Cheat Sheet Module 4

Cheat Sheet Module 4#

  • Type-II Error - Accept \(H_0\) when it is False (or when \(H_a\) is true)

  • 1 million = \(10^6\)

Run Lengths#

  • “In-control” Average Run Length \(ARL_{\alpha} = \frac{1}{\alpha}\)

  • “Out-of-control” Average Run Length \(ARL_{1-\beta} = \frac{1}{1-\beta}\)

    • Detection Power \( = 1-\beta\)

    • Low \(\beta \implies \) High Detection Power \(\implies\) Low “Out-of-Control” Run Length

If ‘discrepancy’ for \(H_a\) is known \(\delta\)#

  • For “two-sided” alternate \(\beta = \Phi(z_{\alpha/2} - \delta \sqrt{n}/\sigma) - \Phi(-z_{\alpha/2} - \delta \sqrt{n}/\sigma)\)

    • Useful for calculating sample size (n)!

Natural Tolerance Limits (NTL)#

  • \(UNTL = \mu + 3\sigma\)

  • \(LNTL = \mu - 3\sigma\)

Process Capability Ratio (only for stable processes)#

Replace with estimates if true value is not available

  • \(T\) - “Target” for \(\mu\)

  • \(C_{pu} = \frac{USL - \mu}{3 \sigma}\) or \(C_{pl} = \frac{\mu - LSL}{3 \sigma}\)

  • One-sided

    • Fall-out = \(Pr(x>USL)\) or \(Pr(x<LSL)\)

  • Two-sided

    • Process Capability \(C_{pk} = \min(C_{pu},C_{pl}\))

    • Taguchi Capability Measure \(C_{pm} = \frac{USL - LSL}{6\sqrt{\sigma^2 + (\mu - T)^2}}\)

    • Potential Capability \(C_p = \frac{USL-LSL}{6 \sigma}\)

    • Fall-out = \(Pr(x>USL) + Pr(x<LSL)\)

    • Fall-out (only for a centered process) = \(2 \times (1 - \Phi(3C_{pk})) = 2 \times \Phi(-3C_{pk})\)

  • Common Terminology

    • Capable Process \(C_p > 1\)

    • Incapable Process \(C_p < 1\)

    • Six-Sigma Process \(C_p = 2.0\)

Standard Normal CDF Values#

Some values beyond the table

  • \(\Phi(value \,\, less \,\, than \,\, -6) \approx 0\)

  • \(\Phi(-6) = 9.866 \times 10^{-10}\)

  • \(\Phi(-5) = 2.867 \times 10^{-7}\)

  • \(\Phi(-4) = 3.167 \times 10^{-5}\)

Contents