Hypothesis Testing in JMP
One sample t-test
Used to compare mean of 1 population/sample to a hypothesized value
Null hypothesis states that population/sample mean is equal to hypothesized value
[Analyze-> Distribution, select TEST MEAN]
Two sample t-test
Used to compare means of 2 populations/samples with each other. Continuous variable (Y) v/s 2-level single categorical variable/factor (X)
Null hypothesis states that means of the 2 populations/samples are equal to each other
[Analyze -> Fit Y by X, select MEANS/ANOVA/POOLED-T or T-TEST]
One Way Anova
Used to compare means of 3 or more populations/samples. Continuous variable (Y) v/s single categorical variable with 3 or more levels (X)
Null hypothesis states that means of all populations/samples/levels are equal to each other.
[Analyze -> Fit Y by X, select MEANS/ANOVA + COMPARE MEANS]
Two Way (Factorial) ANOVA
Used to study effects of 2 categorical variables/factors and their interaction on single response. Continuous variable (Y) v/s two categorical variables (X1, X2)
Null hypothesis states that factors do not have a significant effect
[Analyze -> Fit Model, select LS MEANS PLOT]
JMP Data Types & Graphs
Variables are either Continuous or Categorical (Nominal or Ordinal).
Continuous (Y) v/s Continuous (X) --> BIVARIATE
Continuous (Y) v/s Categorical (X) --> ONE WAY ANOVA
Categorical(Y) v/s Categorical (X) --> CONTINGENCY
Categorical (Y) v/s Continuous (X) --> LOGISTIC
Why is the Exponential Distribution special?
1. Beta = 1
2. Constant failure rate (or, hazard rate = lambda) -> used to model useful life portion of the bathtub curve.
3. R(t+T) = R(t)
4. A 3-parameter Weibull(eta, beta, gamma) is the same as a 2-parameter exponential (with beta = 1 & eta = MTTF = 1/lambda).
5. A 1-parameter Weibull (eta, beta=1, gamma=0) is the same as 1-parameter exponential (with eta = MTTF = 1/lambda)
6. R(t=MTTF) = 36.8% & Q(t=MTTF) = 63.2%.
Hypothesis tests: How?
1. Define problem
2. Develop null & alternate hypotheses
3. Set up test parameters (1-sided v/s 2-sided, choose distribution & significance level or alpha)
4. Calculate test statistic & corresponding p-value
5. Compare p-value with alpha & interpret results
Hypothesis tests: Which & When?
Test of Means:
1-sample or 2-sample: Use z-test for n>=30 or when population variance is known, else use t-test
> 2-samples: Use ANOVA
Test of Variances:
1-sample: Use Chi-square test
2-samples: Use F-ratio test
Test of Proportions:
1-sample or 2-sample: Use z-test
>2-samples: Use Chi-square test
Distributions
Distributions for Attribute/Finite Data:
Hypergeometric: Probability of r rejects in n sample size for N population size with d total rejects. (Intended for small, finite, well characterized populations)
Binomial: Probability of r rejects in n sample size, where n < 10% of N population size, where chance of success in any given trial always stays the same (p)(Intended for large population sizes)
Poisson: Probability of r rejects (=defects or events) in infinite population size, for a given failure rate (lambda). (Intended for n->infinity & p->0)
Binomial distribution approximates Hypergeometric distribution for large N.
Poisson distribution approximates Binomial distribution when N tends to infinity.
Distributions for Continuous Data: Normal, Lognormal, Exponential, Weibull
SPC/ Control Charts
Control charts are used to differentiate & identify special causes of variation from those that are common cause related. These may be freaks/outliers, drifts, shifts, stratification, recurring patterns & systematic variation.
For variable data, use I-MR (for n=1), X(bar)-R (for n = 2 to 10) or X(bar)- s (for n>10)
For attribute data:
1. Count/proportion of defectives is estimated through binomial distribution. For constant sample size(n), estimate count of defectives using np chart, while for variable sample size, estimate proportion of defectives using p-charts.
2. Count/rate of defects is estimated through poisson distribution. For constant sample size(n), estimate count of defects using c-chart, while for variable sample size, estimate rate of defects using u-chart.
Six Sigma & Process Variation
For a normal distribution:
-Approx 68% of variation is contained within +/- 1sigma
-Approx 95% of variation is contained within +/- 2sigma
-Approx 99.7% of variation is contained within +/- 3sigma
Cp = 1 when +/- 3 sigma is contained within spec limits.
Cp = 1.33 when +/- 4 sigma is contained within spec limits.
