Formulas and Equation Sheet by Topic on Metrology | MEGR 3282, Study notes of Mechanical Engineering

Download Formulas and Equation Sheet by Topic on Metrology | MEGR 3282 and more Study notes Mechanical Engineering in PDF only on Docsity! Equation Sheet for MEGR 3282: SPC & Metrology Last modified: 11 February, 2003 Page 1 of 7 Distributions Name pdf mean variance Hypergeometric ( ) ( )Dnx n N xn DN x D xp ,min...,2,1,0= ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = N nD ⎟⎠ ⎞ ⎜ ⎝ ⎛ − − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − 1 1 N nN N D N nD Binomial ( ) ( ) nxpp x n xp xnx ...,2,1,01 =−⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = − np ( )pnp −1 Poisson ( ) ...,1,0 ! == − x x exp xλλ λ λ Normal ( ) ∞<<∞−= ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −− xexf x 2 2 1 2 1 σ µ πσ µ 2σ Exponential ( ) 0≥= − xexf xλλ λ 1 2 1 λ Equation Sheet for MEGR 3282: SPC & Metrology Last modified: 11 February, 2003 Page 2 of 7 Two-sided confidence intervals statistic Interval mean (variance known) n Zx n Zx σµσ αα 22 +≤≤− mean (variance unknown) n Stx n Stx nn 1,21,2 −− +≤≤− αα µ variance ( ) ( ) 2 1,21 2 2 2 1,2 2 11 −−− Χ − ≤≤ Χ − nn SnSn αα σ difference between means (variance known) 2 2 2 1 2 1 221 nn Zxx σσα +−− 21 µµ −≤ 2 2 2 1 2 1 221 nn Zxx σσ α ++−≤ difference between means (variance unknown) 21 2,221 11 21 nn Stxx pnn +−− −+α 21 µµ −≤ 21 2,221 11 21 nn Stxx pnn ++−≤ −+α where ( ) ( ) 2 11 21 2 22 2 11 −+ −+− = nn SnSnS p Ratio of variances (for normal distributions) 1,1,22 2 2 1 2 2 2 1 1,1,212 2 2 1 1212 −−−−− ≤≤ nnnn FS SF S S αα σ σ Equation Sheet for MEGR 3282: SPC & Metrology Last modified: 11 February, 2003 Page 5 of 7 Hypothesis tests (continued): Tests on variances of normal distributions Hypothesis Criteria for Rejection Test Statistic 2 0 2 0 : σσ =H 2 0 2 1 : σσ ≠H 2 1,2 2 0 −Χ>Χ nα or 2 1,21 2 0 −−Χ<Χ nα 2 0 2 0 : σσ =H 2 0 2 1 : σσ <H 2 1,1 2 0 −−Χ<Χ nα 2 0 2 0 : σσ =H 2 0 2 1 : σσ >H 2 1, 2 0 −Χ>Χ nα ( ) 2 0 2 2 0 1 σ Sn − =Χ 2 2 2 10 : σσ =H 2 2 2 11 : σσ ≠H 1,1,20 21 −− > nnFF α or 1,1,210 21 −−− < nnFF α 22 2 1 0 S SF = 2 2 2 10 : σσ =H 2 2 2 11 : σσ <H 1,1,0 12 −− > nnFF α 2 1 2 2 0 S SF = 2 2 2 10 : σσ =H 2 2 2 11 : σσ >H 1,1,0 21 −− > nnFF α 2 2 2 1 0 S SF = Equation Sheet for MEGR 3282: SPC & Metrology Last modified: 11 February, 2003 Page 6 of 7 Variables control charts: µ, σ given. Quality Characteristic Center Line Control Limits x µ σµ A± R σ2d σ2DUCL = , σ1DLCL = S σ4c σ6BUCL = , σ5BLCL = µ, σ not given (unknown). x x RAx 2± , or SAx 3± R R RDUCL 4= , RDLCL 3= S S SBUCL 4= , SBLCL 3= Attributes control charts: Quality Characteristic Center Line Control Limits Fraction non-conforming (p given) p ( ) n ppp −± 13 Fraction non-conforming (p not given) p ( ) n ppp −± 13 Number non-conforming (p given) np ( )pnpnp −± 13 Number non-conforming (p not given) pn ( )ppnpn −± 13 Non-conformities (c given) c cc 3± Non-conformities (c not given) c cc 3± Average number of non-conformities per unit u n uu 3± Equation Sheet for MEGR 3282: SPC & Metrology Last modified: 11 February, 2003 Page 7 of 7 Confidence Interval for PCR 1 ˆ 1 ˆ 2 1,2 2 1,21 − Χ ≤≤ − Χ −−− n RCPPCR n RCP nn αα Hypothesis Testing from PCR 10.0== βα 05.0== βα Sample Size, n )( )( lowPCR highPCR )(lowPCR C )( )( lowPCR highPCR )(lowPCR C 10 1.88 1.27 2.26 1.37 20 1.53 1.20 1.73 1.26 30 1.41 1.16 1.55 1.21 40 1.34 1.14 1.46 1.18 50 1.30 1.13 1.40 1.16 60 1.27 1.11 1.36 1.15 70 1.25 1.10 1.33 1.14 80 1.23 1.10 1.30 1.13 90 1.21 1.10 1.28 1.12 100 1.20 1.09 1.26 1.11