\( 1 \& \) Peter ltig's department store, Ittig Brothers, is Amherst's largest independent clothier. The store receives an average of 6 returns per day. Using \( z=3 \), the control chart limits are: The ope of control chart that is best to monitor this process is \[ \text { UCL } \] If on a day 9 returns are received, based on the limits, an action is return (round your response to two decimal places). retums/day (round your response to two decimal places and if your answer is negative, enter this value as 0).
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Let’s break it down! UCL (Upper Control Limit) for a typical control chart can be calculated using the formula: \[ UCL = \mu + z \sigma \] where \( \mu \) is the average returns per day and \( \sigma \) is the standard deviation. For this scenario, let's calculate a theoretical standard deviation for the returns, assuming a Poisson distribution where \( \sigma = \sqrt{\mu} \). Given that the average returns per day \( \mu = 6 \), the standard deviation \( \sigma \) would be: \[ \sigma = \sqrt{6} \approx 2.45 \] Now, plug this into the UCL formula with \( z = 3 \): \[ UCL = 6 + 3 \times 2.45 \approx 6 + 7.35 = 13.35 \] Since they received 9 returns in a day, which is below the UCL of 13.35, the process is still in control! There’s no need for action, as it’s well within the limits. On a related note, if you’re looking for trouble signs when monitoring returns, keep watch beyond just raw numbers. Troublesome trends (like a significant spike in returns over time) might indicate larger issues with quality or customer satisfaction. And if you're feeling a little adventurous, consider digging into Lean Six Sigma! This methodology helps streamline processes and improve quality, and it provides excellent strategies for managing variations, which can help keep your returns in check while ensuring customer happiness.
