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ENGM 720 - Lecture 06

- Multiple Comparisons,
- 7 Tools of Ishikawa

Assignment

- Reading
- Chapter 4.5, Chapter 5 5.2, 5.4
- Finish reading
- Review for Exam I
- Covers material through hypothesis tests and

seven tools - Assignments
- Obtain the Hypothesis Test (Chart ) Tables
- Access Previous Assignment Solutions Prepare

Notebook - Download Assignment 4 Assignment 4 Solutions

Multiple Comparisons

- Analysis-of-variance (ANOVA) is a statistical

method used to test hypotheses regarding more

than two sample means. - For a one-factor experiment the hypothesis tested

is

Multiple Comparisons

- The strategy in an analysis of variance is to

compare the variability between sample means to

the variability within sample means. If they are

the same, the null hypothesis is accepted. If the

variability between is bigger than within, the

null hypothesis is rejected.

Null Hypothesis

Alternative Hypothesis

Definitions

- An experimental unit is the item measured during

an experiment. The errors in these measurements

are described by random variables. - It is important that the error in measurement be

the same for all treatments (random variables be

independent and have the same distribution). - The easiest way to assure the error is the same

for all treatments is to randomly assign

experimental units to treatment conditions.

Definitions

- The variable measured in an experiment is called

the dependent variable. - The variable manipulated or changed in an

experiment is called the independent variable. - Independent variables are also called factors,

and the sample means within a factor are called

levels or treatments.

Definitions

- Random samples of size n are selected from each

of k different populations. The k different

populations are classified on the basis of a

single criterion or factor. (one-factor and k

treatments) - It is assumed that the k populations are

independent and normally distributed with means

µ1, µ2, ... , µk, and a common variance s2. - Hypothesis to be tested is

Definitions

- A fixed effects model assumes that the treatments

have been specifically chosen by the

experimenter, and our conclusions apply only to

the levels chosen - Fixed Effect Statistical Modelwhere eij are

independent and identically distributed N(0,s2). - Because the fixed effects model assumes that the

experiment is performed in a random manner, a

one-way ANOVA with fixed effects is often called

a completely randomized design.

Overall Mean

ith Treatment Effect

Error in Measurement

Observed Value

Definitions

- For a fixed effects model, if we restrict
- Thenis equivalent to

Analysis of the Fixed Effects Model

. . .

. . .

. . .

. . .

Analysis of the Fixed Effects Model

- Sum of Squares TreatmentsThe sum of squares

treatments is a measure of the variability

between the factor levels. - Sum of Squares ErrorsThe error sum of squares

is a measure of the variability within the

factor levels.

Sum of Squares Errors (SSE)

Factor level 1

Factor level 2

Factor level 3

X3

X1

X2

Sum of Squares Treatments (SSTr)

Analysis of the Fixed Effects Model

- P-valuesThe plausibility of the null

hypothesis (that the factor level means are all

equal)depends upon the relative size of the

sum of squares for treatments (SSTr) tothe sum

of squares for errors (SSE).

Analysis of the Fixed Effects Model

- Sum of Squares Partition for One Factor

LayoutIn a one factor layout, the total

variability in the data observations is measured

by the total sum of squares (SST) which is

defined to be

Total Sum of Squares SST

Treatment Sum of Squares SSTr

Error Sum of Squares SSE

Analysis of the Fixed Effects Model

- Sum of Squares Partition for One Factor

LayoutThis can be partitioned into two

components SST SSTr SSE,where the sum of

squares for treatments (SSTr) - measures the variability between the factor

levels, - and the sum of squares for error (SSE)
- measures the variability within the factor

levels.

