEVEN128 TRIALS FOR FULL FACTORIAL

THE TABLE SHOWS NUMBER OF TRIALS WITH SINGLE REPLICATION



TATA MOTORS LIMITED

STAT-DOE-CREATE FACTORIAL DESIGN-DESIGNS gives the followingScreen gives the following screen with all alternatives

32 trials with seven factors corresponds to ¼ fraction as (1/4)*27=25=32



TATA MOTORS LIMITED

What is resolution?

R CONFOUNDING

III 1-2

IV 1-3 2-2

V 1-4 2-3

Table shows that for resolution III all main effects are confounded withTwo way interaction.For resolution IV all main effects are confounded with three way interactions and all two way interaction are confounded with each other

Confounding implies that Calculated effects are due toCombinations determinedby degree of resolution.

1-main effect2-2way interaction3-3way interaction4-4way interaction



TATA MOTORS LIMITED

Confounding effect due to fractional factorial

When one is choosing a fractional factorial experiment a certain amount of Confounding is introduced with main effect for example in a ¼fraction experiment with 7 factors involving 25 trials (A,B,C,D,E,F,G,H) two of the 5 factors will have Confounding.Two of the factors i.e. G and H will be Confounded with ABCD and ABDE respectively. ALL THE CONFOUNDING DUE TO RESOLUTION AND FRACTIONALFACTORIAL ARE SHOWN IN MINITAB BY ALIAS STRUCTUREAND DESIGN GENERATORS AND BEFORE PROCEEDING ONE MUSTEVALUATE WHETHER KNOWN SIGNIFICANT INTERACTION WILL BE LOST OR NOT.



TATA MOTORS LIMITED

Prediction Model for Standard Deviation Step 1Prepare the reduce matrix in standard order. Then transfer the mean & standard deviation calculated earlier by using 1 way ANOVA ( Using std. Order as factor) in this reduce matrix.

Mean & standard Deviation is achieved from –Way ANOVA



TATA MOTORS LIMITED

Step 2Use menus:Stat /Doe /Factorial / Define Custom Factorial Design ...

This is done in order for Minitab to recognise the new design .

Select the design

Select the factor

Select the 2 level design



TATA MOTORS LIMITED

Select these option for std. Order , run order ,centre point

& block

Now new Design recognized by Minitab



TATA MOTORS LIMITEDStep 3Run the Factorial Analysis Using St. dev as Response Use menus:Stat /Doe /Factorial /AnalyzeFactorial Design ...

Select Term Menu

Include the model Having up to 3 Level of interaction



TATA MOTORS LIMITED

Rule : Look For the coefficients With magnitude that is

roughly greater then half of the constants

Vendor *Heat Interaction is a significant Dispersion Effect

Model with Coefficient value plugged inSt dev = 3.094+0.795*Vendor +0.972*Heat –1.503*(Vendor*Heat)

Step 4Built The prediction model



TATA MOTORS LIMITED

Step 5

Explore Prediction model

• Optimize centre & spread

• Explore “ Trade Off” vs. goals (CTQs)

•Improve Setting

•Track The Result

Step6

Determine the factors settings

The six sigma solution should be trade off between target Mean & Variation



TATA MOTORS LIMITED

RESPONSE SURFACE •Used to determine how a response is affected by a set of quantitative variables/ factors over some specified region.

•For a given number of variables response surface analysis techniques require more trials than two level fractional factorial design techniques.Hence the number of variables considered in an experiment may first need to be reduced through either technical considerations or fractional factorial experiments.

•Information on response surface is used to optimise the settings of a processto give a maximum or minimum response.

•Knowledge of the response surface analysis techniques can help in choosingsettings for a process so that day to day variations which are typically foundIn a manufacturing environment which will have a minimum effect on Degradation of product quality.



TATA MOTORS LIMITED



TATA MOTORS LIMITED



TATA MOTORS LIMITED

Variable Search Variable Search



TATA MOTORS LIMITED

VARIABLE SEARCH

OBJECTIVE :

PINPOINTS “RED X” , “PINK X” etc.. CAPTURES

THE MAGNITUDE OF ALL IMPORTANT MAIN

EFFECTS AND INTERACTION EFFECTS.

