Question

# 1 An x̅ chart with a sample size of 4 is used to control the mean...

1

An x̅ chart with a sample size of 4 is used to control the mean of a normally distributed quality characteristic. It is known that process standard deviation is 8. The upper and lower control limits of the chart are 147 and 123 respectively. Assume the process mean shifts to 121.

What is the probability that this shift is detected on the first subsequent sample?

What is expected number of samples taken before the shift is detected?

2

The lower and upper specification limits of a process are given as 779 and 791 respectively. The process standard deviation is estimated to be 1.8. The process mean has shifted to 782.

What is the potential capability ratio of the process?

What is the actual capability ratio of the process?

3

A process has a target mean of 91. If the process mean shifts to 93, we want to detect this shift using a cusum chart of which control limits are ±8. Suppose we collect individual measurements from the process and the first observed value is 91.8.

What is the first C+ value to be plotted on the cusum chart if we use a 50% headstart?
What is the first C+ value to be plotted on the cusum chart without using a headstart?

4

Sample   1   2   3
Defectives   7   8   12
Sample_size   50   90   160

The above table shows the number of nonconforming (defective) units in the three samples. Based on the given sample data, we want to setup a control chart to monitor the fraction nonconforming (defective) of the process.

What is the center line of the chart?

What is the first point, corresponding to the first sample, to be plotted on the chart?

1)

The null and alternate hypothesis are:

H0: The process is in control.

Ha: The process is out of control.

Required probability = P(shift is detected on the first subsequent sample)

Now,

Now, we know

So, Required probability =

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