(a) (i) What are control charts by variables and control charts by attributes? (ii) Derive the control limits for the construction of control charts for the mean and variability based on sample standard deviation.
(b) (i) State the assumptions involved under sampling inspection plans by variables and describe the operating procedure of a single sampling plan by variables under the assumption of normal distribution for a quality characteristic. (ii) Establish the relationship between the fraction defective and the acceptance probability under a single sampling plan by variables when the quality characteristic follows a normal distribution with mean μ and variance σ², where σ² is unknown, and when an upper specification limit is specified. Using the relationship, obtain the formula for finding the parameters of the sampling plan.
(c) (i) Given a system consisting of n components, define the state vector and the structure function of the system. What do they indicate? (ii) Defining (1) a series system, (2) a parallel system and (3) a k-out-of-n system, obtain the associated expressions for the structure functions and the reliability functions.