Question

2. A process has a X-bar control chart with UCL=17.4, LCL=12.79 using a 3- sigma limit. The subgroup sample size is 3. (a) Estimate the standard deviation of the process, sigma hat. (6 pts) (b) Suppose that the process mean shifts to 13. Find the probability that this shift will be detected on the next sample. (8 pts) (c) Calculate the average run length (ARL) before detecting the shift. (6 pts)

Answer 1

Solution ; Given bela Given process has x-bap Contao/ Chart Given uch=1704 Ich = 12.79 subgroup sample size (n) = 3. Answer :- a) from the contral chait table CX) for n=3, Az = 1.023 de = 1.693 at Az i. Uch- 17-4 i. Uch = 0+1 023 ū - Az ő in LCL 12.79 Lch är 1.023 Ř - After solving these two cauation = 15.095, and R = 2.953

now å - d2 @ = 2.953 - 1.33) 1.693 =/33/ is the standard deviation af the process sigma hat @ - 1.331 b) Giren é = 13; i where u: shifted mean :: P[ No detection there is a shift] = p[lcke izvel/pe-13] -p = P[12-79<# < 14:4) = 13] =p119.79-13 <F-13 2 17.4-13 ålin å lith al = p[ -0.91 <2< 4.4 lin 1.331/ 1-331 In p[-0.273<2<5.737 = $(5-73) – $(-0.275) •99-70.5-PLOC 220.973]] f de to symmetic = .99-C0.5-1064) normal coreve) nature at

hobga ß 5964 Now p[ detection of dext sample ) = BCl-B) =(.5964) Cl-.5964) .241 cj ARL ㅗ I-P hobs.nl = 2.48 2.48 ARL That mean on average we can seey ee the second zed point we will get a point out side the cortfo/ limits Plz Civene thumbupa