OPENS UP TOLERANCES OF ALL UNIMPORTANT

VARIABLES TO REDUCE COST.



TATA MOTORS LIMITED

THE FOUR STAGES OF COMPONENTS/VARIABLESEARCH

To quantity the magnitudes of the important main causes and their interaction effect.

4Factorial Analysis

To verify that the important causes are truly important and that the unimportant causes are truly unimportant.

3Capping Run

To eliminate all unimportant causes and their associated interaction effects.

2Elimination

To determine if Red “X”, Pink “X” are among the causes being considered

1Ball Park

OBJECTIVESTAGE



TATA MOTORS LIMITED

Procedure for Variable SearchStage I )

¾ List the most important input variables or factors or causes :

A,B,C,D,E,F,G,H and so on in descending order of importance of each

factor’s ability to influence the output. The selection can be made by using

prior DOE techniques like Paired comparison or Multivari etc.

¾ Assign two levels to each factor : a high (+) which is most likely to

contributed to good results and a low (-) which represents a most likely

deviation from the high level in day to day production.

¾ Run two experiments, one with all factors at their high levels, the other

with all factors at their low levels. Repeat these two experiments twice more

so that there are three runs with all factors at high levels and three runs

with all factors at low levels.



TATA MOTORS LIMITED

Proc edure for Var iab le Searc h

Apply the D : d bar > 1.25 : 1 rule

where D is the difference between the median of the high readings and the median of

the low readings and d bar is the average of the differences in repeatability within

each set of high and low readings.

Stage I is successful when all three high levels are better than all three low levels and

where the above mentioned ratio is met. If these conditions are not met then look for

any cancellation effect or an important factor that has been left out.



TATA MOTORS LIMITED

Proc edure for Var iab le Searc hSt age I I

Run a pair of tests with the low level of the most important factor, A, I.e. AL

along with the high levels of all the remaining factors labeled RH and with the

high level of the most important factor, A, I.e. AH, along with the low levels of all

the remaining factors RL. Calculate the high side and low side control limits. The

formula is Median + 2.776 ( d bar/ 1.81 ).



TATA MOTORS LIMITED

Procedure for Variable SearchPossible Results :

I) If both pairs of tests AL RH and AHRL falls within the control limits for all

high level combination and all low level combination respectively, factor A,

along with all of its associated interaction effects, is unimportant and can be

eliminated from further study.

II) If there is a complete reversal, I.e. if ALRH falls within the control limits of all

low level combination and AHRL falls within the control limits of all high level

combination, A is a main effect Red X. the rest of the factors – B,C,D etc.

are all unimportant and can be eliminated.

III ) If either of both pairs of tests show results outside the low side and high side

control limits respectively, but not a complete reversal, factors A along with

its associated interaction effects can not be eliminated. A plus some other

factor or factors must be considered.



TATA MOTORS LIMITED

Proc edure for Var iab le Searc hStage3

If the results are I)and III)in Stage 1 repeat steps with next factor B .

I)If the result is complete reversal B is red x and no further search is required.

II)If results is partial swing then a capping run with A is made at high and low levels of

A and B i.e. with ALBLRH and AHBHRL if in stage1 A had shown partial swing.

complete reversal indicates end of experiment and no further search is required.

Partial swing indicates search is not complete and third factor is to be selected and

and the steps I and ii) repeated until a reversal is Obtained with the new factor or a

capping run of the new factor with the old factors which had caused partial swing in

earlier stages.

Normally search does not go beyond capping run with three factors.a three factor

capping run. ( Generally four factor capping run is rare ).

Finally draw up a factorial analysis using the factors identified to quantify the main

effects and interaction effects of the important factors.



TATA MOTORS LIMITED

VARIABLE SEARCH CASE STUDY : THE PRESS BRAKE

IN A METAL STAMPING FORMING OPERATION,PARTS PRODUCED ON THE

PRESS BRAKES COULD NOT BE HELD TO A + 0.005 INCHES TOLERANCE

(OR PROCESS WIDTH OF 0.010 INCHES). VARIATION AS HIGH AS + 0.010

INCHES WERE BEING MEASURED

A VARIABLE SEARCH EXPERIMENT WAS THEN CONDUCTED. THE OBJECTIVE

WAS TO BRING THE PROCESS UNDER CONTROL, CONSISTENTLY, TO

+ 0.005 INCHES OR CLOSER.

