Question

A process is controlled with a fraction nonconforming control chart with three-sigma limits, ? = 100. Suppose that the center line = 0.10.

(a) Suppose that the percentage of conforming units in sample ? is ?? . What distribution should ?? follow? Use a Normal distribution to approximate the distribution of ?? . Specify the mean and the variance of this Normal distribution.

(b) Find the upper and lower control limit for this fraction nonconforming chart.

(c) Find the equivalent control chart for the number nonconforming. Specify the plot variable, the center line and the upper and lower control limits.

(d) If the process fraction of nonconforming increases by 5% (shifts to 0.15). what out-ofcontrol distribution does the fraction of nonconforming follows? Use Normal approximation. Find the probability of type II error.

(e) If the process fraction of nonconforming decrease by 5%. Use Poisson approximation due to the small value of ?. What is the probability of type II error?

Answer #1

A control chart for fraction nonconforming indicates that the
current process average is 0.03. The sample size is constant at 200
units. a) Find the three-sigma control limits for the control
chart. b) What is the probability that a shift in the process
average to 0.08 will be detected on the first subsequent sample?
(Hint: use normal approximation) c) What is the probability that
this shift will be detected on the second sample taken after the
shift?

A fraction nonconforming control chart with n = 400 has the
following parameters: UCL = 0.0962, Center line = 0.0500, LCL =
0.0038
a. Find the width of the control limits in standard deviation
units.
b. Suppose the process fraction nonconforming shifts to 0.15.
What is the probability of detecting the shift on the first
subsequent sample?

A fraction nonconforming control chart with center line 0.10,
UCL = 0.19, and LCL = 0.01 is used to control a process.
Find the average run length if the process fraction noncon-
forming shifts to 0.20.

A control chart indicates that the current process
fraction nonconforming is 0.02. If 50 items are
inspected each day, what is the probability of detect-
ing a shift in the fraction nonconforming to 0.04 on
the first day after the shift? By the end of the third
day following the shift?

Determine the fraction of nonconforming product control chart
upper limit when the standard given is 0.01 proportion of
nonconfroming
Determine the Average Run Length (ARL) of a x-bar chart with
limits where the process has shifted 0.25 times the standard
deviation in one direction

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...

A process is in control and normally distributed with ? control
chart limits of 45 and 15. The subgroup size is 4. Suppose the
process variance suddenly triples while process mean remains
unchanged. What is the probability that the first subsequent
subgroup average will fall outside the control limits? What are the
? probability and ARL? Suppose the process variance suddenly
triples while process mean shifts downward to 10. What are the β
probability and ARL now?

HCH Manufacturing has decided to use a p-Chart with 2-sigma
control limits to monitor the proportion of defective steel bars
produced by their production process. The quality control manager
randomly samples 250 steel bars at 12 successively selected time
periods and counts the number of defective steel bars in the
sample.
Sample Defects
1 7
2 10
3 14
4 8
5 9
6 11
7 9
8 9
9 14
10 11
11 7
12 8
Step 1 of...

A process is in control and normally distributed with ? control
chart limits of 45 and 15. The subgroup size is 4. Suppose the
process variance suddenly triples while process mean remains
unchanged. What is the probability that the first subsequent
subgroup average will fall outside the control limits? What are the
? probability and ARL?

A quality manager asked his team to implement p control chart
for a process that was recently introduced. The team collected
samples of size n = 100 parts hourly over a period of 30 hours and
determined the proportion of nonconforming parts for each sample.
The mean proportion of nonconforming parts for 30 samples turned
out to be 0.025.
Determine the upper and lower control limits for the p chart.
(2 Points)
Discuss how you will use the p Chart...

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