Question

10 pts Question 2 2. [10 points] For a given service process, order response time data provides a mean value of 16 and a standard deviation of 4 as point estimates. Tolerance (that is Specification) limits are specified as UTL (USL) - 26 and LTL (LSL) = 4 a) What percent of orders that this enterprise processes do not meet the response time specifications? Assume any response time that falls outside the tolerance (specification) limits is a defective b) Overall, at what Sigma Level is this process operating at? c) If you assume a 1.5 sigma shift for the process mean from the nominal in the long-term (like what the industry folks practice), then what Sigma Level is this process operating at?

using excel

Answer 1

For this exercise I am assuming the data has a normal distribution a) To find this percentage we have to standardize the limits into a standard normal distribution. We do this by following the formula: Y-A Z= (1) Where Y is the value we want to standardize, i and o are the estimators of the mean and the standard deviation, respectively We denote by U, the standardized upper limit and naturally, L the standard- ized lower limit. Using the fact that f = 16 and ô = 4, by (1) we have U 2.5 L 3 (NOTE: This can be done in excel using the normalization function used by inputing one point of data, the mean fi and the standard deviation ô) Now, using a normal table servation Y should be above have to find the probability that any given ob- or below the upper and lower limits, respectively we P(Y>UTL) P(Z > U£) = P(Z >> 2.5) = 0.0062 P(Y < LTL) P(Z < L.) = P(Z < -3) = P(Z >3) 0.0013 So the probability that Y is outside these limits is the sum of the above values: P(LTL Y>UTL) = P(Y > UTL)+P(Y < LTL)= 0.0062+0.0013 = 0.0075 b) Using the above standard normalized values for UTL and LTL ily read that the upper sigma level is 2.5 while the lower is 3. When both specification limits are different from one another we take the lower one(the one closest to the mean). In our case we take 2.5 to be our sigma level we can eas- c) A 1.5 sigma shift from the mean simply means that over time the process will shift to one side or the other. Since it is not specified in the problem we do both cases i) When the shift is to the right or ) When the shift is to the left. II i) If the shift is to the right, this means that our by 1.5 standard deviations to the mean so that the sigma level for this speci- fication level becomes 1; it also away by 1.5 standard deviations so that its corresponding sigma level becomes 4.5. Thus our overall sigma level will be 1 i) If the shift is to the left, this means that our lower limit will be closer by 1.5 to the mean yielding a sigma level of 1.5 whilst the sigma level for the upper limit will become further away by 1.5 becoming 4. Thus our overall sigma level in this case is 1.5 upper limit will be closer eans that our lower limit will be further T