Cp = 1.50 when +/- 4.5 sigma is contained within spec limits.
Cp = 1.67 when +/- 5 sigma is contained within spec limits.
Cp = 2.00 when +/- 6 sigma is contained within spec limits.
Acceptance sampling: LTPD & AQL
AQL = definition of a threshold good lot.
LTPD = definition of a threshold bad lot.
The sampling plan is designed around the AQL/LTPD such that it defines:
1. MAX chance of ACCEPTING lots of quality that is equal or worse than LTPD. This chance/risk is BETA or CONSUMER's RISK.
2. MAX chance of REJECTING lots of quality that is equal or better than AQL. This chance/risk is ALPHA or PRODUCER's RISK.
Alpha (Probability of rejection) is usually set to 0.05. This equates to 95% chance/confidence of acceptance.
Beta (Probablility of acceptance) is usually set to 0.10. This equates to 90% chance/confidence of rejection.
Power, Confidence, Error, Significance
Reject null hypothesis when true (false positive) = alpha or Type 1 error
Accept null hypothesis when false (or false negative) = beta or Type 2 error
Reject null hypothesis when false: POWER = (1-beta)
Accept null hypothesis when true : CONFIDENCE (= 1-alpha)
At high power, beta is small => alpha is large => likely that p-value will be < alpha (significance level). Most effects tend to be deemed significant.
At low power, beta is large => alpha is small => likely that p-value will be > alpha (significance level). Most effects tend to be deemed insignificant.
Six Sigma : Process & Design
Process: Aims to reduce process variation Define: Plan, scope, charter, schedule, team, objectives, milestones, deliverables Measure: MSA, GR&R, Process Capability, Yields Analyze: Hypothesis tests, ANOVA, PFMEA, Process Maps (KPIV/KPOV) Improve: DoE Control: SPC, Control ChartsDesign: Aims to reduce cycle time and need for rework Define: Plan, scope, charter, schedule, team, objectives, milestones, deliverables Measure: Baseline, benchmark, functional parameters, specs & margins Analyze: DFMEA, Risk analysis, GAP analysis Develop: Deliver design Optimize: DfX - tradeoffs Validate: Prototype builds
Firefighting through methodical madness/8D
1. Develop Team 2. Define Problem: Failure rate, lots affected, establish scope 3. Containment: Raise red flags, lots on hold, generate documentation, reliability assessment, sampling plans, increased checks & balances 4. Problem analysis: Process mapping, history tracking, establish commonalities & dependencies, consult FMEA, RCA/5W/5M, failure analysis, establish hypotheses, develop CAPA theories (short-term/mid-term/long-term) 5. Verify corrective actions: Engineering studies to duplicate problem and verify effectiveness of CA 6. Implement corrective action: Release lots, provide disposition, soft ramp through full release with increased sampling, document lessons learnt 7. Implement preventive action: Mid-term/long-term actions to prevent any recurrences in future 8. Congratulate team
Continuous (Y) v/s Continuous (X) --> BIVARIATE
Continuous (Y) v/s Categorical (X) --> ONE WAY ANOVA
Categorical(Y) v/s Categorical (X) --> CONTINGENCY
Categorical (Y) v/s Continuous (X) --> LOGISTIC
Why is the Exponential Distribution special?
1. Beta = 1
2. Constant failure rate (or, hazard rate = lambda) -> used to model useful life portion of the bathtub curve.
3. R(t+T) = R(t)
4. A 3-parameter Weibull(eta, beta, gamma) is the same as a 2-parameter exponential (with beta = 1 & eta = MTTF = 1/lambda).
5. A 1-parameter Weibull (eta, beta=1, gamma=0) is the same as 1-parameter exponential (with eta = MTTF = 1/lambda)
6. R(t=MTTF) = 36.8% & Q(t=MTTF) = 63.2%.
Hypothesis tests: How?
1. Define problem
2. Develop null & alternate hypotheses
3. Set up test parameters (1-sided v/s 2-sided, choose distribution & significance level or alpha)
4. Calculate test statistic & corresponding p-value
5. Compare p-value with alpha & interpret results
Hypothesis tests: Which & When?
Test of Means:
1-sample or 2-sample: Use z-test for n>=30 or when population variance is known, else use t-test
> 2-samples: Use ANOVA
Test of Variances:
1-sample: Use Chi-square test
2-samples: Use F-ratio test
Test of Proportions:
1-sample or 2-sample: Use z-test
>2-samples: Use Chi-square test
Distributions
Distributions for Attribute/Finite Data:
Hypergeometric: Probability of r rejects in n sample size for N population size with d total rejects. (Intended for small, finite, well characterized populations)
Binomial: Probability of r rejects in n sample size, where n < 10% of N population size, where chance of success in any given trial always stays the same (p)(Intended for large population sizes)
Poisson: Probability of r rejects (=defects or events) in infinite population size, for a given failure rate (lambda). (Intended for n->infinity & p->0)
Binomial distribution approximates Hypergeometric distribution for large N.
