Chapter16 - CHAPTER 16 Section 16.1 1 All ten values of the quality statistic are between the two control limits so no out-of-control signal is

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469 CHAPTER 16 Section 16.1 1. All ten values of the quality statistic are between the two control limits, so no out-of-control signal is generated. 2. All ten values are between the two control limits. However, it is readily verified that all but one plotted point fall below the center line (at height .04975). Thus even though no single point generates an out-of-control signal, taken together, the observed values do suggest that there may be a decrease in the average value of the quality statistic. Such a “small” change is more easily detected by a CUSUM procedure (see section 16.5) than by an ordinary chart. 3. P(10 successive points inside the limits) = P(1 st inside) x P(2 nd inside) x…x P(10 th inside) = (.998) 10 = .9802. P(25 successive points inside the limits) = (.998) 25 = .9512. (.998) 52 = .9011, but (.998) 53 = .8993, so for 53 successive points the probability that at least one will fall outside the control limits when the process is in control is 1 - .8993 = .1007 > .10. Section 16.2 4. For Z, a standard normal random variable, ( 29 995 . = - c Z c P implies that ( 29 ( 29 9975 . 2 005 . 995 . = + = = Φ c Z P c . Table A.3 then gives c = 2.81. The appropriate control limits are therefore s m 81 . 2 ± . 5. a. P(point falls outside the limits when s m m 5 . 0 + = ) + = + < < - - = s m m s m s m 5 . 3 3 1 0 0 0 when n X n P ( 29 n Z n P 5 . 3 5 . 3 1 - < < - - - = ( 29 0301 . 9699 . 1 882 . 1 12 . 4 1 = - = < < - - = Z P . b. - = + < < - - s m m s m s m 0 0 0 3 3 1 when n X n P ( 29 n Z n P + < < + - - = 3 3 1 ( 29 2236 . 24 . 5 76 . 1 = < < - - = Z P c. ( 29 ( 29 6808 . 47 . 1 47 . 7 1 2 3 2 3 1 = - < < - - = - < < - - - Z P n Z n P
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Chapter 16: Quality Control Methods 470 6. The limits are ( 29( 29 80 . 00 . 13 5 6 . 3 00 . 13 ± = ± , from which LCL = 12.20 and UCL = 13.80. Every one of the 22 x values is well within these limits, so the process appears to be in control with respect to location. 7. 95 . 12 = x and 526 . = s , so with 940 . 5 = a , the control limits are 70 . 13 , 20 . 12 75 . 95 . 12 5 940 . 526 . 3 95 . 12 = ± = ± . Again, every point ( 29 x is between these limits, so there is no evidence of an out-of-control process. 8. 336 . 1 = r and 325 . 2 5 = b , yielding the control limits 72 . 13 , 18 . 12 77 . 95 . 12 5 325 . 2 336 . 1 3 95 . 12 = ± = ± . All points are between these limits, so the process again appears to be in control with respect to location. 9. 54 . 96 24 07 . 2317 = = x , 264 . 1 = s , and 952 . 6 = a , giving the control limits 17 . 98 , 91 . 94 63 . 1 54 . 96 6 952 . 264 . 1 3 54 . 96 = ± = ± . The value of x on the 22 nd day lies above the UCL, so the process appears to be out of control at that time. 10.
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This note was uploaded on 03/30/2008 for the course STAT 211 taught by Professor Parzen during the Spring '07 term at Texas A&M.

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Chapter16 - CHAPTER 16 Section 16.1 1 All ten values of the quality statistic are between the two control limits so no out-of-control signal is

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