SIX FACTORS WERE SELECTED, IN DESCENDING ORDER OF PERCEIVED

IMPORTANCE, AND THE HIGH AND LOW LEVELS

FOR EACH FACTOR DETERMINED.



TATA MOTORS LIMITED

VARIABLE SEARCH CASE STUDY: THE PRESS BRAK E

NOT ALIGNEDTHINSOFT

BOWEDAIR FORMAT ANGLE

ALIGNEDTHICKHARDFLAT

COIN FORMLEVEL

A. PUNCH & DIE ALIGNMENTB. METAL THICKNESSC. METAL HARDNESSD. METAL BOWE. RAM STROKEF. HOLDING MATERIAL

LOWHIGHFACTORS

RESULTS : NUMBER BELOW ARE EXPRESSED IN DEVIATION FROMNOMINAL IN MULTIPLES OF 0.001 INCHES ( LOWER

TOLERANCE ARE BETTER AND VICE VERSA ).STAGE 1 ALL HIGH LEVELS ALL LOW LEVELS INITIAL 4 471st REPLICATION 4 612nd REPLICATION 3 68D = ( 61 – 4 ) = 57 d = ( 1 + 21 ) / 2 = 11 SO D/d = 5.2

(STAGE 1 SUCCESSFUL)CONTROL LIMITS = MEDIAN + 2.776 d/ 1.81 = MEDIAN + 16.87SO CONTROL LIMITS ( LOW LEVEL ) = 44.13 TO 77.87SO CONTROL LIMITS (HIGH LEVEL) = -12.87 TO 28.87



TATA MOTORS LIMITED


STAGE - 2

F IMPORTANT WITH ANOTHER FACTOR.

-12,87 TO 20.8744.13 TO 77.87

461

7318

FLRH

FHRL

1112

E NOT IMPORTAN-12,87 TO 20.8744.13 TO 77.87

461

750

ELRH

EHRL

910

D IMPORTANT WITH ANOTHER FACTOR

-12,87 TO 20.8744.13 TO 77.87

461

2330

DLRH

DHRL

78

C NOT IMPORTANT-12,87 TO 20.8744.13 TO 77.87

461

772

CLRH

CHRL

56

B NOT IMPORTANT-12,87 TO 20.8744.13 TO 77.87

461

547

BLRH

BHRL

34

A NOT IMPORTANT-12,87 TO 20.8744.13 TO 77.87

461

372

AL RH

AL RH

12

CONCLUSIONCONTROL LIMITS

MEDIANRESULTSCOMBINATIONTEST



TATA MOTORS LIMITED


STAGE – 3 : CAPPING RUN

•COMPLETE REVERSAL•END OF TEST

44.13 TO 77.876170DLFLRH#

•DF INTERACTION MAY BE PRESENT•R UNIMPORTANT

-12,87 TO 20.8744DHFHRL#

CONCLUSIONCONTROL LIMITS

MEDIANRESULTSCOMBINATIONTEST

# : DENOTES CAPPING RUN



TATA MOTORS LIMITED

VARIABLE SEARCH CASE STUDY: THE PRESS BRAK ESTAGE 4 : FACTORIAL ANALYSIS

47 7261 4768 7270 50

MEDIAN = 64.5

7330

MEDIAN = 51.5

2318

MEDIAN = 20.5

4 34 53 74 7

MEDIAN = 4

DLOWDHIGH

FHIGH

FLOW

72 55.585

24.5

116

68.5



TATA MOTORS LIMITED


MAIN EFFECT : D = (20.5+64.5) – (4+51.5) = (85-55.5 ) = 14.75 PINK ‘X ’2 2

MAIN EFFECT : F = (51.5+64.5) – (4+20.5) = (116-24.5) = 45.75 RED ‘X ’2 2

DF INTERACTION = (51.5+20.5) – (4+64.5) = (72-68.5) = 1 .752 2



TATA MOTORS LIMITED

DESIGN OF EXPERIMENTS

GREEN ‘Y’ : BURNING MARKS ON BALL TRACK OF STEERING NUT(407 / 207)PROBLEM HISTORY : (87 Rej. 19570 Prod.) i.e. 15% OF TOTAL REJN.FOR APR TO OCT.COST OF REJECTION : Rs 138 /- APPROX. PER PIECE.COST OF REJN. FOR BURNING MARKS : 1715 /- PER MONTHDOE TECHNIQUE USED : VARIABLE SEARCHLIKERT SCALE : TO CONVERT ATTRIBUTE DATA INTO VARIABLE DATA.

