Statistical Analysis

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Session VI Statistical Analysis
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Manufacturing Scenario

• Aluminum castings
• Important factor Hardness
• Measured with Brinell units
• Possibly affected by
• Machine and/or Operator
• Chemistry (Iron, Zinc, Manganese)
• Physics (Pressure, Temperature)
• Minimum acceptable hardness is 70 HB

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Design of Experiments

• Many possible type of designs (random, blocked,
  Latin square, etc.)
• Should be driven by a theory or hypothesis
• Make sure that if the hypothetical effect is in
  fact present,
• the design used has a good chance of detecting
  it, (small chance of Type II error more on that
  later) and
• there will be no other reasonable explanation (a
  key driver of DOE)

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Design of Experiments

• Include measurement of factors you would like to
  test for
• No need for independent variables that do not
  vary
• Matched pairs gt independent samples

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Design of Experiments

• One possible approach for the aluminum problem

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• Process is highly variable
• Some castings do no meet the 70 HB target

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Inferential Statistics

• Estimation
• Confidence Intervals
• Sample Size Determination
• Design of Experiments
• Hypothesis Testing
• Classical Method
• p-Values
• Analysis of Variance
• Regression Analysis
• Analysis of Variance in Regression
• High Level Measures
• Hypothesis Tests for Independent Variables

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Estimation

• Fundamental difference between Probability and
  Statistics
• Probability is making an inference about an
  unknown sample from a known population (useful
  for developing theory)
• Statistics is making an inference about an
  unknown population from a known sample (useful in
  the real world)
• Estimation is a statistical tool using sample
  data to make a probabilistic statement about some
  unknown population parameter
• Mean
• Variance
• Proportion
• Differences between Parameters

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Estimation Confidence Intervals
General form of a confidence interval Measure
of Central Tendency ? (Number of Standard
Errors)(Measure of Dispersion)
sample mean, sample proportion, etc.
usually z or t
standard error of mean, etc.
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Estimation Confidence Intervals
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Example Confidence Interval for Population Mean
We are 95 confident that the true population
mean is between 69.75 and 84.27 HB.
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Estimation Sample Size
Our sample of 9 castings has a confidence
interval 15 HB wide maybe too wide for
managerial decision making. How many data would
we need to have a 95 confidence interval within
1 HB?
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Estimation Sample Size
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Estimation Sample Size

• We would need 343 observations (assuming the
  standard deviation is no more than 9.441 HB).
• Slightly different formula for proportions

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Hypothesis Testing Classical Method
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Hypothesis Testing Classical Method
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Example Step 1
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Example Step 2
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Example Step 3
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Example Step 3
T distribution centered on 80
5 probability In lower tail
Critical value 1.86 standard errors
below Hypothesized mean
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Example Step 4
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Example Step 4
Observed value 0.95 standard errors
below Hypothesized mean
Critical value 1.86 standard errors
below Hypothesized mean
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Hypothesis Testing p-values

• Note that the classical method only yields a
  reject or do not reject decision
• Not helpful in situations where different people
  have different tolerances for Type I Error risk
• We would like to know
• How far from the hypothesized value was it, in
  standardized terms? (provided by the test
  statistic)
• How unlikely would this result be, if the null
  hypothesis were true? (provided by the p-value)

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If the null hypothesis is true, we would see a
sample mean this low or lower 18.5 of the time.
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Hypothesis Testing p-values

• English translation If the true mean were really
  80 HB, we would see a sample mean this far below
  80 or farther 18.5 of the time.
• Since our alpha is 5, we dont consider this to
  be strong evidence against the null hypothesis

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Hypothesis Testing Type II Errors

• Any time we fail to reject, we might be
  committing a Type II Error.
• In this case, maybe the true mean is less than 80
  and our sample didnt provide enough information
  for us to realize this.
• What if the true mean had drifted down to 75HB?
  Would our test be able to detect this shift?

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Hypothesis Testing Type II Errors
Hypothesized distribution centered on 80 HB
True distribution centered on 75 HB
Critical value 1.86 standard errors below
hypothesized mean 0.27 standard errors below
true mean of 75 HB
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Hypothesis Testing Type II Errors
60.3 chance of not rejecting a false hypothesis!
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Hypothesis Testing Analysis of Variance

• Useful for testing for differences between more
  than two means
• The F test, named for Fisher

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The F Distribution
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The F Distribution

• Has only one tail cant be negative
• Central to ANOVA and regression analysis
• Based on the ratio of explained to unexplained
  variability
• Two degrees of freedom numbers

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The F Test

• Null Hypothesis Three types of machines produce
  aluminum castings with equal mean hardness.
• Alternative Hypothesis At least one of the
  machines produces aluminum castings with mean
  hardness not equal to the others.
• Test Statistic F
• Decision Rule Critical Value depends on
  numerator and denominator degrees of freedom, and
  our acceptable risk of Type I Error.

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One-way ANOVA
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One-way ANOVA
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One-way ANOVA
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One-way ANOVA
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One-way ANOVA
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One-way ANOVA
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One-way ANOVA
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One-way ANOVA
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One-way ANOVA
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One-way ANOVA
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One-way ANOVA
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One-way ANOVA
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One-way ANOVA
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One-way ANOVA
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One-way ANOVA
Tools Data Analysis (need to have Analysis
ToolPak installed Tools Add-Ins)
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One-way ANOVA
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One-way ANOVA
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Two-way ANOVA
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Two-way ANOVA
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ANOVA

• Advantages
• Good for qualitative (categorical) data
• Can easily handle multiple categories
• Flexible in terms of sample sizes
• Disadvantages
• Not especially useful for continuous data (or
  discrete data with many possible values)
• Most ANOVA procedures can be done equivalently
  using regression, which is not true in reverse

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Regression Analysis
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Regression Analysis
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Correlation Analysis
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Correlation Analysis
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Regression Analysis
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Regression Analysis
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Regression Analysis
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Regression Analysis
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Regression Analysis
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Regression Analysis ANOVA
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Regression Analysis High-level Measures
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Regression Analysis Tests for Independent
Variables
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Model Building

• Enter
• Start with one independent variable (logically,
  the one with the strongest correlation with the
  dependent variable)
• Add new independent variables one by one, in
  order of correlation strength to the dependent
  variable
• Try to maximize Adjusted R square
• Remove
• Start with all possible independent variables
• Remove independent variables (logically, on the
  basis of highest p-value)
• Try to maximize Adjusted R square
• In Both Procedures
• Watch out for multicollinearity

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7 Variables
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6 Variables
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5 Variables
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4 Variables (a)
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4 Variables (b)
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4 Variables (c)
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Summary of Six Models
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Regression Analysis

• Problems with these data
• Multicollinearity between zinc and machine types
• Too few degrees of freedom (because of sample
  size)
• Possible Type I and Type II errors

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Conclusions?

• Preliminary Findings
• Machines matter (Burugula machine really sucks)
• Possible interactions between operators and
  machines
• Pressure also seems to matter
• Zinc?
• Next Steps
• Look for root causes of low pressure
• Study best practices of operators
• Collect more data
• Whats the deal with Zinc?
• DOE to avoid problems with multicollinearity
• Test theories about pressure
• Test theories about operator/machine interactions