hello i currently have this question posted below: You work as a quality enginee

Question

hello i currently have this question posted below:

You work as a quality engineer for a cereal company. The company has just started a new quality improvement program. They want you to construct a control chart for the average weight of the boxed cereals. The scheme you devised is the following: collect 5 boxes of cereal every 15 minutes, take the average weight and plot that average on a control chart with 3-sigma control limits. You also decided to check the control chart every hour (every 4 samples) and conclude that the process is out-of-control if you have at least one point outside the control limits. What is the type I error for this rule

so far i have calculated that the type 1 error for this rule is 0.002699796 based on the 3 sigma. this is what im suppose to find or is there more?

also what happens if the rule is changed to: if there are at least 2 points beyond the control limits within an hour. Calculate the type I error for this new rule and comment on whether this new strategy will indeed work in reducing the false alarms?

Explanation / Answer

For 3 sigma limits, the probability of type 1 error on each boxed cereal is 0.0027

Now, we check 4 samples and if al least one is beyond the control limits, we conclude process is out of control.

Required probability = 1-P(all 4 boxes are in control) = 1-(0.9973)^4 = 0.0108. So, type 1 error is approx 1.1%

Now we want at least 2 out of 4 boxes to be out of control

Hence required probability = 1-P(all 4 boxes are in control) - P(1 box out of control) = 1-(0.9973)^4 - 4C1*(0.9973)^3*0.0027 = 0.00004

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