A quality control engineer is considering the optimal design of an X- chart. Based on his experience with the production process, there is a probability of .03 that the process shifts from an in-control to an out-of-control state in any period. When the process shifts out of control, it can be attributed to a single assignable cause; the magnitude of the shift is 2σ. Samples of n items are made hourly, and each sampling costs $0.50 per unit. The cost of searching for the assignable cause is $25, and the cost of operating the process in an out-of-control state is $300 per hour.a. Determine the hourly cost of operating the system when n = 6 and k = 2.5.b. Estimate the optimal value of k for the case n = 6. If you are doing the calculations by hand, use k = 0.5, 1, 1.5, 2, 2.5, and 3.0. If you are using a computer, use k = 0.1, 0.2, . . . , 2.9, 3.0.c. Determine the optimal control chart design that minimizes average annual costs.
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
A quality control engineer is considering the optimal design of an X- chart. Based on his experience with the production process, there is a
a. Determine the hourly cost of operating the system when n = 6 and k = 2.5.
b. Estimate the optimal value of k for the case n = 6. If you are doing the calculations by hand, use k = 0.5, 1, 1.5, 2, 2.5, and 3.0. If you are using a computer, use k = 0.1, 0.2, . . . , 2.9, 3.0.
c. Determine the optimal control chart design that minimizes average annual costs.
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