4HEAVY BURNING MARKS

3MEDIUM BURNING MARKS

2LIGHT BURNING MARKS

1NO BURNING MARKS

VARIABLE SCALEATTRIBUTE SCALE



TATA MOTORS LIMITED

VARIABLE SEARCH CASE STUDY : STEERING NUT BURNING MARK S PROBLEM

FIXED FACTORS : QUILL R/OUT, DIAMOND ROLL FACE R/OUT, BACKLASH OFLEAD SCREW & BOX NUT, COOLENT FLOW WERE OPTIMISED

90150000.061817GRADE:RA8017VF8

80120000.0226S17 (SUPLHUR TREATED)GRADE : RA8017VF8

A. WORK HEAD FEEDB. WHEEL R.P.MC. DRESSING DEPTH OF CUTD. NO. OF PASSESE. GRINDING WHEEL

WORSTBESTFACTORS



TATA MOTORS LIMITED

RESULTS : NUMBERS BELOW ARE CONVERSION OF ATTRIBUTE DATA INTOVARIABLE ( LIKERT SCALE )

STAGE I ALL BEST LEVELS ALL WORST LEVELINTIAL 2 31st REPLICATION 2 32nd REPLICATION 2 2

D = (3 – 2) = 1 d + (0+1) /2 = 0.5 So, D/d = 2.0STAGE I SUCCESSFUL

CONTROL LIMITS = MEDIAN + 2.776 d / 1.81 = MEDIAN + 0.767

SO CONTROL LIMITS ( WORST LEVEL ) = 2.233 TO 3.767

SO CONTROL LIMITS ( BEST LEVEL ) = 1.233 TO 2.767



TATA MOTORS LIMITED


STAGE 2

E IS NOT IMPORTANT

1.233 TO 2.7672.233 TO 3.767

23

1.53

EWRB

EBRW

910

D IS IMPORTANT1.233 TO 2.7672.233 TO 3.767

23

31

DWRB

DBRW

78

C IS NOT IMPORTANT

1.233 TO 2.7672.233 TO 3.767

23

1.53

CWRB

CBRW

56

B IS IMPORTANT1.233 TO 2.7672.233 TO 3.767

23

31.5

BWRB

BBRW

34

A IS IMPORTANT1.233 TO 2.7672.233 TO 3.767

23

31

AWRB

ABRW

12

CONCLUSIONCONTROL LIMITSMEDIANRESULTSCOMBINATIONTEST



TATA MOTORS LIMITED


STAGE 4 : FACTORIAL ANALYSIS

3.00 2.00 3.003.00 3.00 3.00

2.001.003.00

1.503.002.001.50 2.00 2.001.00 1.50 2.00Media =1.75

DWORSTDBESTDWORSTDBEST

AWROSTABEST

BBEST

BWORST

Media =3



TATA MOTORS LIMITED

RE

D ‘X

’

E:- GRINDING WHEEL

D – NO. OF PASSES

C:- DRESSING DEPTH OF CUT

B:- WHEEL R.P.M

A:- FEED

A B D A X B A X D B X D AX BXD-1 -1 -1 1 1 1 -1 31 -1 -1 -1 -1 1 1 1-1 1 -1 -1 1 -1 1 1.51 1 -1 1 -1 -1 -1 2-1 -1 1 1 -1 -1 1 21 -1 1 -1 1 -1 -1 3-1 1 1 -1 -1 1 -1 31 1 1 1 1 1 1 1.75

-1.75 -0.75 2.25 0.25 1.25 0.25 -4.75Effect

Pin

k X

Documents

Proble Solving