Final answer:
The Average Run Length (ARL) of an x-bar chart can be determined using the formula: ARL = (1/p) * [1 + Φ((L-μ)/σ)].
Step-by-step explanation:
The Average Run Length (ARL) of an x-bar chart with limits where the process has shifted 0.25 times the standard deviation in one direction can be determined using the formula:
ARL = (1/p) * [1 + Φ((L-μ)/σ)]
where L is the shift magnitude, μ is the mean of the process, σ is the standard deviation, and Φ is the cumulative distribution function. To calculate ARL, substitute the given values into the formula and solve for the ARL.
For example, if the mean of the process is 50, the standard deviation is 2, and the shift magnitude L is 0.25, the ARL can be calculated as follows:
ARL = (1/0.25) * [1 + Φ((0.25-50)/2)]
ARL = 4 * [1 + Φ(-24.875)]
ARL = 4 * [1 + 0]
ARL = 4 * 1
ARL = 4
Therefore, the Average Run Length (ARL) of this x-bar chart is 4.