2011-2012 Semester 1 Exam Question 1 is one of the most challenging ones, the suggested answer is provided below. Other questions of the exam are very much the “same” as that of previous years.
Q1. The customer specification of the thickness of a product is 1000±3 mm, i.e., a product is considered defective if its thickness is below 997 mm or above 1003 mm. The production process, when under control, produces the products with thicknesses following a normal distribution with a mean = 1000 mm and a standard deviation of 1 mm. However, due to wearing and tearing, the process mean thickness may change and deviate from the target 1000 mm while the standard deviation stays the same. If the process mean thickness drifts to a position at which the process produces more than 5% defective products, it is considered unacceptable and whenever found so the process should be stopped. A random sample of size 9 is taken from the process from time to time and the sample mean X is used to test and to decide if the process needs to be stopped. The management has decided to set the Type II error probability β as β = 0.01. A. What will be the defective rate the process produces if process mean thickness = 1000 mm; if = 1002 mm? B. Describe the Type II error in words in the context of this problem? C. Find the critical cutting points for X such that if the sample mean X goes beyond the critical cutting points, the process is stopped. Note that β = 0.01. D. What is the implied α under the decision rule determined in C when the process mean thickness is at the target 1000mm?
Ans:
A. If = 1000, = 1. Note that a product is defective if its x > 1003 or x < 997. x > 1003 z = (1003 – 1000)/1 = 3 Prob(x > 1003) = Prob( z > 3) = 0.0013 x < 997 z = (997 – 1000)/1 = - 3 Prob( x < 997) = 0.0013
Thus, defective rate =