Poisson distribution approximates Binomial distribution when N tends to infinity.
Distributions for Continuous Data: Normal, Lognormal, Exponential, Weibull
SPC/ Control Charts
Control charts are used to differentiate & identify special causes of variation from those that are common cause related. These may be freaks/outliers, drifts, shifts, stratification, recurring patterns & systematic variation.
For variable data, use I-MR (for n=1), X(bar)-R (for n = 2 to 10) or X(bar)- s (for n>10)
For attribute data:
1. Count/proportion of defectives is estimated through binomial distribution. For constant sample size(n), estimate count of defectives using np chart, while for variable sample size, estimate proportion of defectives using p-charts.
2. Count/rate of defects is estimated through poisson distribution. For constant sample size(n), estimate count of defects using c-chart, while for variable sample size, estimate rate of defects using u-chart.
Six Sigma & Process Variation
For a normal distribution:
-Approx 68% of variation is contained within +/- 1sigma
-Approx 95% of variation is contained within +/- 2sigma
-Approx 99.7% of variation is contained within +/- 3sigma
Cp = 1 when +/- 3 sigma is contained within spec limits.
Cp = 1.33 when +/- 4 sigma is contained within spec limits.
Cp = 1.50 when +/- 4.5 sigma is contained within spec limits.
Cp = 1.67 when +/- 5 sigma is contained within spec limits.
Cp = 2.00 when +/- 6 sigma is contained within spec limits.
Acceptance sampling: LTPD & AQL
AQL = definition of a threshold good lot.
LTPD = definition of a threshold bad lot.
The sampling plan is designed around the AQL/LTPD such that it defines:
1. MAX chance of ACCEPTING lots of quality that is equal or worse than LTPD. This chance/risk is BETA or CONSUMER's RISK.
2. MAX chance of REJECTING lots of quality that is equal or better than AQL. This chance/risk is ALPHA or PRODUCER's RISK.
Alpha (Probability of rejection) is usually set to 0.05. This equates to 95% chance/confidence of acceptance.
Beta (Probablility of acceptance) is usually set to 0.10. This equates to 90% chance/confidence of rejection.
Power, Confidence, Error, Significance
Reject null hypothesis when true (false positive) = alpha or Type 1 error
Accept null hypothesis when false (or false negative) = beta or Type 2 error
Reject null hypothesis when false: POWER = (1-beta)
Accept null hypothesis when true : CONFIDENCE (= 1-alpha)
At high power, beta is small => alpha is large => likely that p-value will be < alpha (significance level). Most effects tend to be deemed significant.
At low power, beta is large => alpha is small => likely that p-value will be > alpha (significance level). Most effects tend to be deemed insignificant.
Six Sigma : Process & Design
Process: Aims to reduce process variation Define: Plan, scope, charter, schedule, team, objectives, milestones, deliverables Measure: MSA, GR&R, Process Capability, Yields Analyze: Hypothesis tests, ANOVA, PFMEA, Process Maps (KPIV/KPOV) Improve: DoE Control: SPC, Control ChartsDesign: Aims to reduce cycle time and need for rework Define: Plan, scope, charter, schedule, team, objectives, milestones, deliverables Measure: Baseline, benchmark, functional parameters, specs & margins Analyze: DFMEA, Risk analysis, GAP analysis Develop: Deliver design Optimize: DfX - tradeoffs Validate: Prototype builds
Firefighting through methodical madness/8D
1. Develop Team 2. Define Problem: Failure rate, lots affected, establish scope 3. Containment: Raise red flags, lots on hold, generate documentation, reliability assessment, sampling plans, increased checks & balances 4. Problem analysis: Process mapping, history tracking, establish commonalities & dependencies, consult FMEA, RCA/5W/5M, failure analysis, establish hypotheses, develop CAPA theories (short-term/mid-term/long-term) 5. Verify corrective actions: Engineering studies to duplicate problem and verify effectiveness of CA 6. Implement corrective action: Release lots, provide disposition, soft ramp through full release with increased sampling, document lessons learnt 7. Implement preventive action: Mid-term/long-term actions to prevent any recurrences in future 8. Congratulate team
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