Analysis of the Fixed Effects Model

- Sum of Squares Partition for One Factor Layout
- On an intuitive level, the plausibility of the

null hypothesis that the factor level means (µi)

are all equal depends upon the relative size of

the sum of squares for treatments (SSTr) to the

sum of squares for error (SSE)

Possibly Likely the Same

Definitely NOT Likely the Same

Definitely VERY Likely the Same

Analysis of the Fixed Effects Model

- F-Test for One Factor LayoutIn a one factor

layout with k levels and n replications gives a

total sample size kn N, the treatments are said

to have k - 1 degrees of freedom and the error is

said to have N - k degrees of freedom. Mean

squares are obtained by dividing a sum of squares

by its respective degrees of freedom so

thatand

Analysis of the Fixed Effects Model

- F-Test for One Factor LayoutA p-value for the

null hypothesis that the factor level means µi,

are all equal is calculated as - p-value P(X ³ F)
- where the F-statistic isand the random

variable X has a Fk-1, N - k distribution.

Analysis of the Fixed Effects Model

ANOVA Example

- The tensile strength of a synthetic fiber used to

make cloth for shirts is of interest to a

manufacturer. It is suspected that strength is

affected by the percentage of cotton in the

fiber. - Five levels of cotton percentage are of interest

15, 20, 25, 30, and 35. - Five observations are to be taken at each level

of cotton percentage and the 25 total

observations are to be run in random order.

ANOVA Example

RANDOMIZATIONPROCEDURE

. . .

. . .

. . .

ANOVA Example

Tensile Strength of Synthetic Fiber (lb/in2)

ANOVA Example

5-1 4

25-5 20

25-1 24

Critical Points for the F-Distribution

Alpha 0.05

ANOVA Example

ANOVA Example

Ishikawas Magnificent Seven Tools

- The Seven Tools are
- Histogram / Stem Leaf Diagram
- Cause Effect (Fishbone) Diagram
- Defect Concentration Diagram
- Check Sheet
- Scatter (Plot) Diagram
- Pareto Chart
- Control Chart - not covered on exam!
- The tools were not invented by Ishikawa, but were

very successfully put into methodical use by him - The first six are used before starting to use the

seventh - They are also reused when needed to find an

assignable cause

Ishikawas Tools Histogram

- A histogram is a bar chart that takes the shape

of the distribution of the data. The process for

creating a histogram depends on the purpose for

making the histogram. - One purpose of a histogram is to see the shape of

a distribution. To do this, we would like to

have as much data as possible, and use a fine

resolution. - A second purpose of a histogram is to observe the

frequency with which a class of problems occurs.

The resolution is controlled by the number of

problem classes.

Histogram of Lab 01 Results

Ishikawas Tools Fishbone Diagram

- Cause Effect diagram constructed by

brainstorming - Identified problem at the head
- Connects potential causes along the spine
- Sub-causes are listed along the major bones
- Man
- Material
- Method
- Machine
- Environment

Cause Effect Diagram, Cont.

- The purpose of the cause and effect diagram is to

obtain as many potential influencers of a

process, so that the problem solving can take a

more directed approach.

Ishikawas Tools Defect Diagram

- A defect concentration diagram graphically

records the frequency of a defect with respect to

product location. - Obtain a digital photo or multi-view part print

showing all product faces. - Operator tallies the number and location of

defects as they occur on the diagram.

Ishikawas Tools Check Sheet

- Check sheets are used to collect data (values or

pieces of information) in a consistent manner. - List each of the known / possible problems
- Record each occurrence including time-orientation.

Ishikawas Tools Scatter Plot

- A scatter plot shows the relationship between any

two variables of interest - Plot one variable along the X-axis and the other

along the Y-axis - The presence of a relationship can be inferred or

ruled out, but it cannot determine if a cause and

effect relationship exists

Ishikawas Tools Pareto Chart

- 80 of any problem is the result of 20 of the

potential causes - Histogram categories are sorted by the magnitude

of the bar - A line graph is overlaid, and depicts the

cumulative proportion of defects - Quickly identifies where to focus efforts

Use of Ishikawas Tools

- Removing special causes of variation
- Preparation for
- hypothesis tests
- control charts
- process improvement

Questions Issues